From c17cf12fc3bb923e6f388acb2fa3334e6374e386 Mon Sep 17 00:00:00 2001
From: douden <49307154+douden@users.noreply.github.com>
Date: Mon, 4 Nov 2024 08:27:01 +0100
Subject: [PATCH] Publish pre-Q2 2024-2025
---
.gitlab-ci.yml | 3 +-
.gitlab-ci.yml.bak | 27 +
.gitmodules | 2 +-
Appendices/ComplexNumbers.md | 121 +-
Appendices/InverseMatrixTheorem.md | 5 +-
Appendices/SVDProof.md | 100 -
Chapter1/Cross_Product.md | 34 +-
Chapter1/Images/Fig-InnerProduct-SameProj.svg | 339 ++-
Chapter1/Inner_Product.md | 136 +-
Chapter1/Lines_and_Planes.md | 42 +-
Chapter1/Vectors.md | 34 +-
Chapter2/Images/Fig-LinInd-Examplein2D.svg | 23 +-
Chapter2/Images/Fig-LinInd-Examplein3D.svg | 76 +-
Chapter2/LinearCombinations.md | 32 +-
Chapter2/LinearIndependence.md | 94 +-
Chapter2/LinearSystems.md | 55 +-
Chapter2/MatrixVectorProduct.md | 40 +-
Chapter2/SolutionSets.md | 35 +-
Chapter3/GeometryofLinearTransformations.md | 92 +-
Chapter3/Injectivity_and_surjectivity.md | 100 +-
Chapter3/LUdecomp.md | 1909 +++++++++--------
Chapter3/Linear_Transformations.md | 62 +-
Chapter3/MatrixInverse.md | 90 +-
Chapter3/MatrixOperations.md | 87 +-
Chapter4/BasisAndDimension.md | 82 +-
Chapter4/ChangeOfBasis.md | 58 +-
Chapter4/Images/Fig-TheGame.svg | 16 +
Chapter4/Subspaces_of_Rn.md | 51 +-
Chapter5/DetGeometric.md | 527 -----
Chapter5/DeterminantsExtras.md | 88 +-
Chapter5/DeterminantsGeometric.md | 46 +-
Chapter5/DeterminantsViaCofactors.md | 29 +-
Chapter5/DeterminantsViaRowReduction.md | 116 +-
Chapter6/CharPolynomial.md | 27 +-
Chapter6/ComplexEigenvalues.md | 29 +-
Chapter6/Diagonalizability.md | 41 +-
Chapter6/EigenvaluesEigenvectors.md | 36 +-
Chapter7/GramSchmidt.md | 36 +-
Chapter7/LeastSquares.md | 79 +-
Chapter7/OrthoBase.md | 26 +-
Chapter7/OrthoComp.md | 55 +-
Chapter7/Orthogonality.md | 332 ---
Chapter8/Images/Fig-SVD-Decomposition.svg | 376 +++-
Chapter8/QuadraticForms.md | 41 +-
Chapter8/SingularValueDecomp.md | 36 +-
Chapter8/SymmetricMatrices.md | 47 +-
Chapter9/DynSystContinuous.md | 27 +-
Chapter9/DynSystDiscrete.md | 22 +-
.../Fig-DynSystContinuous-Trajectories.svg | 445 ++++
Chapter9/MarkovChains.md | 11 +-
Chapter9/PowerMethod.md | 11 +-
README.md | 2 +-
README.md.bak | 51 +-
_ext/applet.py | 12 +-
_static/extra_css_config.css | 263 +++
_toc.yml | 1 -
add_darklight_grasple.py | 46 +
install.sh | 12 +-
sphinx-grasple | 2 +-
59 files changed, 3933 insertions(+), 2684 deletions(-)
create mode 100644 .gitlab-ci.yml.bak
delete mode 100644 Appendices/SVDProof.md
create mode 100644 Chapter4/Images/Fig-TheGame.svg
delete mode 100644 Chapter5/DetGeometric.md
delete mode 100644 Chapter7/Orthogonality.md
create mode 100644 Chapter9/Images/Fig-DynSystContinuous-Trajectories.svg
create mode 100644 add_darklight_grasple.py
diff --git a/.gitlab-ci.yml b/.gitlab-ci.yml
index 9a5c6ed..ff5e02c 100644
--- a/.gitlab-ci.yml
+++ b/.gitlab-ci.yml
@@ -3,7 +3,8 @@ stages:
- deploy
variables:
GIT_SUBMODULE_STRATEGY: recursive
-
+ GIT_SUBMODULE_UPDATE_FLAGS: --remote
+
jupyter-build:
stage: build
image: python:slim
diff --git a/.gitlab-ci.yml.bak b/.gitlab-ci.yml.bak
new file mode 100644
index 0000000..9a5c6ed
--- /dev/null
+++ b/.gitlab-ci.yml.bak
@@ -0,0 +1,27 @@
+stages:
+ - build
+ - deploy
+variables:
+ GIT_SUBMODULE_STRATEGY: recursive
+
+jupyter-build:
+ stage: build
+ image: python:slim
+ script:
+ - sh install.sh
+ - jupyter-book clean .
+ - jupyter-book build .
+ artifacts:
+ paths:
+ - _build/
+
+pages:
+ stage: deploy
+ image: busybox:latest
+ script:
+ - mv _build/html public
+ artifacts:
+ paths:
+ - public
+ rules:
+ - if: $CI_COMMIT_BRANCH == $CI_DEFAULT_BRANCH
diff --git a/.gitmodules b/.gitmodules
index d56b648..6f834eb 100644
--- a/.gitmodules
+++ b/.gitmodules
@@ -1,3 +1,3 @@
[submodule "sphinx-grasple"]
path = sphinx-grasple
- url = https://github.com/dbalague/sphinx-grasple.git
+ url = https://github.com/TeachBooks/Sphinx-Grasple-public.git
\ No newline at end of file
diff --git a/Appendices/ComplexNumbers.md b/Appendices/ComplexNumbers.md
index 757455f..d5fd876 100644
--- a/Appendices/ComplexNumbers.md
+++ b/Appendices/ComplexNumbers.md
@@ -14,7 +14,6 @@ where $a\neq0$. Previously you probably learned that Equation {eq}`Eq:ComplexNum
:::{math}
:label: Eq:ComplexNumbers:abc-sol
-
x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}.
:::
@@ -65,7 +64,10 @@ Let $a$ be a _positive_ real number. Then the two numbers $x_-=-ai$ and $x_+=ai$
::::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:roots`
+:class: myproof, dropdown
+
+<!-- :::{prf:proof} -->
First we consider $x_-=-ai$ and take its square:
@@ -193,7 +195,9 @@ zw &= (ac-bd)+(ad+bc)i, \\
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:ops`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:ops`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:ops` -->
We prove the four results by working each out separately. We start with the _addition_:
@@ -268,7 +272,9 @@ If $z$ and $w$ are a complex numbers, then the following identities hold:
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:conjops`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:conjops`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:conjops` -->
We show each of the identities, one after the other, where we assume $z=a+bi$ and $w=c+di$, $a,b,c,d\in\mathbb{R}$:
@@ -340,7 +346,9 @@ z\overline{z} &= \Re{z}^2+\Im{z}^2.
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:conjparts`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:conjparts`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:conjparts` -->
We show each of the identities, one after the other, where we assume $z=a+bi$, $a,b\in\mathbb{R}$:
@@ -382,7 +390,9 @@ Assume $z\in\mathbb{C}$. $z\in\mathbb{R}$ if and only if $z=\overline{z}$.
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:realz`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:realz`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:realz` -->
Assume $z\in\mathbb{C}$.
@@ -422,12 +432,9 @@ The reason for introducing complex numbers is to ensure that more equations have
We already solved quadratic equations using a new technique called _completing the square_ and in this section you will learn more ways to solve equations.
-You may think that introducing a new set of numbers as solutions to certain equations can be a never-ending process. When you introduce more numbers, you get more equations (now we need not only solve $x^2=-1$, but also $x^2=i$), which need new solutions, etcetera.
-It turns out that if you restrict yourself to polynomial equations, this is not the case. This statement is the Fundamental Theorem of Algebra:
-
-**Fundamental Theorem of algebra**
+We introduced complex numbers to give the equation $x^2+1 = 0$ a solution. It appears that something much stronger holds, namely, that every polynomial equation with coefficients in $\mathbb{C}$, for instance $(1+i)x^4 - 2x^2 + x = 10i$, has solutions in $\mathbb{C}$. This is the content of the following theorem.
-::::{prf:theorem}
+::::{prf:theorem} Fundamental Theorem of algebra
:label: Thm:ComplexNumbers:fundamental
Consider a polynomial $p(z)$ of degree $n$,
@@ -488,7 +495,9 @@ $$
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:uniquezeros`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:uniquezeros`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:uniquezeros` -->
Assume $\{z_1,z_2,\ldots,z_k\}$ is the set of _unique_ zeros of a polynomial $p$ of degree $n$. Then following {prf:ref}`Thm:ComplexNumbers:fundamental`, we can write
@@ -506,7 +515,7 @@ Because $p(z_1)=0$ for $j\in\{1,\ldots,k\}$, we must have that $\alpha_1\in\{1,\
:::{math}
:label: Eq:ComplexNumbers:factorsb1
-$p(z) = a_n (z-z_1)^{\alpha_1}(z-b*{\alpha_1+1}) \cdots (z-b_n).$
+p(z) = a_n (z-z_1)^{\alpha_1}(z-b*{\alpha_1+1}) \cdots (z-b_n).
:::
@@ -515,7 +524,7 @@ We can repeat the above argument for $z_2$: we must have that $\alpha_2\in\{1,\l
:::{math}
:label: Eq:ComplexNumbers:factorsb2
-$p(z) = a_n (z-z_1)^{\alpha_1}(z-z_2)^{\alpha_2}(z-b_{\alpha_1+\alpha_2+1}) \cdots (z-b_n).$
+p(z) = a_n (z-z_1)^{\alpha_1}(z-z_2)^{\alpha_2}(z-b_{\alpha_1+\alpha_2+1}) \cdots (z-b_n).
:::
@@ -545,7 +554,9 @@ If $p(z)=0$, then $p(\overline{z})=0$ as well, and the algebraic multiplicities
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:realpoly`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:realpoly`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:realpoly` -->
Consider a polynomial $p$ of degree $n$, $\sum_{j=0}^n a_j z^j$, where the coefficients $a_n, a_{n-1}, \ldots, a_0$ are real valued numbers and $a_n\neq 0$.
@@ -670,7 +681,7 @@ The polar coordinates of a point in the complex plane $\C$ are the distance $r=|
:::
Notice that the argument is not uniquely defined, as you can always go a full circle extra and add $2\pi$ radians to the angle. For example, the number $1$ has argument 0 (as it is on the positive real axis), but also $2\pi$, $4\pi$, and $-2\pi$ (etc.). In order to make a uniform choice, we sometimes work with the principal value of the argument, which is by definition the unique value of the argument between $-\pi$ and $\pi$.
-We write the principal value using a capital A. Thus we have $-\pi < \text{Arg} z \leq \pi$.
+We write the principal value using a capital A. Thus we have $-\pi < \Arg{z} \leq \pi$.
:::: {prf:example}
@@ -791,17 +802,33 @@ We recognize this product as the number with modulus $|zw|=rs$ and argument $\ar
If you take the complex conjugate of a complex number $z$, the modulus remains the same and the argument is negated:
<ul>
-<li>$|\overline{z}| = |z|$,</li>
+<li>
+
+$|\overline{z}| = |z|$,
-<li>$\arg(\overline{z}) = -\arg(z)$.</li>
+</li>
+
+<li>
+
+$\arg(\overline{z}) = -\arg(z)$.
+
+</li>
</ul>
If you multiply two complex numbers $z$ and $w$, you multiply the moduli and add the arguments:
<ul>
-<li>$|zw| = |z| \cdot |w|$,</li>
+<li>
+
+$|zw| = |z| \cdot |w|$,
+
+</li>
+
+<li>
-<li>$\arg(zw) = \arg(z) + \arg(w)$.</li>
+$\arg(zw) = \arg(z) + \arg(w)$.
+
+</li>
</ul>
If you divide the complex number $z$ by the complex number $w\neq0$ you divide the modulus of $z$ by the modulus of $w$ and subtract the argument of $w$ from the argument of $z$:
@@ -822,7 +849,9 @@ $\arg\left(\frac{z}{w}\right) = \arg(z) - \arg(w)$.
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:polarmultdiv`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:polarmultdiv`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:polarmultdiv` -->
_Proof for conjugation_
@@ -901,7 +930,9 @@ $$
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:re`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:re`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:re` -->
The proof is relatively straight forward:
@@ -924,7 +955,9 @@ $$
::::
-::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:re`
+::::{admonition} Proof of {prf:ref}`Thm:ComplexNumbers:re_diff`
+:class: dropdown, myproof
+<!-- ::::{dropdown} Proof of {prf:ref}`Thm:ComplexNumbers:re_diff` -->
The proof is again straight forward:
@@ -1061,15 +1094,6 @@ The three solutions from {prf:ref}`Ex:ComplexNumbers:threesolutions`.
We can generalize the method for solving $z^n=w$ from the example above:
-```{applet}
-:url: appendix/complex_numbers
-:fig: Images/Fig-ComplexNumbers-threesolfig.svg
-:name: Fig:ComplexNumbers:general
-:position: 0, -1
-
-The n solutions from {prf:ref}`Ex:ComplexNumbers:threesolutions` generalized.
-```
-
```{prf:algorithm} Solving $z^n=w$
<ol>
@@ -1584,7 +1608,8 @@ $z^3=4\cos(\frac{\pi}{6})+4i\sin(\frac{\pi}{6})$
## Solutions
-::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:eval_aplusbi` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ComplexNumbers:eval_aplusbi`
+:class: solution, dropdown
<ol type="a">
@@ -1652,7 +1677,8 @@ $3 + 4i$
::::
-::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:complete` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ComplexNumbers:complete`
+:class: solution, dropdown
<ol type="a">
@@ -1678,7 +1704,8 @@ $2, -6$
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:division` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:division`
+:class: solution, dropdown
<ol type="a">
@@ -1704,7 +1731,8 @@ $h(z)$ is not a polynomial.
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:roots` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:roots`
+:class: solution, dropdown
<ol type="a">
@@ -1730,7 +1758,8 @@ The roots of $p(z)$ are $3+i, 3-i, 1, -2$, all with multiplicity 1.
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:argmod` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:argmod`
+:class: solution, dropdown
<ol type="a">
@@ -1750,7 +1779,8 @@ $|z|=2\sqrt{3}$ and $\Arg{z}=\frac{\pi}{6}$ or $\arg(z)=\frac{\pi}{6}+2k\pi$ for
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:polarform` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:polarform`
+:class: solution, dropdown
<ol type="a">
@@ -1770,7 +1800,8 @@ $6 \cos(\frac{-\pi}{6})+ i 6 \sin(\frac{-\pi}{6})$
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:polar_abi` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:polar_abi`
+:class: solution, dropdown
<ol type="a">
@@ -1808,7 +1839,8 @@ $1-\sqrt{3} i$
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:polarform_calc` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:polarform_calc`
+:class: solution, dropdown
<ol type="a">
@@ -1828,7 +1860,8 @@ $z=4e^{-\frac16\pi i}$ and $w=2\sqrt{2}e^{-\frac14\pi i}$, thus $zw=8 \sqrt{2} \
:::::
-::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:solve_aplusbi` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ComplexNumbers:solve_aplusbi`
+:class: solution, dropdown
<ol type="a">
@@ -1848,7 +1881,8 @@ $i, -\frac{1}{2}\sqrt{3}-i\frac{1}{2}, \frac{1}{2}\sqrt{3}-i\frac{1}{2}$
::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:DeMoivre` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:DeMoivre`
+:class: solution, dropdown
$\cos(4\theta)=\cos^4(\theta)+\sin^4(\theta)-6\cos^2(\theta)\sin^2(\theta)$
@@ -1858,7 +1892,8 @@ $\sin(4\theta)=4\cos^3(\theta)\sin(\theta)-4\cos(\theta)\sin^3(\theta)$.
:::::
-:::::{dropdown} Solution to {numref}`Exc:ComplexNumbers:solve_euler` (_click to show_)
+:::::{admonition} Solution to {numref}`Exc:ComplexNumbers:solve_euler`
+:class: solution, dropdown
<ol type="a">
diff --git a/Appendices/InverseMatrixTheorem.md b/Appendices/InverseMatrixTheorem.md
index 6d5f840..c21e9cd 100644
--- a/Appendices/InverseMatrixTheorem.md
+++ b/Appendices/InverseMatrixTheorem.md
@@ -15,7 +15,7 @@ For an $n\times n$ matrix $A$, the following are equivalent:
\label{It:Appendices:InvDef}
\item There exists a matrix $B$ with $AB=I$.
-\item There esists a matrix $B$ with $BA=I$.
+\item There exists a matrix $B$ with $BA=I$.
\item The linear system $A\vect{x}=\vect{b}$ has a unique solution for any $\vect{b}$ in $\R^{n}$.
\item $A$ is row equivalent to the identity matrix.
\item $A$ has linearly independent columns.
@@ -36,7 +36,8 @@ For an $n\times n$ matrix $A$, the following are equivalent:
::::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:Appendices:InverseMatrixTheorem`
+:class: myproof
For the equivalence of {itemref}`It:Appendices:InvDef` through {itemref}`It:Appendices:InvDefColSpanRn`, see {prf:ref}`Thm:MatrixInv:InvertibilityCharacterizations` and {numref}`Exc:BasisDim:RankABLeqRankA`. Statement {itemref}`It:Appendices:InvIffFullRank` is part of {prf:ref}`Thm:BasisDim:RankThm`. {prf:ref}`Thm:DetRowReduction:Invertibility` says precisely that invertibility is equivalent to {itemref}`It:Appendices:InvIffDetNeq0`. For {itemref}`It:Appendices:InvIffZeroNoEV`, see {prf:ref}`Prop:EigenValues:SingularMatrix`.
diff --git a/Appendices/SVDProof.md b/Appendices/SVDProof.md
deleted file mode 100644
index 87c342f..0000000
--- a/Appendices/SVDProof.md
+++ /dev/null
@@ -1,100 +0,0 @@
-(Sec:ProofSVD)=
-
-# Proof of the existence of SVD
-
-This appendix is devoted to construct the proof of {prf:ref}`Thm:SVD:ExistenceSVD`. Some new definitions will be given.
-
-::::{prf:definition} Norm of a Linear Transformation
-
-Let $T:\mathbb{R}^n\to \mathbb{R}^m$ be a linear transformation with standard matrix $A$. We define the **matrix norm** of $A$ as
-
-$$
-\norm{A} = \max_{{\mathbf{v}\in \mathbb{R}^n},\norm{\mathbf{v}}=1} \norm{A\mathbf{v}}.
-$$
-
-We define the the **norm of a linear transformation** $T$
-as the norm of its standard matrix. That is
-
-$$\norm{T} = \norm{A}$$
-
-::::
-
-Observe that the matrix norm is defined by finding a maximum. We will leave to the reader (and for a calculus course) to prove that the function
-
-\begin{align*}
-f:\mathbb{R}^n &&\to&&& \mathbb{R}\\
-\mathbf{v} &&\mapsto&&& f(\mathbf{v}) = \norm{\mathbf{v}}
-\end{align*}
-
-is a continuous function.
-
-:::{prf:remark}
-
-The matrix norm is sometimes called **induced norm**.
-:::
-
-:::{prf:definition}
-
-Suppose that $T:V\subset\mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation with standard matrix $A$. Then we define $\norm{T}$ restricted in $V$ as
-
-$$
- \norm{T_{|V}} = \max_{\mathbf{v}\in V,\norm{\mathbf{v}}=1} \norm{A\mathbf{v}}
-$$
-
-:::
-
-We will proof {prf:ref}`Thm:SVD:ExistenceSVD` by using a similar algorithm to {prf:ref}`Alg:SVD:SVDalgorithm`. The proof follows the geometric interpretation described in {numref}`Subsec:SVDGeometrically`. If the reader is not familiar with it, we recommend understanding the idea before continuing.
-
-## Construction of $V$ and $U$
-
-Suppose that $A$ is the standard matrix of a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ with rank $p$.
-
-Since the construction will be iterative we will define, for convenience, $V^{(1)} = \mathbb{R}^n$.
-
-Let's consider "the unit sphere" in $V^{(1)}$. That is
-
-$$
-S^{(1)} = \lbrace \mathbf{v} \in V^{(1)} \,|\, \norm{\mathbf{v}}=1\rbrace
-$$
-
-Now choose $\mathbf{v}_1\in V^{(1)}$ such that $\norm{T} = \norm{A\mathbf{v}_1} = s_1$
-
-If $s_1=0$ then we are done as $A\mathbf{v}=\mathbf{0}$ for all $\mathbf{v}\in V^{(1)}$.
-
-Let's suppose that $s_1>0$ and define $\mathbf{u}_1 = \frac{1}{s_1}A\mathbf{v}_1$. Notice that $\mathbf{u}_1\in \mathbb{R}^m$.
-
-Next, consider the unit sphere in the subspace $V^{(2)}=\Span{\mathbf{v}_1}^\perp$:
-
-$$
-S_2 = \lbrace \mathbf{v}\in V^{(2)}\subset\mathbb{R}^n\,|\, \norm{\mathbf{v}}=1\rbrace,
-$$
-
-and the restriction of $T$ onto the subspace $V^{(2)}$. Then choose $\mathbf{v}_2$ such that $s_2 = \norm{T_{|V^{(2)}}} = \norm{A\mathbf{v}_2}$.
-
-Notice that, by construction, $\mathbf{v}_1 \perp \mathbf{v}_2$. If $s_2=0$ then we are done, as $A\mathbf{v} = \mathbf{0}$ for all $\mathbf{v}\in V^{(2)}$. Suppose that $s_2 >0$, and define $\mathbf{u}_2 = \frac{1}{s_2}A\mathbf{v}_2$.
-
-Observe that since $S^{(2)} \subset S^{(1)}$ we have that $\displaystyle\max_{\mathbf{v}\in S^{(2)},\norm{\mathbf{v}}=1} \norm{A\mathbf{v}} \le \max_{\mathbf{v}\in S^{(1)},\norm{\mathbf{v}}=1}\norm{A\mathbf{v}}$. Therefore, $s_2 \le s_1$.
-
-In addition, we can prove that $\mathbf{u}_1\perp \mathbf{u}_2$.
-
-:::{exercise}
-
-Prove that $\mathbf{u}_1\perp \mathbf{u}_2$. Notice that we cannot use the same argument as in {numref}`Subsec:SVD:Algorithm` since we can not use {prf:ref}`Propo:SVD:singularvalues` (it assumes that the singular value decomposition exists, and this is what we are trying to prove).
-
-:::
-
-Suppose that we repeat this process $r$ times. That is, we found $\mathbf{v}_1, \dots, \mathbf{v}_r$, and $\mathbf{u}_1,\dots,\mathbf{u}_r$, and the values $s_1>s_2>\cdots>s_r>0$. Now we define $V^{(r+1)} = \Span{\mathbf{v}_1,\dots,\mathbf{v}_r}^{\perp}$ and we consider again the unit sphere in this subspace:
-
-$$
-S^{(r+1)} = \lbrace \mathbf{v}\in V^{(r+1)}\,|\, \norm{\mathbf{v}}=1\rbrace.
-$$
-
-Choose $s_{r+1} = \norm{T_{|V^{(r+1)}}}$. If $s_{r+1}=0$ we are done as $A\mathbf{v}=\mathbf{0}$ for all $\mathbf{v}\in V^{(r+1)}$. Else, we continue the process until either $s_{r}=0$ for some $r$, or until we find a basis for $\mathbb{R}^n$, i.e., $r=n$.
-
-Now we take the vectors $\lbrace\mathbf{v}_1,\dots,\mathbf{v}_r\rbrace$. If $r < n$, we complete the set until we have an orthonormal basis $\mathcal{V}$ of $\mathbb{R}^n$. Then we do the same with the vectors $\lbrace\mathbf{u}_1,\dots,\mathbf{u}_r\rbrace$ by completing the set, if needed, to an orthonormal basis $\mathcal{U}$ of $\mathbb{R}^m$.
-
-The values $s_1 > s_2 > \cdots > s_r > 0$ are the singular values (and 0 if we had to complete the set of vectors).
-
-What remains to prove is that we have exactly $p$ non-zero singular values.
-
-To see that, we consider the set $\lbrace \mathbf{u}_1, \dots, \mathbf{u}_r \rbrace$.
diff --git a/Chapter1/Cross_Product.md b/Chapter1/Cross_Product.md
index 7a58198..2e6ea9b 100644
--- a/Chapter1/Cross_Product.md
+++ b/Chapter1/Cross_Product.md
@@ -45,6 +45,7 @@ This is no coincidence, as we will see in {prf:ref}`Prop:CrossProduct:Cportho`.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c5058abb-3d5b-4e8c-b836-40aeff08a301?id=65634
:label: grasple_exercise_1_3_A
:dropdown:
@@ -61,7 +62,8 @@ If $\mathbf{u}$ and $\mathbf{v}$ are vectors in $\mathbb{R}^3$, then $\mathbf{u}
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:Cportho`
+:class: myproof
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^3$ such that
@@ -87,6 +89,7 @@ Knowing that the cross product of two vectors is orthogonal to these vectors doe
::::{figure} Images/Fig-CrossProduct-Righthandrule.svg
:name: Fig:CrossProduct:RightHandRule
+:class: dark-light-same
The right-hand rule. Adapted from Acdx, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons.
::::
@@ -108,7 +111,8 @@ where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:NormCrossProduct`
+:class: myproof
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^3$ such that
@@ -147,23 +151,27 @@ Notice some similarities between the formula for the length of the cross product
We can derive some interesting geometrical results from {prf:ref}`Prop:CrossProduct:NormCrossProduct`.
::::{prf:proposition}
+:label: Prop:CrossProduct:Parallel
Two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel if and only if $\mathbf{u}\cp \mathbf{v}=\mathbf{0}$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:Parallel`
+:class: myproof
Let $\mathbf{u}$ and $\mathbf{v}$ be two non-zero vectors. First of all, the vector $\mathbf{u}\cp \mathbf{v}$ is equal to the zero vector if and only if $\norm{\mathbf{u} \cp \mathbf{v}}=0$. Since $\norm{\mathbf{u}}$ and $\norm{\mathbf{v}}$ are both not equal to zero, it follows from {prf:ref}`Prop:CrossProduct:NormCrossProduct` that $\norm{\mathbf{u} \cp \mathbf{v}}=0$ if and only if $\sin(\theta)=0$, where $\theta$ is the angle between the vectors. This means that $\mathbf{u}\cp \mathbf{v}=\mathbf{0}$ if and only if $\theta$ is equal to either $0$ or $\pi$, which is equivalent to saying that $\mathbf{u}$ and $\mathbf{v}$ have the same direction or the opposite direction. In both cases the vectors are parallel.
::::
::::{prf:proposition}
+:label: Prop:CrossProduct:AreaParallelogram
If $\mathbf{u}$ and $\mathbf{v}$ are vectors in $\mathbb{R}^3$, then $\norm{\mathbf{u} \cp \mathbf{v}}$ is equal to the area of the parallelogram spanned by $\mathbf{u}$ and $\mathbf{v}$.
:::{figure} Images/Fig-CrossProduct-Area.svg
:name: Fig:CrossProduct:AreaParallelogram
+:class: dark-light
Parallelogram spanned by two vectors.
@@ -171,7 +179,8 @@ Parallelogram spanned by two vectors.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:AreaParallelogram`
+:class: myproof
The area of a parallelogram is equal to the product of the length of its base and its height. As we can see in {numref}`Figure %s <Fig:CrossProduct:AreaParallelogram>` the length of the base of the parallelogram is equal to $\norm{\mathbf{u}}$ and the height is equal to $\norm{\mathbf{h}}$. If we look at the right-angled triangle $OPP'$ we see that $\norm{\mathbf{h}}=\norm{\mathbf{v}}\sin{\theta}$. This means that the area of the parallelogram is equal to $\norm{\mathbf{u}} \norm{\mathbf{v}} \sin(\theta)$ (because we use an angle between $0$ and $\pi$ we can omit the absolute-value signs) and thus to $\norm{\mathbf{u} \cp \mathbf{v}}$.
@@ -205,6 +214,7 @@ What is the area of the triangle with vertices $(2,1,0)$, $(2,2,2)$ and $(3, 1,
:::{figure} Images/Fig-CrossProduct-TrianglePQR.svg
:name: Fig:CrossProduct:AreaTriangle
+:class: dark-light
Area of the triangle $PQR$.
@@ -257,7 +267,8 @@ $(\mathbf{v_1}+\mathbf{v_2})\cp\mathbf{v_3} = \mathbf{v_1}\cp\mathbf{v_3}+\mathb
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:RulesCrossProduct`
+:class: myproof
Let $\mathbf{v_1}$, $\mathbf{v_2}$ and $\mathbf{v_3}$ be vectors in $\mathbb{R}^3$ such that
@@ -319,6 +330,7 @@ is equal to the value $ad-bc$. Such an expression is called a _determinant_.
The entries of the cross product of two vectors can also be computed using determinants.
::::{prf:proposition}
+:label: Prop:CrossProduct:dets
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^3$ such that
@@ -340,7 +352,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:CrossProduct:dets`
+:class: myproof
This follows from the definition.
@@ -383,6 +396,7 @@ $$
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c5058abb-3d5b-4e8c-b836-40aeff08a301?id=65634
:label: grasple_exercise_1_3_1
:dropdown:
@@ -391,6 +405,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/529702ff-6fc3-46ab-a148-7d93d081870b?id=63138
:label: grasple_exercise_1_3_2
:dropdown:
@@ -399,6 +414,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/48448d89-c286-45c5-9af4-780329a8821f?id=65637
:label: grasple_exercise_1_3_3
:dropdown:
@@ -407,6 +423,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f6c1bb4b-e63e-492e-910a-5a8c433de281?id=75093
:label: grasple_exercise_1_3_4
:dropdown:
@@ -415,6 +432,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/84b635e4-2278-4882-915d-6f8b253213a3?id=78749
:label: grasple_exercise_1_3_5
:dropdown:
@@ -423,6 +441,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6b660feb-fc36-47a0-bf86-e424d28edf6f?id=63354
:label: grasple_exercise_1_3_6
:dropdown:
@@ -431,6 +450,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8a0fc383-dd10-4f31-931a-c62c0d650bd9?id=63479
:label: grasple_exercise_1_3_7
:dropdown:
@@ -439,6 +459,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3e62cc2d-5860-43c2-b8aa-e54ab3a9a981?id=79268
:label: grasple_exercise_1_3_8
:dropdown:
@@ -447,6 +468,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/122013a2-1012-4203-99c7-ed5deafd82a4?id=78786
:label: grasple_exercise_1_3_9
:dropdown:
diff --git a/Chapter1/Images/Fig-InnerProduct-SameProj.svg b/Chapter1/Images/Fig-InnerProduct-SameProj.svg
index e7e842b..eaa3ef8 100644
--- a/Chapter1/Images/Fig-InnerProduct-SameProj.svg
+++ b/Chapter1/Images/Fig-InnerProduct-SameProj.svg
@@ -1,21 +1,43 @@
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- xmlns="http://www.w3.org/2000/svg"
- viewBox="0 0 816 1056"
- height="1056"
- width="816"
+ viewBox="0 0 378.43271 386.04504"
+ height="386.04504"
+ width="378.43271"
xml:space="preserve"
id="svg2"
- version="1.1"><metadata
+ version="1.1"
+ sodipodi:docname="Fig-InnerProduct-SameProj.svg"
+ inkscape:version="1.3 (0e150ed6c4, 2023-07-21)"
+ xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+ xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+ xmlns="http://www.w3.org/2000/svg"
+ xmlns:svg="http://www.w3.org/2000/svg"
+ xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+ xmlns:cc="http://creativecommons.org/ns#"
+ xmlns:dc="http://purl.org/dc/elements/1.1/"><sodipodi:namedview
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+ inkscape:window-maximized="1"
+ inkscape:current-layer="svg2"
+ inkscape:clip-to-page="false" /><metadata
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+ transform="matrix(1.3333333,0,0,-1.3333333,-138.90421,907.58073)"
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diff --git a/Chapter1/Inner_Product.md b/Chapter1/Inner_Product.md
index 075afe5..3c8e934 100644
--- a/Chapter1/Inner_Product.md
+++ b/Chapter1/Inner_Product.md
@@ -28,8 +28,9 @@ in the plane, which we denote by $\norm{\mathbf{v}}$, can be computed using the
:::{figure} Images/Fig-InnerProduct-Length-2D.svg
:name: Fig:InnerProduct:Length-2D
+:class: dark-light
-The length of a vector via Pythagoras' Theorem
+The length of a vector via Pythagoras' Theorem.
:::
:::{applet}
@@ -37,8 +38,9 @@ The length of a vector via Pythagoras' Theorem
:fig: Images/Fig-InnerProduct-length-3D.svg
:name: Fig:InnerProduct:length-3D
:status: approved
+:class: dark-light
-The length of a vector via Pythagoras' Theorem
+The length of a vector via Pythagoras' Theorem.
:::
Using this theorem twice we find a similar formula for the length of a vector
@@ -74,8 +76,9 @@ we find that
:::{figure} Images/Fig-InnerProduct-perp-non-perp.svg
:name: Fig:InnerProduct:perp-non-perp
+:class: dark-light
-Perpendicular versus non-perpendicular
+Perpendicular versus non-perpendicular.
:::
Let us now turn our attention to another important geometric concept, namely that of
@@ -95,18 +98,14 @@ There is another way to look at this, which will be useful for the definition of
% \norm{\mathbf{v}+\mathbf{w}} \neq \norm{\mathbf{v}-\mathbf{w}}.
%$$
-:::{figure} Images/Fig-InnerProduct-diagonal-parallelogram.svg
+```{applet}
+:url: dot_product/diagonal_parallelogram
+:fig: Images/Fig-InnerProduct-diagonal-parallelogram.svg
:name: Fig:InnerProduct:diagonal-parallelogram
+:class: dark-light
The parallelogram spanned by $\vect{v}$ and $\vect{w}$ and its diagonals. How should you choose $\vect{v}$ and $\vect{w}$ such that the diagonals have the same length?
-:::
-
-%::: OLD: two figures rectangle / non-rectangle
-%:::{figure} Images/Fig-InnerProduct-diagonal-parallelogram.svg
-%:name: Fig:InnerProduct:diagonal-parallelogram%
-%
-%Diagonal of a rectangle versus diagonal of a parallelogram
-%:::
+```
In the picture on the right the vectors are not perpendicular and
@@ -155,7 +154,7 @@ The derivation is completely analogous to the one above, only now we have one ex
So to check 'algebraically' whether two vectors are perpendicular we just have to compute $a_1b_1 +a_2b_2\, (\,+\,a_3b_3\,)$
and see whether this is equal to 0.
-This expression is called the *dot product* (or *inner product*) of the vectors $\mathbf{v}$ and $\mathbf{w}$. We denote it by $\mathbf{v}\ip\mathbf{w}$.
+This expression is called the _dot product_ (or _inner product_) of the vectors $\mathbf{v}$ and $\mathbf{w}$. We denote it by $\mathbf{v}\ip\mathbf{w}$.
Note that the dot product of a general vector $\mathbf{v}=\begin{bmatrix} a_{1}\\a_{2}\\a_{3}\end{bmatrix}$ in $\mathbb{R}^3$ with itself gives
$$
@@ -182,7 +181,7 @@ Using the dot product the concepts length and perpendicular easily carry over to
::::{prf:definition}
:label: Dfn:InnerProduct:DotProduct
-The **dot product** (or *inner product*) of two vectors
+The **dot product** (or _inner product_) of two vectors
$\mathbf{v}=\begin{bmatrix}a_{1}\\a_{2}\\ \vdots\\a_{n}\end{bmatrix}$ and
$\mathbf{w}=\begin{bmatrix}b_{1}\\b_{2}\\ \vdots\\b_{n}\end{bmatrix}$ in $\mathbb{R}^n$ is defined as
@@ -242,7 +241,8 @@ iii. $(\mathbf{v}_1+\mathbf{v}_2)\ip\mathbf{v}_3 = \mathbf{v}_1\ip\mathbf{v}_3+\
iv. $\mathbf{v}\ip\mathbf{v} \geq 0$, and $\mathbf{v}\ip\mathbf{v} = 0 \iff \mathbf{v} = \mathbf{0}$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:RulesInnerProduct`
+:class: myproof
The first three properties follow from the corresponding properties of real numbers. For instance, for the first rule we simply use that $ab = ba$ holds for the product of real numbers $a$ and $b$.
@@ -305,7 +305,8 @@ if and only if all the squares are 0, which only happens if each entry $a_i$ is
Prove property iii.
:::
-:::{dropdown} Solution to {numref}`Exc:InnerProduct:CheckPropInnerProd` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:InnerProduct:CheckPropInnerProd`
+:class: solution, dropdown
Let
@@ -349,7 +350,8 @@ $$
:::
-:::{dropdown} Solution to {numref}`Exc:InnerProduct:(v-w)(v+w)` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:InnerProduct:(v-w)(v+w)`
+:class: solution, dropdown
First of all, because of rule i. and rule iii. of {prf:ref}`Prop:RulesInnerProduct`
it holds that
@@ -393,7 +395,8 @@ and explain why it is called the _parallelogram rule_.
:::
-:::{dropdown} Solution to {numref}`Exc:InnerProduct:PargramRule` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:InnerProduct:PargramRule`
+:class: solution, dropdown
Again it's a chain of identities using basic properties of the dot product.
@@ -458,6 +461,7 @@ and conclude that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, $\mathbf{u}$ and
:::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/59912254-6fc8-43c7-9c44-1ea7eab1c236?id=62409
:label: grasple_exercise_1_2_1T
:dropdown:
@@ -476,7 +480,8 @@ Suppose $\mathbf{v} \in \mathbb{R}^n$. Then $\mathbf{v}\perp\mathbf{v} \i
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:InnerProduct:vDotv=0Impliesv=0`
+:class: myproof
By definition
@@ -499,11 +504,14 @@ Let $\mathbf{n}$ be any nonzero vector in the plane.
The set of vectors that are orthogonal to $\mathbf{n}$ all lie on a line through the origin. (See {numref}`Figure %s <Fig:InnerProduct:PerpendicularLine>`.) If we agree that $\mathbf{0}\perp\mathbf{n}$, it will be the whole line.
The vector $\mathbf{n}$ is often said to be a _normal_ vector to the line.
-:::{figure} Images/Fig-InnerProduct-PerpendicularLine.svg
+```{applet}
+:url: dot_product/perpendicularline
+:fig: Images/Fig-InnerProduct-PerpendicularLine.svg
:name: Fig:InnerProduct:PerpendicularLine
+:class: dark-light
-Vectors orthogonal to a nonzero vector $\mathbf{n}$ in the plane
-:::
+Vectors orthogonal to a nonzero vector $\mathbf{n}$ in the plane.
+```
::::
@@ -534,8 +542,9 @@ $$
:url: dot_product/innerproduct_projectionvectorline
:fig: Images/Fig-InnerProduct-ProjectionVectorLine.svg
:name: Fig:InnerProduct:ProjectionVectorLine
+:class: dark-light
-Projection of a vector $\mathbf{w}$ onto a nonzero vector $\mathbf{v}$
+Projection of a vector $\mathbf{w}$ onto a nonzero vector $\mathbf{v}$.
:::
:::{prf:proposition}
@@ -551,7 +560,8 @@ $$
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:InnerProduct:UniqueProjection`
+:class: myproof
With the rules of the dot product the vector $\mathbf{w}$ is easily constructed. <BR>
Starting from
@@ -631,6 +641,7 @@ as required.
:::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/88c460cd-36ee-49b0-8fb8-d29b55ad253a?id=84822
:label: grasple_exercise_1_2_2T
:dropdown:
@@ -656,7 +667,8 @@ $$
:::
-::::{dropdown} Solution to {numref}`Exc:InnerProduct:SameProjectionThenWhat` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:InnerProduct:SameProjectionThenWhat`
+:class: solution, dropdown
Suppose $\text{proj}_{\mathbf{v}}(\mathbf{w}_1) = \text{proj}_{\mathbf{v}}(\mathbf{w}_2) $. Thus $\dfrac{\mathbf{w}_1\ip\mathbf{v}}{\mathbf{v}\ip\mathbf{v}} \mathbf{v} = \dfrac{\mathbf{w}_2\ip\mathbf{v}}{\mathbf{v}\ip\mathbf{v}} \mathbf{v}$.
@@ -692,6 +704,7 @@ so indeed $(\mathbf{w}_1 - \mathbf{w}_2)$ and $\vect{v}$ are orthogonal.
:::{figure} Images/Fig-InnerProduct-SameProj.svg
:name: Fig:InnerProduct:SameProj
+:class: dark-light
Two vectors $\vect{w}_1$, $\vect{w}_2 $ with the same projection onto $\vect{v}$.
@@ -769,8 +782,9 @@ The first two of these properties are very easy to prove. The proof of the trian
:fig: Images/Fig-InnerProduct-TriangleInequality.svg
:name: Fig:InnerProduct:TriangleInequality
:position: 2,2
+:class: dark-light
-The Triangle Inequality
+The Triangle Inequality.
:::
::::{prf:example}
@@ -835,11 +849,13 @@ $$
:::{figure} Images/Fig-InnerProduct-Distance.svg
:name: Fig:InnerProduct:Distance
+:class: dark-light
-The distance between two vectors
+The distance between two vectors.
:::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5bc4274c-56a0-461b-bd3d-9f8bdb8f44e0?id=69740
:label: grasple_exercise_1_2_2
:dropdown:
@@ -912,7 +928,8 @@ $$
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:InnerProduct:UnitVectorForv`
+:class: myproof
Assume that $\mathbf{v} \neq \mathbf{0}$.
For $\mathbf{u} = k\mathbf{v}$, with $\norm{\mathbf{u}} = 1$ and $k > 0$ to hold, we must have
@@ -957,14 +974,6 @@ $$
:::
-%\begin{figure}
-%\begin{center}
-%\includegraphics{Images/Fig-InnerProduct-TriangleInequality.pdf}
-%\caption{The Triangle Inequality}
-%\label{Fig:InnerProduct:TriangleInequality}
-%\end{center}
-%\end{figure}
-
Interestingly, Pythagoras' theorem also holds in $\mathbb{R}^n$.
:::{prf:theorem}
@@ -980,7 +989,8 @@ $$
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:InnerProduct:PythagorasInRn`
+:class: myproof
This follows quite straightforwardly from the properties of the dot product.
@@ -1068,7 +1078,8 @@ $$
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:InnerProduct:Cauchy-Schwarz` ({prf:ref}`Cauchy-Schwarz Inequality <Thm:InnerProduct:Cauchy-Schwarz>`)
+:class: myproof
There are many ways to prove the Cauchy-Schwarz inequality. There is even a whole book devoted to it: "Cauchy Schwarz master class" by J.M. Steele.
@@ -1172,6 +1183,7 @@ With this inequality established, the Triangle Inequality
{eq}`Item:Prop:InnerProduct:TriangleInequality` is easily proved. Let's repeat it, and prove it.
:::{prf:theorem}
+:label: Thm:InnerProduct:TriangleInequality
For any two vectors in $\mathbb{R}^n$,
@@ -1182,7 +1194,8 @@ $$
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:InnerProduct:TriangleInequality`
+:class: myproof
Since all terms involved are non-negative we may as well show that the inequality holds for the squares:
@@ -1227,13 +1240,16 @@ $$
The first motivation to consider the dot product came from the question of perpendicularity of two vectors in the plane or in $\R^3$.
Perpendicularity of two vectors means that the angle between them is equal to $\frac12\pi$.
-Below we will show that it is possible to express the angle between *any* two (nonzero) vectors into dot products. And use this to define the concept of angle in a general space $\R^n$.
+Below we will show that it is possible to express the angle between _any_ two (nonzero) vectors into dot products. And use this to define the concept of angle in a general space $\R^n$.
-:::{figure} Images/Fig-InnerProduct-AngleAndProjection.svg
+```{applet}
+:url: dot_product/angleandprojection
:name: Fig:InnerProduct:AngleAndProjection
+:fig: Images/Fig-InnerProduct-AngleAndProjection.svg
+:class: dark-light
-Angle between two vectors
-:::
+Angle between two vectors.
+```
First we will show a geometrical characterization of the dot product that holds in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$.
@@ -1278,11 +1294,12 @@ $$
:::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:InnerProduct:DotProdGeometric`
+:class: myproof
We will derive formula {eq}`Eq:InnerProduct:GeometricDefinition`.
Assume that $\mathbf{v}$ and $\mathbf{w}$ are nonzero vectors.
-Recall the formula of the orthogonal projection of $\mathbf{w}$ onto $\mathbf{v}$,
+Recall the formula of the orthogonal projection of $\mathbf{w}$ onto $\mathbf{v}$,
$$
@@ -1353,7 +1370,7 @@ is the length of the orthogonal projection of $\vect{w}$ onto $\vect{v}$.
::::
-:::{exercise}
+:::{prf:example}
:label: Ex:InnerProduct:AnglesInMethaneMolecule
In a methane molecule $\ce{CH_4}$ the four $\ce{H}$-atoms are positioned in a perfectly symmetrical way around the $\ce{C}$-atom.
@@ -1399,7 +1416,7 @@ $$
\varphi = \angle(\mathbf{v},\mathbf{w}) = \arccos\left(\dfrac{\mathbf{v}\ip\mathbf{w}}{\norm{\mathbf{v}} \norm{\mathbf{w}}} \right).
$$
-This definition makes sense, since the Cauchy-Schwarz inequality ({prf:ref}`Thm:InnerProduct:Cauchy-Schwarz`)
+This definition makes sense, since the Cauchy-Schwarz inequality ({prf:ref}`Thm:InnerProduct:Cauchy-Schwarz`)
implies
$$
@@ -1443,6 +1460,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/59912254-6fc8-43c7-9c44-1ea7eab1c236?id=62409
:label: grasple_exercise_1_2_3
:dropdown:
@@ -1451,6 +1469,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7b49e0f5-ae8b-4e92-8878-665dc080b7ee?id=65601
:label: grasple_exercise_1_2_4
:dropdown:
@@ -1459,6 +1478,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c8b4eed4-179f-42ab-9ec9-07f66445c960?id=69482
:label: grasple_exercise_1_2_5
:dropdown:
@@ -1467,6 +1487,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b5a4e1c0-92ca-4307-9eb0-25a3a5807fc7?id=62415
:label: grasple_exercise_1_2_6
:dropdown:
@@ -1475,6 +1496,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/34bbb9e1-207e-4c06-8686-1c32b3f3d0aa?id=78751
:label: grasple_exercise_1_2_8
:dropdown:
@@ -1483,6 +1505,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/30a7abfe-9d40-4faa-a848-83bd67e024a0?id=62406
:label: grasple_exercise_1_2_7
:dropdown:
@@ -1491,6 +1514,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7dc339bb-fe79-4eb9-914c-ea1a7ca85a85?id=69737
:label: grasple_exercise_1_2_9
:dropdown:
@@ -1499,6 +1523,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8de90b0e-e89a-49a6-aa63-1b1e39f6e98e?id=79262
:label: grasple_exercise_1_2_10
:dropdown:
@@ -1507,6 +1532,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d4dd1154-a3ec-497e-bc73-1cd96529f0e7?id=69741
:label: grasple_exercise_1_2_11
:dropdown:
@@ -1515,6 +1541,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c2242315-7e4f-463b-b3cf-09e9e15c8b2b?id=69739
:label: grasple_exercise_1_2_12
:dropdown:
@@ -1523,14 +1550,16 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/67334454-d109-45a2-b640-545041ff896d?id=62416
:label: grasple_exercise_1_2_13
:dropdown:
-:description: Find $\text{proj}*{\mathbf{v}}(\mathbf{w})$ in $\R^2$.
+:description: Find $\text{proj}_{\mathbf{v}}(\mathbf{w})$ in $\R^2$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9705b078-6c91-42c6-9768-8a043115b881?id=62658
:label: grasple_exercise_1_2_14
:dropdown:
@@ -1539,6 +1568,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/531d3be2-dd62-4c21-b023-70e0b63809be?id=78747
:label: grasple_exercise_1_2_15
:dropdown:
@@ -1547,6 +1577,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/161ecdf6-4cfb-41ba-bc16-685fe8532471?id=62414
:label: grasple_exercise_1_2_16
:dropdown:
@@ -1555,6 +1586,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c4d2743f-5f14-4812-9531-1a40c28c15cb?id=62413
:label: grasple_exercise_1_2_17
:dropdown:
@@ -1563,6 +1595,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/407cb45d-2baf-4b0d-a1eb-6e51186e19f3?id=69738
:label: grasple_exercise_1_2_18
:dropdown:
@@ -1571,6 +1604,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c4c1c609-b1dd-4588-865f-53d7e8221f88?id=62689
:label: grasple_exercise_1_2_19
:dropdown:
@@ -1579,10 +1613,10 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2a2423c3-0907-40b7-bd5f-7607baf7cc09?id=62668
-:label: grasple*exercise_1_2_20
+:label: grasple_exercise_1_2_20
:dropdown:
-:description: What to conclude
-from $\text{proj}_{\mathbf{v}}(\mathbf{w}_1 ) = \text{proj}_{\mathbf{v}}(\mathbf{w}\_2)$?
+:description: What to conclude from $\text{proj}_{\mathbf{v}}(\mathbf{w}_1 ) = \text{proj}_{\mathbf{v}}(\mathbf{w}\_2)$?
::::
diff --git a/Chapter1/Lines_and_Planes.md b/Chapter1/Lines_and_Planes.md
index 25d315e..77706d8 100644
--- a/Chapter1/Lines_and_Planes.md
+++ b/Chapter1/Lines_and_Planes.md
@@ -27,6 +27,7 @@ The point $(4, 1)$, for example, is a point on the line with equation $2x-3y=5$
::::{figure} Images/Fig-LinesAndPlanes-LineInPlane.svg
:name: Fig:LinesAndPlanes:LineInPlane
+:class: dark-light
The line $2x-3y=5$ in the plane.
::::
@@ -37,6 +38,7 @@ Two lines can have a point of intersection. Let $\mathcal{L}_1$ and $\mathcal{L}
::::{figure} Images/Fig-LinesAndPlanes-PointIntersection.svg
:name: Fig:LinesAndPlanes:PointIntersection
+:class: dark-light
Intersecting lines.
::::
@@ -45,6 +47,7 @@ In the previous example there was exactly one point of intersection. This is not
::::{figure} Images/Fig-LinesAndPlanes-ParallelLines.svg
:name: Fig:LinesAndPlanes:ParallelLines
+:class: dark-light
Parallel lines.
::::
@@ -53,6 +56,7 @@ Finally, let us take a look at the line $\mathcal{L}_2$, which was defined by th
::::{figure} Images/Fig-LinesAndPlanes-CoincidingLines.svg
:name: Fig:LinesAndPlanes:CoincidingLines
+:class: dark-light
Coinciding lines.
::::
@@ -90,11 +94,14 @@ $$
is a vector parallel to the same line.
-::::{figure} Images/Fig-LinesAndPlanes-VectorEquation.svg
+```{applet}
+:url: lines_and_planes/vector_equation
+:fig: Images/Fig-LinesAndPlanes-VectorEquation.svg
:name: Fig:LinesAndPlanes:VectorEquation
+:class: dark-light
The line $\mathcal{L}_1$: $\vect{x} = \vect{v}_0 + r\vect{u}$.
-::::
+```
How do we obtain all vectors on the line $\mathcal{L}_1$? Let us start with the vector
@@ -159,6 +166,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b110d6af-8cf7-4385-92f4-6f0ee364c860?id=63490
:label: grasple_exercise_1_4_1
:dropdown:
@@ -185,6 +193,7 @@ $$
::::{figure} Images/Fig-LinesAndPlanes-NormalEquationLine.svg
:name: Fig:LinesAndPlanes:NormalEquationLine
+:class: dark-light
Line $\mathcal{L}$ with normal vector $\vect{n}$.
::::
@@ -220,6 +229,7 @@ Let us try to find a Cartesian equation for the line $\mathcal{L}$ through the p
:::{figure} Images/Fig-LinesAndPlanes-OrthogonalLines.svg
:name: Fig:LinesAndPlanes:OrthogonalLines
+:class: dark-light
A line orthogonal to $\mathcal{L}_1$.
:::
@@ -270,6 +280,7 @@ As we can see in {numref}`Figure %s <Fig:LinesAndPlanes:NormalEquationPlaneOrigi
:fig: Images/Fig-LinesAndPlanes-NormalEquationPlaneOrigin.svg
:name: Fig:LinesAndPlanes:NormalEquationPlaneOrigin
:status: approved
+:class: dark-light
A plane through the origin.
::::
@@ -281,6 +292,7 @@ Now let $P$ be the point with coordinates $(0, 2, 1)$ and take an arbitrary poin
:fig: Images/Fig-LinesAndPlanes-NormalEquationPlane.svg
:name: Fig:LinesAndPlanes:NormalEquationPlane
:status: approved
+:class: dark-light
A plane through the point $P$.
::::
@@ -296,6 +308,7 @@ What is the Cartesian equation of this plane? If the coordinates of $Q$ are $(x,
This means that $Q$ is on the plane $\mathcal{P}$ through $P$ and orthogonal to $\mathbf{n}$ if and only if its coordinates satisfy $2x+y+3z-5=0$ or $2x+y+3z=5$. Hence, we obtain a Cartesian equation of $\mathcal{P}$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9b9d1faa-013e-4c46-8863-53c3fdfeff4a?id=79284
:label: grasple_exercise_1_4_2
:dropdown:
@@ -390,6 +403,7 @@ We can find directional vectors of a plane by taking vectors that connects two d
:fig: Images/Fig-LinesAndPlanes-DirectionalVectorsPlane.svg
:name: Fig:LinesAndPlanes:DirectionalVectorsPlane
:status: approved
+:class: dark-light
A parametric vector equation of a plane.
:::
@@ -461,8 +475,9 @@ It is of course possible that two planes have no points in common at all. Take a
:fig: Images/Fig-LinesAndPlanes-TwoDisjointPlanes.svg
:name: Fig:LinesAndPlanes:TwoDisjointPlanes
:status: approved
+:class: dark-light
-Two planes without a common point in common.
+Two planes without a point in common.
:::
Two planes in $\mathbb{R}^3$ can never have a single point of intersection. They can, however, have infinitely many common points. This can occur when the two planes have a line of intersection, as we can see in {numref}`Figure %s <Fig:LinesAndPlanes:TwoPlanesLineIntersection>`. On the other hand, it is also possible that two planes coincide, see {numref}`Figure %s <Fig:LinesAndPlanes:TwoPlanesCoincide>`. In this case each point on one of the two planes is an intersection point.
@@ -472,6 +487,7 @@ Two planes in $\mathbb{R}^3$ can never have a single point of intersection. They
:fig: Images/Fig-LinesAndPlanes-TwoPlanesLineIntersection.svg
:name: Fig:LinesAndPlanes:TwoPlanesLineIntersection
:status: approved
+:class: dark-light
Two planes with a line of intersection.
:::
@@ -481,6 +497,7 @@ Two planes with a line of intersection.
:fig: Images/Fig-LinesAndPlanes-TwoPlanesCoincide.svg
:name: Fig:LinesAndPlanes:TwoPlanesCoincide
:status: approved
+:class: dark-light
Two planes that coincide.
:::
@@ -494,6 +511,7 @@ First of all, it is possible that there are no points that are on $\mathcal{P}_1
:fig: Images/Fig-LinesAndPlanes-DisjointPlanes.svg
:name: Fig:LinesAndPlanes:DisjointPlanes
:status: approved
+:class: dark-light
Three planes without a point in common.
:::
@@ -505,6 +523,7 @@ In {numref}`Figure %s <Fig:LinesAndPlanes:PlanesPointIntersection>` we see three
:fig: Images/Fig-LinesAndPlanes-PlanesPointIntersection.svg
:name: Fig:LinesAndPlanes:PlanesPointIntersection
:status: approved
+:class: dark-light
Three planes with one point in common.
:::
@@ -516,6 +535,7 @@ There are several circumstances where three planes have an infinite number of po
:fig: Images/Fig-LinesAndPlanes-PlanesLineIntersection.svg
:name: Fig:LinesAndPlanes:PlanesLineIntersection
:status: approved
+:class: dark-light
Three planes with line of intersection.
:::
@@ -538,6 +558,7 @@ Let $\mathcal{L}$ be a line in $\mathbb{R}^3$, $\mathbf{v_0}$ a vector that conn
:fig: Images/Fig-LinesAndPlanes-ParametricLineSpace.svg
:name: Fig:LinesAndPlanes:ParametricLineSpace
:status: approved
+:class: dark-light
The line $\mathcal{L}$ in $\mathbb{R}^3$.
:::
@@ -573,6 +594,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d9a13ea7-786d-4935-8531-2ba3ec41c930?id=67061
:label: grasple_exercise_1_4_3
:dropdown:
@@ -581,6 +603,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a96fb8c3-000c-4d62-be3c-36dc47ced610?id=67694
:label: grasple_exercise_1_4_4
:dropdown:
@@ -589,6 +612,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1c6242a7-85e4-4bef-b05b-4c6693170bfc?id=67268
:label: grasple_exercise_1_4_5
:dropdown:
@@ -597,6 +621,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/74dbf343-63db-46cf-9517-ab2694d616fb?id=71058
:label: grasple_exercise_1_4_6
:dropdown:
@@ -605,6 +630,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c86f64d5-5d64-403e-98f7-9d8e8c4b90fb?id=78903
:label: grasple_exercise_1_4_7
:dropdown:
@@ -613,6 +639,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/42d254e8-e577-4bb1-b365-110f9805c1cf?id=78848
:label: grasple_exercise_1_4_8
:dropdown:
@@ -621,6 +648,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2b9cb96e-caa3-4f1c-8f65-a77be3f70050?id=67065
:label: grasple_exercise_1_4_9
:dropdown:
@@ -629,6 +657,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/35559ca4-9c46-4d25-a6ed-d44c4eb6d33b?id=67067
:label: grasple_exercise_1_4_10
:dropdown:
@@ -637,6 +666,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/67cbf60d-ce34-49f0-b833-715784647873?id=67257
:label: grasple_exercise_1_4_11
:dropdown:
@@ -645,6 +675,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bbb3c082-b8d1-42d6-b718-b37d5ba26363?id=67260
:label: grasple_exercise_1_4_12
:dropdown:
@@ -653,6 +684,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/23faf686-3fb3-4280-aa26-f14be36f5aae?id=78870
:label: grasple_exercise_1_4_13
:dropdown:
@@ -660,6 +692,7 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/0b29e241-4c23-4321-be4c-9a7f8cd80b5c?id=67697
:label: grasple_exercise_1_4_14
:dropdown:
@@ -668,8 +701,9 @@ is thus a possible parametric vector equation of $\mathcal{L}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f02eebc1-b919-4070-a87a-f04bb620cc8d?id=80872
-:label: grasple_exercise_1_4_14
+:label: grasple_exercise_1_4_15
:dropdown:
:description: To give a cartesian equation for a plane containing a given point $A$ and with a given normal vector $\vect{n}$.
diff --git a/Chapter1/Vectors.md b/Chapter1/Vectors.md
index c77e2fb..1a991b3 100644
--- a/Chapter1/Vectors.md
+++ b/Chapter1/Vectors.md
@@ -18,6 +18,7 @@ Consider an arrow $\mathbf{v}_{1}$ in the plane as in {numref}`Figure %s <Fig:Ve
```{figure} Images/Fig-Vectors-ArrowinPlane.svg
:name: Fig:Vectors:ArrowinPlane
+:class: dark-light
Two arrows in the plane.
```
@@ -28,6 +29,7 @@ Of course, such route plans can be combined. Take a look at {numref}`Figure %s <
```{figure} Images/Fig-Vectors-AdditionPlane.svg
:name: Fig:Vectors:AdditionPlane
+:class: dark-light
Geometrical interpretation of addition in the plane.
```
@@ -90,6 +92,7 @@ which is precisely $2\mathbf{v}_{3}$.
We have found that the geometrical notion of an arrow in the plane with a certain length pointing in a certain direction is captured perfectly by the algebraic notion of its vector. Moreover, the natural operations of stretching and combining arrows can easily be done algebraically. In fact, there is no reason why we should restrict ourselves to arrows in the plane. We can just as well take arrows in three dimensional space and glue them together or stretch them. In {numref}`Subsection %s <Subsec:Vectors:ndim>` we will formalize this notion to $n$ dimensions -- and we will see that it is not so strange to have more than $3$ dimensions!
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/553450f1-e960-4bac-9bb4-074fe8106369?id=78688
:label: grasple_exercise_1_1_1
:dropdown:
@@ -172,6 +175,7 @@ We will sometimes call real numbers *scalars*, because we use them to scale vect
:url: vectors/3Daddition
:fig: Images/Fig-Vectors-3Daddition.svg
:status: approved
+:class: dark-light
Geometrical interpretation of addition for three-dimensional vectors.
```
@@ -186,15 +190,8 @@ Note that we only define the sum of two vectors if they have the same size!
This might look a bit scary, but is really just what we did in the plane, just with more numbers now. We now use the term vector instead of arrow. Adding two vectors is again just gluing the second one to the tip of the first one, and taking a scalar multiple of a vector is just stretching it again (or perhaps shrinking it).
-%\begin{app}
-%More than three dimensions? How does that make sense? Well, it is actually not so strange at all! Consider for example a (digital) picture. In order to describe one point of the image, we not only have to give its location but also its colour. The location is given by two numbers: how far along the $x$-axis and how far up the $y$-axis our point is. The colour is usually given in terms of primary colours. Typical primary colours would be red, green, and blue. Other colours are made by mixing certain amounts of red, green, and blue. You can get magenta, for example, by mixing red and blue in equal amounts. To fully describe a point in our picture, we therefore need a vector of length 5: two entries for the location, and three for the colour.
-%\begin{figure}
-%\missingfigure{Applet with e.g. 20x10 squares and five changeable dimensions.}
-%\caption{A 20x10 picture. Every point is described by two numbers giving its location and three numbers giving its colour.}
-%\end{figure}
-%\end{app}
-
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a962fb8c-89b0-4b97-b76e-8f53335cf301?id=70140
:label: grasple_exercise_1_1_2
:dropdown:
@@ -235,13 +232,12 @@ Suppose $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ are vectors in $\mathbb{R
````
-```{prf:proof}
-
-Exercise.
+::::{admonition} Proof of {prf:ref}`Prop:Vectors:BasicRules`
+:class: myproof
-%(Grasple 62390)
+See {numref}`grasple_exercise_1_1_11`.
-```
+::::
For example, the equation {eq}`Item:Vectors:Associativity` tells us that we do not have to worry about bracketing when adding vectors. That is, we can first add $\mathbf{v}_{1}$ to $\mathbf{v}_{2}$ and then add $\mathbf{v}_{3}$ to the result or we can first add $\mathbf{v}_{2}$ to $\mathbf{v}_{3}$ and then add the result to $\mathbf{v}_{1}$. In both cases, we will get the same answer.
@@ -262,7 +258,7 @@ Note that for every molecule of sodium sulfate we need two molecules of carbon i
```{figure} Images/Fig-Vectors-ChemRec.svg
:name: Fig:Vectors:ChemRec
-The chemical reaction given in {eq}`Eq:Vectors:ChemReac`
+The chemical reaction given in {eq}`Eq:Vectors:ChemReac`.
```
There are four different chemical elements involved in this reaction: sodium ($\ce{Na}$), sulfide ($\ce{S}$), oxygen ($\ce{O}$), and carbon. Each of our molecules can be written as a vector of size four with the entries giving the number of sodium, sulfide, oxygen, and carbon atoms (in that order) in the molecule. This gives:
@@ -315,6 +311,7 @@ Suppose we fix an arbitrary point in the plane. Let us call it the _origin_ and
```{figure} Images/Fig-Vectors-PointandVec.svg
:name: Fig:Vectors:PointandVect
+:class: dark-light
The point $P=(-1,2)$ and its associated vector $\mathbf{v}=\begin{bmatrix}-1\\2\end{bmatrix}$.
```
@@ -351,6 +348,7 @@ $$
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/670148b0-07cb-4f0c-861f-0ac6fbc83fb2?id=70141
:label: grasple_exercise_1_1_3
:dropdown:
@@ -359,6 +357,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f85678d3-21f1-484f-a589-4e2fc5b0f76d?id=73610
:label: grasple_exercise_1_1_4
:dropdown:
@@ -367,6 +366,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9c3a037d-7bcb-49c7-a167-baffbae14d46?id=70142
:label: grasple_exercise_1_1_5
:dropdown:
@@ -375,6 +375,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a33e81af-5670-421f-96a0-d4ed40f5e79b?id=74451
:label: grasple_exercise_1_1_6
:dropdown:
@@ -383,6 +384,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b212a5ff-ea16-47a9-bc71-8e2b21944c9d?id=69732
:label: grasple_exercise_1_1_7
:dropdown:
@@ -391,6 +393,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a56eb5e4-62c9-4dfa-9f9c-e53c1f1c913a?id=69479
:label: grasple_exercise_1_1_8
:dropdown:
@@ -399,6 +402,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e1df17bf-cfb6-4dde-ae63-c424b5e149ba?id=73622
:label: grasple_exercise_1_1_9
:dropdown:
@@ -407,6 +411,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ff44880f-1ce0-428e-8bb2-42898b66e76f?id=78691
:label: grasple_exercise_1_1_10
:dropdown:
@@ -415,6 +420,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/65b17e6e-b9e1-44de-9445-578c5ee1f633?id=62390
:label: grasple_exercise_1_1_11
:dropdown:
diff --git a/Chapter2/Images/Fig-LinInd-Examplein2D.svg b/Chapter2/Images/Fig-LinInd-Examplein2D.svg
index 26b11b4..1bd7580 100644
--- a/Chapter2/Images/Fig-LinInd-Examplein2D.svg
+++ b/Chapter2/Images/Fig-LinInd-Examplein2D.svg
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+</symbol>
</g>
<clipPath id="clip1">
<path d="M 3 0 L 161 0 L 161 142.054688 L 3 142.054688 Z M 3 0 "/>
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-<g style="fill:rgb(64.704895%,0%,20.391846%);fill-opacity:1;">
+<path style="fill:none;stroke-width:1.59404;stroke-linecap:round;stroke-linejoin:miter;stroke:rgb(0%,60.78186%,46.665955%);stroke-opacity:1;stroke-miterlimit:10;" d="M 23.057998 23.056696 L 23.057998 134.555185 " transform="matrix(0.996685,0,0,-0.996685,3.096557,144.32402)"/>
+<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,60.78186%,46.665955%);fill-opacity:1;" d="M 26.078125 6.445313 L 23.058594 12.480469 L 26.078125 10.214844 L 29.09375 12.480469 "/>
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<use xlink:href="#glyph1-2" x="17.272412" y="67.435728"/>
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<use xlink:href="#glyph0-1" x="67.646881" y="58.020042"/>
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<use xlink:href="#glyph0-1" x="394.104179" y="130.727239"/>
</g>
<g style="fill:rgb(64.704895%,0%,20.391846%);fill-opacity:1;">
- <use xlink:href="#glyph1-2" x="401.178651" y="132.514296"/>
+ <use xlink:href="#glyph1-4" x="401.178651" y="132.514296"/>
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<g style="fill:rgb(0%,60.78186%,46.665955%);fill-opacity:1;">
- <use xlink:href="#glyph1-3" x="218.847033" y="67.435728"/>
+ <use xlink:href="#glyph1-2" x="218.847033" y="67.435728"/>
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diff --git a/Chapter2/Images/Fig-LinInd-Examplein3D.svg b/Chapter2/Images/Fig-LinInd-Examplein3D.svg
index 43cc254..0d8d899 100644
--- a/Chapter2/Images/Fig-LinInd-Examplein3D.svg
+++ b/Chapter2/Images/Fig-LinInd-Examplein3D.svg
@@ -1,111 +1,111 @@
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diff --git a/Chapter2/LinearCombinations.md b/Chapter2/LinearCombinations.md
index 9eedcca..f6728e0 100644
--- a/Chapter2/LinearCombinations.md
+++ b/Chapter2/LinearCombinations.md
@@ -23,6 +23,7 @@ The vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ are two vectors in the plane $\mat
:fig: Images/Fig-LinearCombinations-LinComb.svg
:name: Fig:LinearCombinations:LinearCombinations
:status: approved
+:class: dark-light
Linear combinations of vectors in the plane.
```
@@ -96,6 +97,7 @@ In this case it is a lot easier to decide whether $\mathbf{b}$ is a linear combi
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ac63b286-09e1-46e5-91fc-952b54436293?id=78560
:label: grasple_exercise_2_2_A
:dropdown:
@@ -104,6 +106,7 @@ In this case it is a lot easier to decide whether $\mathbf{b}$ is a linear combi
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bd263ac1-b906-48dc-a898-d959254d9681?id=70163
:label: grasple_exercise_2_2_B
:dropdown:
@@ -154,6 +157,7 @@ What does the span of a single non-zero vector look like? A linear combination o
:fig: Images/Fig-LinearCombinations-SpanOne.svg
:name: Fig:LinearCombinations:SpanOneVectors
:status: approved
+:class: dark-light
The span of a single non-zero vector.
```
@@ -171,16 +175,12 @@ $$
This looks like the parametric vector equation of a plane. Since the span must contain the zero vector we find that we obtain a plane through the origin like in {numref}`Figure %s <Fig:LinearCombinations:SpanTwoVectors>`.
-:::{figure}
-:name:
-
-:::
-
```{applet}
:url: linear_combinations/span_two_plane
:fig: Images/Fig-LinearCombinations-SpanTwoPlane.svg
:name: Fig:LinearCombinations:SpanTwoVectors
:status: approved
+:class: dark-light
The span of two non-zero, non-parallel vectors.
```
@@ -196,6 +196,7 @@ The span of two non-zero vectors does not need to be a plane through the origin.
:fig: Images/Fig-LinearCombinations-SpanTwoLine.svg
:name: Fig:LinearCombinations:SpanTwoParallelVectors
:status: approved
+:class: dark-light
The span of two non-zero, parallel vectors.
```
@@ -208,16 +209,13 @@ If two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel, then $\mathb
If we start with three non-zero vectors in $\mathbb{R}^3$, then the resulting span may take on different forms. The span of the three vectors in {numref}`Figure %s <Fig:LinearCombinations:SpanThreeVectors1>`, for example, is equal to the entire space $\mathbb{R}^3$. In {numref}`Sec:BasisDim` we will see why this is the case.
-:::{figure}
-:name:
-
-:::
```{applet}
:url: linear_combinations/span_three
:fig: Images/Fig-LinearCombinations-SpanThreeR3.svg
:name: Fig:LinearCombinations:SpanThreeVectors1
:status: approved
+:class: dark-light
The span of three vectors.
```
@@ -229,6 +227,7 @@ On the other hand, if we start with the three vectors that you can see in {numre
:fig: Images/Fig-LinearCombinations-SpanThreePlane.svg
:name: Fig:LinearCombinations:SpanThreeVectors2
:status: approved
+:class: dark-light
The span of three vectors lying in the same plane.
```
@@ -238,6 +237,7 @@ There is also a possibility where the span of three non-zero vectors in $\mathbb
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/676d672c-74fc-4545-99ba-6b308af566ce?id=78542
:label: grasple_exercise_2_2_C
:dropdown:
@@ -304,7 +304,8 @@ If $(\mathbf{e}_1, \ldots, \mathbf{e}_n)$ is the standard basis for $\mathbb{R}^
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinearCombinations:SpanStandardBasis`
+:class: myproof
Take an arbitrary vector $\mathbf{v}$ in $\mathbb{R}^n$ with
@@ -334,6 +335,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9c780d10-9a8f-4fd6-9471-3f1a0e46c009?id=70171
:label: grasple_exercise_2_2_1
:dropdown:
@@ -342,6 +344,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.f74168ff-a448-4420-88d9-ebe7365a00a9?id=70172com/exercises/
:label: grasple_exercise_2_2_2
:dropdown:
@@ -350,6 +353,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b760d9b9-d0ba-4875-b828-397e7a045283?id=70175
:label: grasple_exercise_2_2_3
:dropdown:
@@ -360,6 +364,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
% ------------------------------------------------
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a8175390-3844-408c-b192-c4b05f9beb7b?id=70170
:label: grasple_exercise_2_2_4
:dropdown:
@@ -368,6 +373,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fab5c526-91ed-407b-9faa-645f40c22b8b?id=70169
:label: grasple_exercise_2_2_5
:dropdown:
@@ -376,6 +382,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2167085c-2498-4694-9eac-abfeeb0ec307?id=70162
:label: grasple_exercise_2_2_6
:dropdown:
@@ -386,6 +393,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
% ------------------------------------------------
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/493831d9-ab4a-4f78-b9ea-7b707aa9f4c2?id=70174
:label: grasple_exercise_2_2_7
:dropdown:
@@ -394,6 +402,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c008320d-9d0e-463f-8bb7-344988f10438?id=70176
:label: grasple_exercise_2_2_8
:dropdown:
@@ -402,6 +411,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b4f4dc1f-4f56-41e8-b16d-a2694e90890c?id=70181
:label: grasple_exercise_2_2_9
:dropdown:
@@ -412,6 +422,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
% ------------------------------------------------
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/45bc5527-e79b-4198-b6b7-9b3168d9d1ff?id=70182
:label: grasple_exercise_2_2_10
:dropdown:
@@ -422,6 +433,7 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
% ------------------------------------------------
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7fcebe18-474c-4995-9c81-f1da7ab4cc5e?id=70360
:label: grasple_exercise_2_2_11
:dropdown:
diff --git a/Chapter2/LinearIndependence.md b/Chapter2/LinearIndependence.md
index e600de6..22e6e7e 100644
--- a/Chapter2/LinearIndependence.md
+++ b/Chapter2/LinearIndependence.md
@@ -4,6 +4,7 @@ As we have seen ({prf:ref}`Ex:LinearCombinations:SpanOfOneVector` and {prf:ref}`
::::{figure} Images/Fig-LinInd-Examplein1D.svg
:name: Fig:LinInd:Examplein1D
+:class: dark-light
The set $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2}\right\rbrace$ contains two vectors, but one of them is superfluous. Every vector one can make as a linear combination of $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ can also be made with just $\mathbf{v}_{1}$.
::::
@@ -37,7 +38,8 @@ precisely two vectors, say $\mathbf{u}$ and $\mathbf{v}$. Then $S$ is linearly i
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:LinInd:LinIndforSmallSets`
+:class: myproof
<ul>
<li>
@@ -59,7 +61,7 @@ As you can see from the proof of {prf:ref}`Prop:LinInd:LinIndforSmallSets`, our
::::::{prf:example}
:label: Item:LinInd:LinDepExin2D
-<ul>
+<ol type="i">
<li id="Item:LinInd:LinDepExin2D">
Consider the vectors
@@ -98,8 +100,8 @@ which are shown on the left in {numref}`Figure %s <Fig:LinInd:Examplein2D>`. The
Indeed, if we take an arbitrary vector $\mathbf{v}$ in $\Span{S}$, we can write it as
\begin{align*}
-\mathbf{v}&=c*{1}\mathbf{v}_{1}+c_{2}\mathbf{v}_{2}+c_{3}\mathbf{v}_{3}\\
-&=(c_{2}-c*{1})\mathbf{v}*{2}+(c*{3}+c*{1})\mathbf{v}_{3}
+\mathbf{v}&=c_{1}\mathbf{v}_{1}+c_{2}\mathbf{v}_{2}+c_{3}\mathbf{v}_{3}\\
+&=(c_{2}-c_{1})\mathbf{v}_{2}+(c_{3}+c_{1})\mathbf{v}_{3}
\end{align*}
in view of equation {eq}`Eq:LinInd:LinIndEx1`. This means that $\mathbf{v}$ is also in $\Span{S\setminus\left\lbrace\mathbf{v}_{1}\right\rbrace}$ and consequently that $\mathbf{v}_{1}$ is linearly dependent on $\mathbf{v}_{2}$ and $\mathbf{v}_{3}$.
@@ -114,25 +116,26 @@ Consider now the vectors
$$
\mathbf{v}_{1}=
- \begin{bmatrix}1\\0\end{bmatrix}\quad\mathbf{v}_{2}=
- \begin{bmatrix}2\\0\end{bmatrix}\quad
- \mathbf{v}_{3}=
- \begin{bmatrix}0\\1\end{bmatrix}
+ \begin{bmatrix}1\\0\end{bmatrix}\quad
+ \mathbf{v}_{2}=
+ \begin{bmatrix}0\\1\end{bmatrix}\quad\mathbf{v}_{4}=
+ \begin{bmatrix}2\\0\end{bmatrix}
$$
which are shown on the right in {numref}`Figure %s <Fig:LinInd:Examplein2D>`.
-The set $S=\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\rbrace$ is again linearly dependent since
+The set $S=\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{4}\right\rbrace$ is again linearly dependent since
$$
-\mathbf{v}_{2}=2\mathbf{v}_{1}+0\mathbf{v}_{3}\nonumber
+\mathbf{v}_{4}=2\mathbf{v}_{1}+0\mathbf{v}_{2}\nonumber
$$
-but now the subset $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2}\right\rbrace$ is a linearly dependent subset of $S$. On the other hand, the subsets $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{3}\right\rbrace$ and $\left\lbrace\mathbf{v}_{2},\mathbf{v}_{3}\right\rbrace$ are linearly independent.
+but now the subset $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{4}\right\rbrace$ is a linearly dependent subset of $S$. On the other hand, the subsets $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2}\right\rbrace$ and $\left\lbrace\mathbf{v}_{2},\mathbf{v}_{4}\right\rbrace$ are linearly independent.
::::{figure} Images/Fig-LinInd-Examplein2D.svg
:name: Fig:LinInd:Examplein2D
+:class: dark-light
-The vectors from [i.](#Item:LinInd:LinDepExin2D) on the left and from [ii.](#Item:LinInd:LinDepandIndDepExin2D) on the right. On the left, there is no vector which is a multiple of another vector so every set of two vectors is linearly independent. On the right this is not the case. The vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are multiples of each other and therefore $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2}\right\rbrace$ is linearly dependent.
+The vectors from [i.](#Item:LinInd:LinDepExin2D) on the left and from [ii.](#Item:LinInd:LinDepandIndDepExin2D) on the right. On the left, there is no vector which is a multiple of another vector, so every set of two vectors is linearly independent. On the right this is not the case. The vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{4}$ are multiples of each other and therefore $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{4}\right\rbrace$ is linearly dependent.
::::
</li>
@@ -141,27 +144,29 @@ The vectors from [i.](#Item:LinInd:LinDepExin2D) on the left and from [ii.](#Ite
Put
$$
- \mathbf{v}_{1}=
- \begin{bmatrix}1\\0\\0\end{bmatrix},\quad\mathbf{v}_{2}=
- \begin{bmatrix}0\\1\\0\end{bmatrix},\quad\mathbf{v}_{3}=
- \begin{bmatrix}1\\2\\0\end{bmatrix},\quad \text{and}\quad\mathbf{v}_{4}=
+ \mathbf{w}_{1}=
+ \begin{bmatrix}1\\0\\0\end{bmatrix},\quad\mathbf{w}_{2}=
+ \begin{bmatrix}0\\1\\0\end{bmatrix},\quad\mathbf{w}_{3}=
+ \begin{bmatrix}1\\2\\0\end{bmatrix},\quad \text{and}\quad\mathbf{w}_{4}=
\begin{bmatrix}1\\2\\1\end{bmatrix}.
$$
-The set $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\rbrace$ is linearly dependent. The set $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{4}\right\rbrace$, however, is not. This is illustrated in {numref}`Figure %s <Fig:LinInd:Examplein3D>`.
+The set $\left\lbrace\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}\right\rbrace$ is linearly dependent. The set $\left\lbrace\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{4}\right\rbrace$, however, is not. This is illustrated in {numref}`Figure %s <Fig:LinInd:Examplein3D>`.
::::{figure} Images/Fig-LinInd-Examplein3D.svg
:name: Fig:LinInd:Examplein3D
+:class: dark-light
-The four vectors from [iii.](#Item:LinInd:LinDepExin3D). Note that $\mathbf{v}_{3}$ lies in the plane spanned by $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ but $\mathbf{v}_{4}$ does not. This means that $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\rbrace$ is linearly dependent but $\left\lbrace\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{4}\right\rbrace$ is not.
+The four vectors from [iii.](#Item:LinInd:LinDepExin3D). Note that $\mathbf{w}_{3}$ lies in the plane spanned by $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ but $\mathbf{w}_{4}$ does not. This means that $\left\lbrace\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}\right\rbrace$ is linearly dependent but $\left\lbrace\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{4}\right\rbrace$ is not.
::::
</li>
-</ul>
+</ol>
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e7ff6fad-218f-4583-907f-514b3980698a?id=70195
:label: grasple_exercise_2_5_txt1
:dropdown:
@@ -170,6 +175,7 @@ The four vectors from [iii.](#Item:LinInd:LinDepExin3D). Note that $\mathbf{v}_{
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/96efb1e1-8994-4067-88b2-a24fb58c63cb?id=70196
:label: grasple_exercise_2_5_txt2
:dropdown:
@@ -178,6 +184,7 @@ The four vectors from [iii.](#Item:LinInd:LinDepExin3D). Note that $\mathbf{v}_{
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/956e2076-9232-4b43-aad9-ddbbf8252a71?id=70197
:label: grasple_exercise_2_5_txt3
:dropdown:
@@ -214,12 +221,12 @@ If $T$ is linearly independent and $S\subseteq T$, then $S$ is linearly independ
We leave the verifications of these statements to the reader.
-::::::{prf:exercise}
+:::{exercise}
:label: Ex:LinInd:LinDepSets
-Prove {prf:ref}`Prop:LinInd:LinDepSets`
+Prove {prf:ref}`Prop:LinInd:LinDepSets`.
-::::::
+:::
But how do you determine whether a set of vectors is linearly independent or not? Like so many problems in linear algebra, it comes down to solving a system of linear equations, as {prf:ref}`Prop:LinInd:LinIndisNonTrivSol` shows.
@@ -231,7 +238,7 @@ A set $\left\lbrace\mathbf{v}_{1},...,\mathbf{v}_{k}\right\rbrace$ of vectors in
:::{math}
:label: Eq:LinInd:VecEqisZero
-c*{1}\mathbf{v}*{1}+\cdots +c*{k}\mathbf{v}*{k}=\mathbf{0}
+c_{1}\mathbf{v}_{1}+\cdots +c_{k}\mathbf{v}_{k}=\mathbf{0}
:::
@@ -239,7 +246,8 @@ has a non-trivial solution. That is, a solution where not all $c_i$ are equal to
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:LinInd:LinIndisNonTrivSol`
+:class: myproof
If $\left\lbrace\mathbf{v}_{1},...,\mathbf{v}_{k}\right\rbrace$ is linearly dependent, one of these vectors, say $\mathbf{v}_{i}$, is linearly dependent on the others, i.e. it is in $\Span{\mathbf{v}_{1},...,\mathbf{v}_{i-1},\mathbf{v}_{i+1},...\mathbf{v}_{k}}$. Therefore, there exist some scalars $c_{1},...,c_{i-1},c_{i+1},...,c_{k}$ such that
@@ -280,7 +288,7 @@ has a non-trivial solution, i.e. if $A$ has a column without a pivot.
Again we leave the verification to the diligent reader.
-::::::{prf:exercise}
+::::::{exercise}
:label: Exc:LinInd:LinDepSets
Prove {prf:ref}`Cor:LinInd:LinIndisColwithoutPivot`
@@ -330,6 +338,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4d23327b-93df-41b8-bc55-481e82ba28c0?id=70201
:label: grasple_exercise_2_5_txt4
:dropdown:
@@ -338,6 +347,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cf7c40ff-3d18-4a98-9bc5-20a4ea263bb0?id=87417
:label: grasple_exercise_2_5_txt5
:dropdown:
@@ -354,7 +364,8 @@ An ordered set $S=(\mathbf{v}_{1},...,\mathbf{v}_{n})$ is linearly dependent if
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:LindInd:LinIndisVectDeponPrevious`
+:class: myproof
Let us assume $\mathbf{v}_{k}=c_{1}\mathbf{v}_{1}+\cdots+c_{k-1}\mathbf{v}_{k-1}$ for some scalars $c_{1},...,c_{k-1}$. An arbitrary element $\mathbf{v}$ of $\Span{S}$ is a linear combination of $\mathbf{v}_{1},...,\mathbf{v}_{n}$, so it is
@@ -404,7 +415,8 @@ Suppose $\mathbf{u}_{1},...,\mathbf{u}_{k}$ and $\mathbf{v}_{1},...,\mathbf{v}_{
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep`
+:class: myproof
Consider the matrices
@@ -435,7 +447,8 @@ Let $S$ be a subset of $\mathbb{R}^{n}$. If there are more than $n$ vectors in $
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:LinInd:MoreRowthanColmeansLinDep`
+:class: myproof
Take distinct vectors $\mathbf{v}_{1},...,\mathbf{v}_{n+1}$ in $S$. $\Span{\mathbf{v}_{1},...,\mathbf{v}_{n+1}}$ is contained in $\Span{\mathbf{e}_{1},..,\mathbf{e}_{n}}$ and $n+1>n$, so $\left\lbrace\mathbf{v}_{1},..,\mathbf{v}_{n+1}\right\rbrace$ is linearly dependent by {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep`. Since this set is contained in $S$, $S$ must be linearly dependent, too, by {prf:ref}`Prop:LinInd:LinDepSets`.
@@ -504,6 +517,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d40a884f-c356-4ba7-a777-92fd5f4fffcd?id=70202
:label: grasple_exercise_2_5_1
:dropdown:
@@ -512,6 +526,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d40a884f-c356-4ba7-a777-92fd5f4fffcd?id=70202
:label: grasple_exercise_2_5_2
:dropdown:
@@ -520,6 +535,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7341c27a-5482-4303-b20f-4a3965c99535?id=70209
:label: grasple_exercise_2_5_3
:dropdown:
@@ -528,6 +544,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/018f7ed5-d1ac-490b-add4-40568f525878?id=70213
:label: grasple_exercise_2_5_4
:dropdown:
@@ -536,6 +553,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c42855ae-f0af-4b52-9a89-fab0e1bdf877?id=87321
:label: grasple_exercise_2_5_5
:dropdown:
@@ -544,6 +562,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d78332f6-0a0b-404a-973d-adc745782ab6?id=70204
:label: grasple_exercise_2_5_6
:dropdown:
@@ -552,6 +571,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9345f478-7f65-4239-a9ea-26929131f010?id=70205
:label: grasple_exercise_2_5_7
:dropdown:
@@ -560,6 +580,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9345f478-7f65-4239-a9ea-26929131f010?id=70205
:label: grasple_exercise_2_5_8
:dropdown:
@@ -568,6 +589,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fce5512f-2e50-4973-8de1-d8d569e497b4?id=70208
:label: grasple_exercise_2_5_9
:dropdown:
@@ -576,6 +598,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fce5512f-2e50-4973-8de1-d8d569e497b4?id=70208
:label: grasple_exercise_2_5_11
:dropdown:
@@ -584,6 +607,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5cb7db99-e91c-4f82-94ce-e69c042e14af?id=70191
:label: grasple_exercise_2_5_12
:dropdown:
@@ -592,6 +616,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4da2f0e7-eef3-4acc-baea-ac689bda49f3?id=87426
:label: grasple_exercise_2_5_13
:dropdown:
@@ -600,6 +625,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cd594e68-f4b5-41fa-839e-139dd5a2c428?id=70198
:label: grasple_exercise_2_5_14
:dropdown:
@@ -608,6 +634,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/221b7ec2-e749-4528-9be9-ee6138e2f13d?id=70199
:label: grasple_exercise_2_5_15
:dropdown:
@@ -616,6 +643,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c277e508-cced-46f8-911f-4fb0dce4bd18?id=70200
:label: grasple_exercise_2_5_16
:dropdown:
@@ -624,6 +652,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/72bf1389-22c8-4588-8f73-4c7215b7cea4?id=70217
:label: grasple_exercise_2_5_17
:dropdown:
@@ -632,6 +661,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/85cbd7f9-9ab3-4e4a-9c19-152453ce0c52?id=68883
:label: grasple_exercise_2_5_18
:dropdown:
@@ -640,6 +670,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b7fa29e0-5d17-4a74-b173-0219f69fb2a3?id=68884
:label: grasple_exercise_2_5_19
:dropdown:
@@ -648,6 +679,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/804d59bd-3813-484b-b0b9-f79a1d6921c2?id=87427
:label: grasple_exercise_2_5_20
:dropdown:
@@ -656,6 +688,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5f21724d-6e37-456f-b661-3a7bfd83fb39?id=70219
:label: grasple_exercise_2_5_21
:dropdown:
@@ -664,6 +697,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bcd13c4c-50e6-4c8c-be50-abd4ba71ef2f?id=70221
:label: grasple_exercise_2_5_22
:dropdown:
@@ -672,6 +706,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/274a9a74-5977-435e-8374-59e6f93c1262?id=70194
:label: grasple_exercise_2_5_23
:dropdown:
@@ -680,6 +715,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c395902a-87aa-4858-915b-7ddc5513cb85?id=87398
:label: grasple_exercise_2_5_24
:dropdown:
@@ -687,6 +723,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8f89db8f-777f-4f11-9e09-23ddacf7a08d?id=87428
:label: grasple_exercise_2_5_25
:dropdown:
@@ -695,6 +732,7 @@ since a win yielded 2 points, a draw 1 point, and a loss 0 points. This means th
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/43810ef4-d9c7-4097-8d05-f91dd67bbb43?id=68868
:label: grasple_exercise_2_5_26
:dropdown:
diff --git a/Chapter2/LinearSystems.md b/Chapter2/LinearSystems.md
index 703596e..540dae3 100644
--- a/Chapter2/LinearSystems.md
+++ b/Chapter2/LinearSystems.md
@@ -1,8 +1,8 @@
----
+<!-- ---
csp:
frame-ancestors: 'self' https://embed.grasple.com/;
---
-
+ -->
(Section:LinSystems)=
# Systems of Linear Equations
@@ -143,6 +143,7 @@ which are both true identities.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8159aed6-a915-4cc1-bf75-f243702de530?id=83000
:label: grasple_exercise_2_1_A
:dropdown:
@@ -518,7 +519,8 @@ Changing the order of the equations.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinSystems:ElimOperations`
+:class: myproof
The correctness of the first operation is illustrated in
{prf:ref}`Ex:LinSystems:EliminationFirst`. One example is by far not a proof, but the explanation given there can be generalized/formalized.
@@ -1059,6 +1061,7 @@ Namely, in matrix $A_4$ the second row is a non-zero row that is below the all-z
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a6512aec-5f2d-4e3c-9c26-a0a22815480e?id=86165
:label: grasple_exercise_2_1_D
:dropdown:
@@ -1141,6 +1144,7 @@ In fact we simplified the system and the matrix along parallel paths. From now o
In later chapters we will also apply row reduction to matrices in other contexts, i.c. for other purposes.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/100dee65-a3fa-4747-9806-425aabb199f1?id=86189
:label: grasple_exercise_2_1_B
:dropdown:
@@ -1452,6 +1456,7 @@ the first and the third are echelon matrices and only the third is a reduced ech
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ec78c509-6e28-441b-bace-9d2b24f14d63?id=70366
:label: grasple_exercise_2_1_C
:dropdown:
@@ -1600,7 +1605,7 @@ $$
\left[\begin{array}{rrrr}2 & -1 & -1 & 2\\1 & 2 & 4 & 4\\4 & -2 & -4 & 6
\end{array}\right]\begin{array}{l}
[R_1] \\
-{[2R_1]} \\
+{[2R_2]} \\
{[R_3]} \\
\end{array} \quad\sim
\left[\begin{array}{rrrr}2 & -1 & -1 & 2\\2 & 4 & 8 & 8\\4 & -2 & -4 & 6
@@ -1617,7 +1622,7 @@ $$
\end{array}\right]\begin{array}{l}
[R_1] \\
{[R_2]} \\
-{[(-\nicefrac12)R_2]} \\
+{[(-\nicefrac12)R_3]} \\
\end{array} \quad\sim
\left[\begin{array}{rrrr}2 & -1 & -1 & 2\\0 & 5 & 9 & 6\\0 & 0 & 1 & -1
\end{array}\right]\begin{array}{l}
@@ -1750,6 +1755,7 @@ The word 'elimination' refers to the fact that the zeros that are created in the
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c242961e-d472-49b4-88e0-80cac8c617f9?id=87134
:label: grasple_exercise_2_1_E
:dropdown:
@@ -1773,7 +1779,8 @@ A system of linear equations has either zero, or one, or infinitely many solutio
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Thm:LinSystems:ZeroOneInfSolutions`
+:class: myproof
This just depends on the outcome of the elimination method (i.e. {prf:ref}`Alg:LinSystems:ElimMethod`).
If iii. occurs, the number of solutions is zero; if iv. occurs and there are no free variables, there is just one solution. Lastly, if there is at least one free variable, the solution set automatically contains infinitely many solutions.
@@ -1851,6 +1858,7 @@ Then the first system has a unique solution, the second system has infinitely ma
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fdf71712-2d0a-4bf2-bfb5-cdcac02f55df?id=70143
:label: grasple_exercise_2_1_F
:dropdown:
@@ -1872,7 +1880,8 @@ A linear system of $m$ equations in $n$ unknowns can only have a unique solution
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinSystems:UniqueSolutionImpliesSize`
+:class: myproof
Let
@@ -1912,6 +1921,7 @@ The first exercises are quite straightfordwardly computational.
The remaining exercises tend to be a bit more theoretic.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c8dee65d-a165-4534-89ee-76967d660c9c?id=69544
:label: grasple_exercise_2_1_1
:dropdown:
@@ -1920,6 +1930,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4bd179e9-99a6-436a-a178-9ad77210f86b?id=71057
:label: grasple_exercise_2_1_1B
:dropdown:
@@ -1928,6 +1939,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9bb8381d-f1a2-4e58-8928-4985cce492c4?id=76019
:label: grasple_exercise_2_1_2
:dropdown:
@@ -1936,6 +1948,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/42f38f80-f854-469d-b1e6-893539fd3572?id=82676
:label: grasple_exercise_2_1_3
:dropdown:
@@ -1944,6 +1957,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/0d23f4f3-5798-4a7a-b40e-f163f7e2b37f?id=82667
:label: grasple_exercise_2_1_4
:dropdown:
@@ -1952,6 +1966,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/99536b94-713b-4bae-874c-62958af0f5fe?id=80875
:label: grasple_exercise_2_1_5
:dropdown:
@@ -1960,6 +1975,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ff8bfe99-7a87-4711-b589-7ba70a857a39?id=80876
:label: grasple_exercise_2_1_6
:dropdown:
@@ -1968,6 +1984,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e1ffae46-da26-42b6-98ad-957478b6d58c?id=76653
:label: grasple_exercise_2_1_7
:dropdown:
@@ -1976,6 +1993,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c01e55fb-21d2-4539-a870-353a40d51db0?id=69506
:label: grasple_exercise_2_1_8
:dropdown:
@@ -1984,6 +2002,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/796aca3d-2b17-4e17-bad7-a83c23c88db8?id=69545
:label: grasple_exercise_2_1_9
:dropdown:
@@ -1992,6 +2011,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/447cc6ad-095e-4704-9445-8fcb4e9c4b8e?id=69587
:label: grasple_exercise_2_1_10
:dropdown:
@@ -2000,6 +2020,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/204cb1ad-0608-40ce-bb56-a2a6c6e8f1af?id=69559
:label: grasple_exercise_2_1_11
:dropdown:
@@ -2008,6 +2029,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a05fdf80-1325-4e86-874e-cd858133ad46?id=69558
:label: grasple_exercise_2_1_12
:dropdown:
@@ -2016,6 +2038,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/73ced8a4-d6f8-494f-b58a-9e2f4053cd5b?id=82689
:label: grasple_exercise_2_1_13
:dropdown:
@@ -2024,6 +2047,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2b25fdf4-e662-4859-9f30-8838a1a7079f?id=69562
:label: grasple_exercise_2_1_14
:dropdown:
@@ -2032,6 +2056,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1dad6770-3b71-4c80-ae78-e63d2cfd93e9?id=69563
:label: grasple_exercise_2_1_15
:dropdown:
@@ -2040,6 +2065,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8d249aed-fa38-4a70-8271-8be07187dd06?id=69541
:label: grasple_exercise_2_1_16
:dropdown:
@@ -2048,6 +2074,7 @@ The remaining exercises tend to be a bit more theoretic.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8d249aed-fa38-4a70-8271-8be07187dd06?id=69541
:label: grasple_exercise_2_1_17
:dropdown:
@@ -2058,6 +2085,7 @@ The remaining exercises tend to be a bit more theoretic.
The remaining exercises are a bit more theoretical.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/34663755-68a2-46ec-a3e7-0ad78ba9bdcd?id=71059
:label: grasple_exercise_2_1_T1
:dropdown:
@@ -2066,6 +2094,7 @@ The remaining exercises are a bit more theoretical.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3077fca5-9295-4634-a72d-18eca315de59?id=69743
:label: grasple_exercise_2_1_T2
:dropdown:
@@ -2074,6 +2103,7 @@ The remaining exercises are a bit more theoretical.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e9b7c9da-fe93-46a3-bde7-6bd8c4583aa8?id=68838
:label: grasple_exercise_2_1_T3
:dropdown:
@@ -2084,6 +2114,7 @@ The remaining exercises are a bit more theoretical.
Four exercises about linear systems with a parameter.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e906b587-b749-438a-a97b-4a4ef917a7b2?id=69744
:label: grasple_exercise_2_1_T4A
:dropdown:
@@ -2091,6 +2122,7 @@ Four exercises about linear systems with a parameter.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9a51faa0-98f0-4934-81b5-7d78d7fef7ec?id=69745
:label: grasple_exercise_2_1_T4B
:dropdown:
@@ -2099,6 +2131,7 @@ Four exercises about linear systems with a parameter.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/12852bd2-dfc4-41e4-8e53-c0c5664d2537?id=69746
:label: grasple_exercise_2_1_T4C
:dropdown:
@@ -2107,6 +2140,7 @@ Four exercises about linear systems with a parameter.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/69589418-544a-46c7-9d36-517b3db92bd1?id=69747
:label: grasple_exercise_2_1_T4D
:dropdown:
@@ -2117,6 +2151,7 @@ Four exercises about linear systems with a parameter.
Three exercises about linear systems and pivots.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9e6cf6e3-80d2-4552-b668-cfc3bcdad27a?id=69748
:label: grasple_exercise_2_1_T5B
:dropdown:
@@ -2125,6 +2160,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5eb59a87-6524-4563-9c86-f54e6fdca71d?id=69749
:label: grasple_exercise_2_1_T5C
:dropdown:
@@ -2133,6 +2169,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9fdb7774-56f2-4a04-ae4a-a28fd4d2fd97?id=69750
:label: grasple_exercise_2_1_T5D
:dropdown:
@@ -2141,6 +2178,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d8ed4c96-da3f-4fb4-baa4-77c99cfdfeae?id=70370
:label: grasple_exercise_2_1_T17
:dropdown:
@@ -2149,6 +2187,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/23a43d40-9e2d-4d92-bf63-40519dcb7d65?id=82692
:label: grasple_exercise_2_1_T18
:dropdown:
@@ -2157,6 +2196,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/63dd8fd-fdbd-4d59-b49e-1d52977a1a8e?id=87122
:label: grasple_exercise_2_1_T19
:dropdown:
@@ -2165,6 +2205,7 @@ Three exercises about linear systems and pivots.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dbeb91c3-4834-4204-9b0f-ca7bf4bd5ecd?id=71103
:label: grasple_exercise_2_1_T20
:dropdown:
diff --git a/Chapter2/MatrixVectorProduct.md b/Chapter2/MatrixVectorProduct.md
index 509bd42..f2eae05 100644
--- a/Chapter2/MatrixVectorProduct.md
+++ b/Chapter2/MatrixVectorProduct.md
@@ -90,7 +90,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:MatVecProd:Row-ColumnRule` ({prf:ref}`Row-column rule <Prop:MatVecProd:Row-ColumnRule>`)
+:class: myproof
The vector on the left-hand side of the identity is by definition equal to the linear combination
@@ -151,6 +152,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e9b864bf-de65-4b67-92d2-7075121ae5e5?id=70222
:label: grasple_exercise_2_4_1T
:dropdown:
@@ -159,6 +161,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/848de922-1bd5-48a2-806d-a2b94bd40a4b?id=70223
:label: grasple_exercise_2_4_2T
:dropdown:
@@ -167,6 +170,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8eb9d800-ebd8-4805-8d56-eac5150f405d?id=85094
:label: grasple_exercise_2_4_3 T
:dropdown:
@@ -267,6 +271,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d9a8f246-359c-4666-bb8c-2f573e192e5c?id=68857
:label: grasple_exercise_2_4_4T
:dropdown:
@@ -294,7 +299,8 @@ $A\,(c\mathbf{x}) = c\,A\mathbf{x}$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:MatVecProduct:Linearity`
+:class: myproof
We will prove the first of the two statements; the other statement goes in a similar fashion.
There are several ways to derive the formula. Via the linear combination idea it may be the easiest.
@@ -342,7 +348,8 @@ Prove statement (ii) of the previous proposition.
::::
-:::{dropdown} Solution to {numref}`Exc:MatVecProduct:CheckLinearity(ii)` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:MatVecProduct:CheckLinearity(ii)`
+:class: solution, dropdown
Assume
@@ -504,7 +511,8 @@ where $r_i = p_i - q_i,\,i=1,\ldots,\,m$ \, is inconsistent.
::::
-:::{dropdown} Solution to {numref}`Exc:MatVecProduct:PracticeWithProp` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:MatVecProduct:PracticeWithProp`
+:class: solution, dropdown
We start with some notations.
@@ -557,7 +565,8 @@ The collection $\Span{\mathbf{v}_1, \ldots, \mathbf{v}_k}$ is equal to $\mathbb{
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinearCombinations:SpanSolution`
+:class: myproof
If $\Span{\mathbf{v}_1, \ldots, \mathbf{v}_k}$ is equal to $\mathbb{R}^n$, then each vector $\mathbf{b}$ is a vector in the span of the vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$. This means that we can write $\mathbf{b}$ as a linear combination
@@ -590,7 +599,8 @@ The equation $A \mathbf{x}=\mathbf{b}$ has a solution for each $\mathbf{b}$ in $
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinearCombinations:PivotInEachRow`
+:class: myproof
Suppose that $A$ does not contain a pivot position in each row. By definition of the reduced echelon form we know that the last row of $A$ does not have a pivot position. If $E$ is the reduced echelon form of $A$, then this means that the bottom row of $E$ contains only zeros. Let $\mathbf{e}_n$ be again the vector of which the last entry is equal to 1 and all other entries are equal to zero.
@@ -632,7 +642,8 @@ The matrix $A$ has a pivot position in each row.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinearCombinations:PivotSpanSolution`
+:class: myproof
This follows from {prf:ref}`Prop:LinearCombinations:SpanSolution` and {prf:ref}`Prop:LinearCombinations:PivotInEachRow`.
@@ -676,12 +687,14 @@ Since there are only two pivots in the reduced echelon matrix, we know that $\ma
::::
::::{prf:proposition}
+:label: Prop:LinearCombinations:SpanNotRn
If $\mathbf{v}_1, \dots ,\mathbf{v}_k$ are vectors in $\mathbb{R}^n$ and $k<n$, then the span of $\mathbf{v}_1, \dots ,\mathbf{v}_k$ is not equal to $\mathbb{R}^n$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LinearCombinations:SpanNotRn`
+:class: myproof
Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix $A$. By definition, the matrix $A$ is an $n\times k$ matrix. Let $E$ be the reduced echelon form of $A$. Since $E$ has $k$ columns we know that $E$ can have at most $k$ pivots. Because $k<n$ this means that the number of pivots is less than $n$. Therefore, we find that the number of pivots is less than the number of rows in $E$. This implies that it is impossible for $E$ to have a pivot in each row. {prf:ref}`Prop:LinearCombinations:PivotSpanSolution` now tells us that the span of the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ cannot be equal to $\mathbb{R}^n$.
@@ -690,6 +703,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5708acc0-9a35-429b-85ff-43139eed1722?id=85086
:label: grasple_exercise_2_4_1
:dropdown:
@@ -698,6 +712,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3cb73c25-fa69-4cf1-a686-1e71f2f0bf89?id=85092
:label: grasple_exercise_2_4_2
:dropdown:
@@ -706,6 +721,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/56cf013b-dc6a-4774-ac1e-fa694b16a2a8?id=85089
:label: grasple_exercise_2_4_3
:dropdown:
@@ -714,6 +730,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dba850cd-e353-4339-9811-656a565e7270?id=85091
:label: grasple_exercise_2_4_4
:dropdown:
@@ -722,6 +739,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a5715fe9-74ae-4df5-857f-2c6ed1cc9cdc?id=68889
:label: grasple_exercise_2_4_5
:dropdown:
@@ -730,6 +748,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6332f523-a152-4be7-b160-bb0bab18a4a0?id=69773
:label: grasple_exercise_2_4_6
:dropdown:
@@ -738,6 +757,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cd77f0bd-bd35-4674-9524-38a9446cd076?id=70183
:label: grasple_exercise_2_4_7
:dropdown:
@@ -746,6 +766,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5bdcebc9-3ab6-4b64-9f30-033cb9f79b80?id=76273
:label: grasple_exercise_2_4_8
:dropdown:
@@ -754,6 +775,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5ccac4fe-bf25-471e-b268-5add4b06ecfe?id=76278
:label: grasple_exercise_2_4_9
:dropdown:
@@ -762,6 +784,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e8dfc02b-2628-44f0-8a57-bed3fb0cbb26?id=77658
:label: grasple_exercise_2_4_10
:dropdown:
@@ -770,6 +793,7 @@ Use the vectors $\mathbf{v}_1, \dots ,\mathbf{v}_k$ as the columns for a matrix
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/525d98b9-7cd3-40be-80e9-6d2f65f26002?id=77661
:label: grasple_exercise_2_4_11
:dropdown:
diff --git a/Chapter2/SolutionSets.md b/Chapter2/SolutionSets.md
index 9c047b1..671d243 100644
--- a/Chapter2/SolutionSets.md
+++ b/Chapter2/SolutionSets.md
@@ -82,11 +82,14 @@ $$
to an element of the solution set from {prf:ref}`Example:SolSet:TwoLinesinR3`. The green line in {numref}`Figure %s <Fig:SolSet:TwoLinesinR3>` corresponds to the $a=2$ case.
-:::{figure} Images/Fig-SolSet-TwoLinesinR3.svg
+```{applet}
+:url: solution_sets/two_lines_in_r3
+:fig: Images/Fig-SolSet-TwoLinesinR3.svg
:name: Fig:SolSet:TwoLinesinR3
+:class: dark-light
The solution sets for the two systems of equations from {prf:ref}`Example:SolSet:TwoLinesinR3` and {prf:ref}`Example:SolSet:TwoLinesinR3b`. In blue we see the solution set of the original system {eq}`Eq:SolSet:HomSys2D`, in green that of the system with the non-zero right hand side {eq}`Eq:SolSet:NonHomSys2D`.
-:::
+```
::::
@@ -178,11 +181,15 @@ $$
\begin{bmatrix}1\\0\\1\end{bmatrix}.
$$
-:::{figure} Images/Fig-SolSet-TwoPlanesinR3.svg
+```{applet}
+:url: solution_sets/two_planes_in_r3
+:fig: Images/Fig-SolSet-TwoPlanesinR3.svg
:name: Fig:SolSet:TwoPlanesinR3
+:class: dark-light
The solution set for the two systems of equations from {prf:ref}`Ex:Solset:TwoPlanesinR3`.
-:::
+
+```
Note that, if we had changed the right hand side of our first equation to $a$, $a\neq0$, and the second one to anything but $2a$, the system would have had no solutions at all. The two equations would in that case describe two parallel planes.
@@ -229,7 +236,8 @@ Suppose $(c_{1},...,c_{n})$ is a solution of a linear system. Then $(c_{1}',...,
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:SolSet:SolplusHom`
+:class: myproof
Consider the linear system
@@ -295,7 +303,8 @@ Suppose $c_{1}$ and $c_{2}$ are arbitrary scalars. If $\vect{v}_{1}$ and $\vect{
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:SolSet:SolSetisVecSpa`
+:class: myproof
This is a left as an exercise.
@@ -319,7 +328,7 @@ By putting $c_{2}=0$ in statement [ii.](#Item:SolSet:LinCombinSolSet), we find t
By putting $c_{1}=1=c_{2}$, we find that any sum of two solutions is again a solution.
</li>
-</ol>>
+</ol>
You should keep in mind, however, that this **only** holds for solutions of **homogeneous** systems!
@@ -417,6 +426,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/03d9c585-c51a-41e0-b3ac-33ad9f42cb55?id=83384
:label: grasple_exercise_2_3_1
:dropdown:
@@ -425,6 +435,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1e3c93cb-4fa7-4bf5-b46f-0c631e074d7e?id=83594
:label: grasple_exercise_2_3_2
:dropdown:
@@ -433,6 +444,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/44dcb893-3beb-46a2-bddd-75f830cba5de?id=83499
:label: grasple_exercise_2_3_3
:dropdown:
@@ -441,6 +453,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ce8a45dc-f26f-45ca-a754-35512f882411?id=80874
:label: grasple_exercise_2_3_4
:dropdown:
@@ -449,6 +462,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/633cfb15-272d-40c0-adc6-36f091446d7d?id=83279
:label: grasple_exercise_2_3_5
:dropdown:
@@ -457,6 +471,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/611e08ab-df69-4a14-96d3-b6a9bbda316b?id=83238
:label: grasple_exercise_2_3_6
:dropdown:
@@ -465,6 +480,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9536051d-baba-4b22-94ed-94190e9e6b47?id=83246
:label: grasple_exercise_2_3_7
:dropdown:
@@ -473,6 +489,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/196ac202-23e4-4c94-842b-e50410fedea0?id=83505
:label: grasple_exercise_2_3_8
:dropdown:
@@ -481,6 +498,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/196ac202-23e4-4c94-842b-e50410fedea0?id=87438
:label: grasple_exercise_2_3_9
:dropdown:
@@ -489,6 +507,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/021bb82e-7af3-4c84-86b8-0dcd22bf555b?id=84556
:label: grasple_exercise_2_3_10
:dropdown:
@@ -497,6 +516,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5d723e95-a4e5-4594-970e-6332e4953e73?id=84559
:label: grasple_exercise_2_3_11
:dropdown:
@@ -505,6 +525,7 @@ Hence, in order to turn one molecule of sodium sulfate into sodium sulfide, we m
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/297528c4-7ea0-426b-aaa0-85dbcbfa97af?id=83227
:label: grasple_exercise_2_3_12
:dropdown:
diff --git a/Chapter3/GeometryofLinearTransformations.md b/Chapter3/GeometryofLinearTransformations.md
index bec6e0a..c653483 100644
--- a/Chapter3/GeometryofLinearTransformations.md
+++ b/Chapter3/GeometryofLinearTransformations.md
@@ -32,6 +32,8 @@ $$
can be seen on the left in {numref}`Figure %s <Fig:GeomLinTrans:ProjinR2>`. Let us briefly verify that it really is a linear transformation.
::::::{prf:proposition}
+:label: Prop:Geometry:Projection
+
For any vector $\mathbf{v}$ in $\mathbb{R}^{n}$, the map
$$
@@ -42,7 +44,9 @@ is a linear transformation.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Geometry:Projection`
+:class: myproof
+
The proof is a simple application of the definitions. For any $\mathbf{w}_{1},\mathbf{w}_{2}$ in $\mathbb{R}^{n}$, we have
$$
@@ -78,7 +82,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:MatofProjonLine`
+:class: myproof
The vector
@@ -90,6 +95,7 @@ is a unit vector on $\mathcal{L}$. Using the fact that $\mathbf{u}-\proj_{\mathc
::::{figure} Images/Fig-GeomLinTrans-MatofProjonLine.svg
:name: Fig:GeomLinTrans:MatofProjonLine
+:class: dark-light
The projection of $\mathbf{e}_{1}$ on the line $\mathcal{L}$ that makes an angle $\theta$ with the positive $x$-axis. Note that the length of $T_{\mathcal{L}}(\mathbf{e}_{1})$ is $\cos(\theta)$ since the length of $\mathbf{e}_{1}$ is $1$.
::::
@@ -101,6 +107,7 @@ That the second column is as claimed, too, can be shown analogously. We leave it
Often, you might have not the angle $\mathcal{L}$ makes with the positive $x$ axis, but rather a vector $\mathbf{v}$ on $\mathcal{L}$. In this case, too, you can find the standard matrix of the projection on $\mathcal{L}$ quite easily.
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:ProjMat2
Let $\mathcal{L}$ be a line that passes through the origin in the direction of $\mathbf{v}=\begin{bmatrix} v_{1}\\v_{2}\end{bmatrix}$. The projection $T_{\mathcal{L}}$ has standard matrix
$$
@@ -112,7 +119,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:ProjMat2`
+:class: myproof
It suffices to find the cosine and sine of the angle $\mathcal{L}$ makes with the positive $x$-axis in terms of $v_{1}$ and $v_{2}$. We leave this as an exercise.
@@ -132,7 +140,8 @@ An $n\times n$-matrix $P$ is the standard matrix of a projection if and only if
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:ProjSquaredisProj`
+:class: myproof
We leave this as an exercise.
@@ -150,21 +159,25 @@ this projection is depicted on the right in {numref}`Figure %s <Fig:GeomLinTrans
:url: geom_lin_trans/proj_in_r2
:fig: Images/Fig-GeomLinTrans-ProjinR2.svg
:name: Fig:GeomLinTrans:ProjinR2
+:class: dark-light
On the left an orthogonal projection $T_{1}$ acting on a few selected vectors $\mathbf{u}_{1}$, $\mathbf{u}_{2}$, and $\mathbf{u}_{3}$. On the right a non-orthogonal projection $T_{2}$ acting on some selected vectors $\mathbf{v}_{1}$, $\mathbf{v}_{2}$, and $\mathbf{v}_{3}$. In both cases, the blue line represents the line $\mathcal{L}$ in the direction of $\begin{bmatrix}2\\1\end{bmatrix}$. On the left, every vector $\mathbf{u}_{i}$ is mapped to the closest vector that lies on $\mathcal{L}$. On the right, every vector $\mathbf{v}_{i}$ is mapped to the intersection of $\mathcal{L}$ wih the line through $\mathbf{v}_{i}$ in the direction given by $\begin{bmatrix}-2\\1\end{bmatrix}$.
```
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:MapToLine
+
Let $\mathcal{L}$ be a line through the origin and let $\mathbf{w}$ be a vector not on $\mathcal{L}$. The transformation $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ which maps a vector $\mathbf{u}$ to the intersection of $\mathcal{L}$ and the line through $\mathbf{u}$ in the direction of $\mathbf{w}$ is a linear transformation.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:MapToLine`
+:class: myproof
For any vector $\mathbf{u}$, there is a unique pair of real numbers $(c_{\mathbf{u}},d_{\mathbf{u}})$ such that $\mathbf{u}+c_{\mathbf{u}}\mathbf{w}=d_{\mathbf{u}}\mathbf{v}$. What $T$ does is map $\mathbf{u}$ to $d_{\mathbf{u}}\mathbf{v}$. Hence, for any two vectors $\mathbf{u}_{1},\mathbf{u}_{2}$ in $\mathbb{R}^{2}$, we have
\begin{align*}
-\mathbf{u}*{1}+c*{\mathbf{u}*{1}}\mathbf{w}&=d*{\mathbf{u}*{1}}\mathbf{v}=T(\mathbf{u}_{1})\quad\text{and}\\
-\mathbf{u}_{2}+c*{\mathbf{u}*{2}}\mathbf{w}&=d*{\mathbf{u}*{2}}\mathbf{v}=T(\mathbf{u}_{2}).\\
+\mathbf{u}_{1}+c_{\mathbf{u}_{1}}\mathbf{w}&=d_{\mathbf{u}_{1}}\mathbf{v}=T(\mathbf{u}_{1})\quad\text{and}\\
+\mathbf{u}_{2}+c_{\mathbf{u}_{2}}\mathbf{w}&=d_{\mathbf{u}_{2}}\mathbf{v}=T(\mathbf{u}_{2}).\\
\end{align*}
Clearly, we also have
@@ -216,11 +229,12 @@ gives the projection on $\mathcal{P}$.
```{applet}
:url: geom_lin_trans/3d_proj_on_line
-:fig: Images/Fig-GeomLinTrans-3DProjonLine.svg
+:fig: Images/Fig-GeomLinTRans-3DProjonLine.svg
:name: Fig:GeomLinTrans:3DProj
:status: reviewed
+:class: dark-light
-Projections in three dimensional space. On the left, the projection on a line $\mathcal{L}$, on the right the projection on a plane $\mathcal{P}$.
+Projections on a line $\mathcal{L}$ in three dimensional space.
```
Let us briefly discuss what happens in higher dimensions.
@@ -256,6 +270,7 @@ $$
:fig: Images/Fig-GeomLinTrans-ReflinR2.svg
:name: Fig:GeomLinTrans:ReflinR2
:status: reviewed
+:class: dark-light
The reflection along the line $\mathcal{L}$ in the direction of $\mathbf{v}=\begin{bmatrix}1\\1\end{bmatrix}$. The vectors in red are mapped to the vector in blue by this reflection.
```
@@ -270,6 +285,7 @@ $$
::::{figure} Images/Fig-GeomLinTrans-ReflFromDoubleProj.svg
:name: Fig:GeomLinTrans:ReflFromDoubleProj
+:class: dark-light
Reflection along the line $\mathcal{L}$ can be seen as the transformation $2\proj_{\mathcal{L}}-I$.
::::
@@ -290,11 +306,15 @@ is the **reflection** over $\text{range}(T)$.
Since any reflection is a linear combination of some projection and the identity, we arrive at the following proposition.
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:Reflection
+
Any reflection is a linear transformation.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:Reflection`
+:class: myproof
+
A reflection is by definition a sum of scaled linear transformations. As such, it is again a linear transformation.
::::::
@@ -302,11 +322,15 @@ A reflection is by definition a sum of scaled linear transformations. As such, i
The following proposition guarantees that, as you would expect, applying a reflection twice leaves you back where you started.
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:TwoReflections
+
If $M$ is the standard matrix of a reflection, then $R^{2}=I$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:TwoReflections`
+:class: myproof
+
We know that $M=2P-I$ where $P$ is the standard matrix of some projection. By {prf:ref}`Prop:GeomLinTrans:ProjSquaredisProj`, we have $P^{2}=P$ and therefore
$$
@@ -328,7 +352,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:MatofReflinPlane`
+:class: myproof
Exercise. For the second equality, remember the trigonometric identities
@@ -342,9 +367,10 @@ For large $n$ it is hard to picture what a reflection in $n$-dimensional space d
```{applet}
:url: geom_lin_trans/3d_refl_along_plane
-:fig: Images/Fig-GeomLinTrans-3DReflalongPlane.svg
+:fig: Images/Fig-GeomLinTRans-3DReflalongPlane.svg
:name: Fig:GeomLinTrans:3DReflalongPlane
:status: reviewed
+:class: dark-light
Reflection along the plane $\mathcal{P}$ in $\mathbb{R}^{3}$.
@@ -361,7 +387,8 @@ $$S(\vect{w}_{1})\cdot S(\vect{w}_{2})=\vect{w}_{1}\cdot\vect{w}_{2}.$$
::::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:ReflDotProd`
+:class: myproof
By definition, there is an orthogonal projection with standard matrix $P$ such that $S(\vect{w})=(2P-I)\vect{w}$. We assume $P$ is the projection on the span of a single vector $\vect{v}$. If there are more, the computations become considerably messier, but neither harder nor more enlightening.
@@ -396,7 +423,9 @@ Rotations are linear transformations.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:RotsAreLinTrans`
+:class: myproof
+
Let $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a rotation. Because of {prf:ref}`Prop:InnerProduct:DotProdGeometric`, we have $\mathbf{v}_{1}\ip \mathbf{v}_{2}=T(\mathbf{v}_{1})\ip T(\mathbf{v}_{2})$. A tedious but not terribly hard calculation now shows that, for every $\mathbf{v}_{1},\mathbf{v}_{2}$ in $\mathbb{R}^{n}$,
$$
@@ -420,6 +449,7 @@ In fact, the proof of {prf:ref}`Prop:GeomLinTrans:RotsAreLinTrans` only uses the
The name _rotation_ is inspired by the following observation about rotations in the plane.
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:RotMatR2
For any real number $\theta$, the rotation over the angle $\theta$ in the plane has standard matrix
@@ -436,12 +466,14 @@ This is indeed the standard matrix of a rotation.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:RotMatR2`
+:class: myproof
Suppose we take the vector $\mathbf{e}_{1}$ and rotate it (counterclockwise) over an angle $\theta$. Where do we end up? By definition, the $x$-coordinate of our new location will be $\cos(\theta)$ and its $y$-coordinate will be $\sin(\theta)$. Similarly, if we start with the vector $\mathbf{e}_{2}$ and rotate that over the angle $\theta$, the $x$-coordinate of our new point will be $-\sin(\theta)$. This is illustrated in {numref}`Figure %s <Fig:GeomLinTrans:RotinPlane>`.
::::{figure} Images/Fig-GeomLinTrans-RotinPlane.svg
:name: Fig:GeomLinTrans:RotinPlane
+:class: dark-light
The rotation over the angle $\theta$ working on $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$. Note that the distance between $R_{\theta}\mathbf{e}_{2}$ and the vertical axis is $\sin(\theta)$ but, as $R_{\theta}\mathbf{e}_{2}$ lies to the left of the vertical axis, the first entry of $R_{\theta}\mathbf{e}_{2}$ is $-\sin(\theta)$.
::::
@@ -458,7 +490,9 @@ Any rotation in the plane is the composition of two reflections.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:RotisDoubleRefl`
+:class: myproof
+
We will show that the standard matrix $R_{\theta}$ of the rotation $T_{\theta}$ over an angle $\theta$ is the product of the standard matrices of two reflections. The claim follows then from the definition of the matrix product.
Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be two lines in the plane through the origin that make an angle of $\theta/2$ with each other. If we call $\phi/2$ the angle $\mathcal{L}_{1}$ makes with the positive $x$-axis, we can conclude that $\mathcal{L}_{2}$ makes an angle of $\phi/2+\theta/2$ with the positive $x$-axis. From {prf:ref}`Prop:GeomLinTrans:MatofReflinPlane`, we know that the standard matrices of the reflections $T_{\mathcal{L}_{1}}$ and $T_{\mathcal{L}_{2}}$ along $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, respectively, are
@@ -492,6 +526,7 @@ $$
:url: geom_lin_trans/rotisdoublerefl
:fig: Images/Fig-GeomLinTrans-RotisDoubleRefl.svg
:name: Fig:GeomLinTrans:RotisDoubleRefl
+:class: dark-light
{prf:ref}`Prop:GeomLinTrans:RotisDoubleRefl` illustrated. $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are arbitrary lines that make an angle of $\theta/2$ with each other. Composing the reflections along $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ then gives the rotation over the angle $\theta$. This is shown for the particular vector $\mathbf{v}$. Note that the angle $\phi/2$ that $\mathcal{L}_{1}$ makes with the positive $x$ axis is irrelevant to the proof.
```
@@ -543,12 +578,15 @@ so $T$ moves points not on $\mathcal{L}$ parallel to $\mathcal{L}$. Points close
::::::
-::::{figure} Images/Fig-GeomLinTrans-ShearTrans.svg
+```{applet}
+:url: geom_lin_trans/sheartrans
+:fig: Images/Fig-GeomLinTrans-ShearTrans.svg
:name: Fig:GeomLinTrans:ShearTrans
+:class: dark-light
The shear transformation $T$ from Example {numref}`Figure %s <Fig:GeomLinTrans:ShearTrans>` working on the vectors
$\mathbf{e}_{1}=\begin{bmatrix}1\\0\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}-1\\1\end{bmatrix}$. Note how the distance between a vector and the line $\mathcal{L}$ is preserved by $T$. As a consequence, the area of the green and blue parallelogams on the left is the same as that of their respective images on the right.
-::::
+```
::::::{prf:definition}
:label: Dfn:GeomLinTrans:ShearScale
@@ -577,11 +615,15 @@ $$
Note that the scalar $c$ in [ii.](#Item:GeomLinTrans:ShearScale) from {prf:ref}`Dfn:GeomLinTrans:ShearScale` is different for different vectors. For vectors lying on one side of $\mathcal{L}$, it will be positive. For vectors on the other side of $\mathcal{L}$ it will be negative. Moreover, for vectors further away from $\mathcal{L}$, $c$ will be larger than for vectors closer to $\mathcal{L}$.
::::::{prf:proposition}
+:label: Prop:GeomLinTrans:ShearTransDistance
+
If $T$ is a shear transformation fixing the line $\mathcal{L}$ and $\mathbf{v}$ is an arbitrary vector in $\mathbb{R}^{2}$, then the distance from $\mathbf{v}$ to $\mathcal{L}$ is the same as the distance from $T(\mathbf{v})$ to $\mathcal{L}$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:GeomLinTrans:ShearTransDistance`
+:class: myproof
+
The distance between a vector $\mathbf{v}$ and a line $\mathcal{L}$ is the length of $\mathbf{v}-\proj_{\mathcal{L}}(\mathbf{v})$. We find, for arbitrary $\mathbf{v}$ in $\mathbb{R}^{2}$,
\begin{align*}
\lVert T(\mathbf{v})-\proj_{\mathcal{L}}(T(\mathbf{v}))\rVert &= \lVert \mathbf{v}+c\mathbf{w}-\frac{(\mathbf{v}+c\mathbf{w})\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert=\lVert \mathbf{v}-\frac{\mathbf{v}\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert\\
@@ -651,6 +693,7 @@ Shear transformations are widely used to model this kind of displacement of laye
::::{figure} Images/Fig-GeomLinTrans-CardsStack.svg
:name: Fig:GeomLinTrans:CardsStack
+:class: dark-light
A shear transformation applied to a stack of cards.
::::
@@ -658,6 +701,7 @@ A shear transformation applied to a stack of cards.
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1531c9ba-540c-4d64-bddf-169105eaa5ff?id=70393
:label: grasple_exercise_3_3_1
:dropdown:
@@ -666,6 +710,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/51095023-d860-483f-8758-44d2b83d7c9e?id=70394
:label: grasple_exercise_3_3_2
:dropdown:
@@ -674,6 +719,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5eae3328-453b-4065-9829-be8acb10f0fa?id=70421
:label: grasple_exercise_3_3_3
:dropdown:
@@ -682,6 +728,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cf49c839-9eee-4f7b-b459-cfe3edcf530b?id=70422
:label: grasple_exercise_3_3_4
:dropdown:
@@ -690,6 +737,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/aca8c030-4392-4e22-be38-2316f9c483c4?id=70425
:label: grasple_exercise_3_3_5
:dropdown:
@@ -698,6 +746,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e67fb238-1b3e-4cad-bbb2-4126579fa97f?id=78593
:label: grasple_exercise_3_3_6
:dropdown:
@@ -706,6 +755,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/795b979c-e3d9-4f24-80ad-0dfad38c84d2?id=83137
:label: grasple_exercise_3_3_7
:dropdown:
@@ -714,6 +764,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7d10562b-929a-4b1f-9a90-6280e12b9c98?id=85261
:label: grasple_exercise_3_3_8
:dropdown:
@@ -722,6 +773,7 @@ A shear transformation applied to a stack of cards.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ee24dead-8281-493e-9ced-b2f0f9cb1421?id=85263
:label: grasple_exercise_3_3_9
:dropdown:
diff --git a/Chapter3/Injectivity_and_surjectivity.md b/Chapter3/Injectivity_and_surjectivity.md
index 15ef05c..f7cab90 100644
--- a/Chapter3/Injectivity_and_surjectivity.md
+++ b/Chapter3/Injectivity_and_surjectivity.md
@@ -56,8 +56,9 @@ so any vector $\mathbf{v}=\begin{bmatrix}a_{1}\\a_{2}\end{bmatrix}$ with $a_{1}=
::::{figure} Images/Fig-InjSurj-NonInjEx.svg
:name: Fig:InjSurj:NonInjEx
+:class: dark-light
-The transformation $T$ from {prf:ref}`Ex:InjSurj:InjEx` [i.](#Item:InjSurj:NonInjEx) . All vectors on the black line on the left, in particular $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$, are mapped to the vector $\mathbf{u}$ on the right. Similarly, all vectors on the blue line are mapped to $\mathbf{0}$. Since there is not just one vector on these lines, $T$ is not injective.
+The transformation $T$ from {prf:ref}`Ex:InjSurj:InjEx` [i.](#Item:InjSurj:NonInjEx) . All vectors on the <span class="only-light">black</span><span class="only-dark">white</span> line on the left, in particular $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$, are mapped to the vector $\mathbf{u}$ on the right. Similarly, all vectors on the blue line are mapped to $\mathbf{0}$. Since there is not just one vector on these lines, $T$ is not injective.
::::
</li>
@@ -88,11 +89,14 @@ $$
There are no free variables, so if there is a solution to the system $A\vect{x}=\vect{v}$ (which, in this case, is true for all $\vect{v}$) it will be unique. This means that there is only one $\mathbf{x}$ with $T(\mathbf{x})=A\mathbf{x}=\mathbf{u}$, hence $T$ is injective.
-::::{figure} Images/Fig-InjSurj-InjEx.svg
+```{applet}
+:url: injectivity_and_surjectivity/injsurj-injex
+:fig: Images/Fig-InjSurj-InjEx.svg
:name: Fig:InjSurj:InjEx
+:class: dark-light
The transformation $T$ from {prf:ref}`Ex:InjSurj:InjEx` [ii.](#Item:InjSurj:InjEx) and {prf:ref}`Ex:InjSurj:SurjEx` [ii.](#Item:InjSurj:SurjEx) acting on some selected vectors. Note that this transformation scales vectors and rotates them. For any vector $\mathbf{u}$ on the right, you can find only one vector $\mathbf{v}$ on the left such that $T(\mathbf{v})=\mathbf{u}$. Hence $T$ is injective.
-::::
+```
</li>
@@ -129,6 +133,15 @@ has, when it is consistent, a unique solution as there are no free variables.
::::::
+```{applet}
+:url: injectivity_and_surjectivity/injsurj-injex-example
+:fig: Images/Fig-InjSurj-InjEx.svg
+:name: Fig:InjSurj:InjExExample
+:class: dark-light
+
+TODO: with call to action
+```
+
If we perform two actions which can both be undone, then we should be able to undo the combination of those two actions. That this intuitive rule really does hold is essentially the content of {prf:ref}`Prop:InjSurj:InjafterInjisInj`.
::::::{prf:proposition}
@@ -138,7 +151,9 @@ If $S:\mathbb{R}^{l}\to\mathbb{R}^{m}$ and $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ a
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:InjafterInjisInj`
+:class: myproof
+
Take $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $\mathbb{R}^{l}$ such that $(T\circ S)(\mathbf{v}_{1})=(T\circ S)(\mathbf{v}_{2})$, i.e. $T(S(\mathbf{v}_{1}))=T(S(\mathbf{v}_{2}))$. Since $T$ is injective, we must have $S(\mathbf{v}_{1})=S(\mathbf{v}_{2})$. As $S$ is also injective, this in turn implies $\mathbf{v}_{1}=\mathbf{v}_{2}$ which establishes injectivity of $T\circ S$.
::::::
@@ -164,6 +179,15 @@ Note that $S$ is injective by {prf:ref}`Ex:InjSurj:InjEx` [iii.](#Item:InjSurj:I
::::::
+```{applet}
+:url: injectivity_and_surjectivity/injsurj-injex-example2
+:fig: Images/Fig-InjSurj-InjEx.svg
+:name: Fig:InjSurj:InjExExample
+:class: dark-light
+
+TODO: with call to action
+```
+
::::::{prf:proposition}
:label: Prop:InjSurj:CompInjFirstInj
@@ -171,7 +195,9 @@ If $S:\mathbb{R}^{l}\to\mathbb{R}^{m}$ and $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ a
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:CompInjFirstInj`
+:class: myproof
+
Suppose $T\circ S$ is injective but $S$ is not. Then there are $\mathbf{v}_{1}\neq\mathbf{v}_{2}$ in $\mathbb{R}^{l}$ such that $S(\mathbf{v}_{1})=S(\mathbf{v}_{2})$. But then $T(S(\mathbf{v}_{1}))=T(S(\mathbf{v}_{2}))$ which contradicts the assumption that $T\circ S$ is injective.
::::::
@@ -210,7 +236,8 @@ Prove {prf:ref}`Prop:InjSurj:InjChars`.
::::
-:::{dropdown} Solution to {numref}`Exc:InjSurj:InjChars` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:InjSurj:InjChars`
+:class: solution, dropdown
Assume $T$ is injective and $A\vect{x}=\vect{b}$ has solutions $\vect{x}_{1}$ and $\vect{x}_{2}$. Then $T(\vect{x}_{1})=T(\vect{x}_{2})$, so $\vect{x}_{1}=\vect{x}_{2}$ by injectivity of $T$. Similarly, if we assume that $A\vect{x}=\vect{b}$ has at most one solution, then $T(\vect{x}_{1})=T(\vect{x}_{2})$ would imply $\vect{x}_{1}=\vect{x}_{2}$, hence $T$ is injective.
@@ -219,21 +246,27 @@ If $A$ has a column without pivot, then there is a free variable. Consequently,
:::
::::::{prf:corollary}
+:label: Col:InjSurj:mgreatern
If $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ is injective, then $m\leq n$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Col:InjSurj:mgreatern`
+:class: myproof
+
If $T$ is injective, then its standard matrix $A$ has a pivot in every column. Consequently, the number of columns of $A$, which is $m$, must be smaller than or equal to the number of rows of $A$, which is $n$.
::::::
::::::{prf:proposition}
+:label: Col:InjSurj:viszero
+
A linear transformation $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ is injective if and only if $T(\mathbf{v})=\mathbf{0}$ implies $\mathbf{v}=\mathbf{0}$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Col:InjSurj:viszero`
+:class: myproof
Since $T(\mathbf{0})=\mathbf{0}$ for any linear transformation $T$, injectivity of $T$ implies that $T(\mathbf{v})=\mathbf{0}$ only occurs when $\mathbf{v}=\mathbf{0}$.
@@ -283,11 +316,14 @@ $$
so the system $A\mathbf{x}=\mathbf{u}$ can only be solved if $a_{1}+a_{2}=0$. This transformation is therefore not surjective.
-::::{figure} Images/Fig-InjSurj-NonSurjEx.svg
+```{applet}
+:url: injectivity_and_surjectivity/injsurj-nonsurjex
+:fig: Images/Fig-InjSurj-NonSurjEx.svg
:name: Fig:InjSurj:NonSurjEx
+:class: dark-light
The transformation $T$ from {prf:ref}`Ex:InjSurj:SurjEx` [i.](#Item:InjSurj:NonSurjEx) . The green line is the collection of all $\mathbf{b}$ for which the system $A\mathbf{x}=\mathbf{b}$ is consistent where $A$ is the standard matrix of $T$. Since this is not all of $\mathbb{R}^{2}$, $T$ is not surjective.
-::::
+```
</li>
<li id="Item:InjSurj:SurjEx">
@@ -312,11 +348,14 @@ is surjective.
</li>
</ol>
-::::{figure} Images/Fig-InjSurj-InjEx.svg
+```{applet}
+:url: injectivity_and_surjectivity/injsurj-injex
+:fig: Images/Fig-InjSurj-InjEx.svg
:name: Fig:InjSurj:SurjEx
+:class: dark-light
The transformation $T$ from {prf:ref}`Ex:InjSurj:SurjEx` [ii.](#Item:InjSurj:SurjEx) acting on some selected vectors. Note that this transformation scales vectors and rotates them. For any vector $\mathbf{u}$ on the right, you can find a vector $\mathbf{v}$ on the left such that $T(\mathbf{v})=\mathbf{u}$. Hence $T$ is surjective.
-::::
+```
::::::
@@ -327,7 +366,9 @@ If $S:\mathbb{R}^{l}\to\mathbb{R}^{m}$ and $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ a
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:SurjafterSurjisSurj`
+:class: myproof
+
Take $\mathbf{u}$ in $\mathbb{R}^{n}$. As $T$ is surjective, there is a $\mathbf{v}\in\mathbb{R}^{m}$ with $T(\mathbf{v})=\mathbf{u}$. Since $S$ is surjective, there is a $\mathbf{w}$ in $\mathbb{R}^{l}$ with $S(\mathbf{w})=\mathbf{v}$. We find $(T\circ S)(\mathbf{w})=T(\mathbf{v})=\mathbf{u}$, so $T\circ S$ is surjective, as claimed.
::::::
@@ -362,7 +403,8 @@ If $S:\mathbb{R}^{l}\to\mathbb{R}^{m}$ and $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ a
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:CompSurjSecondSurj`
+:class: myproof
Take $\mathbf{u}\in\mathbb{R}^{n}$. Since $T\circ S$ is surjective, there is a $\mathbf{w}$ in $\mathbb{R}^{l}$ with $\mathbf{u}=(T\circ S)(\mathbf{w})=T(S(\mathbf{w}))$. Hence $\mathbf{u}=T(\mathbf{v})$ for $\mathbf{v}=S(\mathbf{w})$, which shows $T$ to be surjective.
@@ -400,20 +442,25 @@ Prove {prf:ref}`Prop:InjSurj:SurjChars`.
::::
-:::{dropdown} Solution to {numref}`Exc:InjSurj:SurjChars` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:InjSurj:SurjChars`
+:class: solution, dropdown
Assume $T$ is surjective and take an arbitrary $\vect{b}$ in $\R^{n}$. Then there is an $\vect{x}$ in $\mathbb{R}^{m}$ such that $T(\vect{x})=\vect{b}$, i.e. $\vect{x}$ is a solution of $A\vect{x}=\vect{b}$. Similarly, if $A\vect{x}=\vect{b}$ has a solution for any $\vect{b}$ in $\R^{n}$, then $\vect{b}=T(\vect{x})$ which establishes surjectivity of $T$.
-If $A$ has a row without pivot, then, for a well-chosen $\vect{b}$, a pivot can appear in the last column of the augmented matrix $[A|\vect{b}]$. This means that $\A\vect{x}=\vect{b}$ has no solutions. Similarly, if $A\vect{x}=\vect{b}$ always has a solution, then $[A|\vect{b}]$ can never have a pivot in the lost column. Consequently, $A$ must have a pivot in every row.
+If $A$ has a row without pivot, then, for a well-chosen $\vect{b}$, a pivot can appear in the last column of the augmented matrix $[A|\vect{b}]$. This means that $A\vect{x}=\vect{b}$ has no solutions. Similarly, if $A\vect{x}=\vect{b}$ always has a solution, then $[A|\vect{b}]$ can never have a pivot in the lost column. Consequently, $A$ must have a pivot in every row.
:::
::::::{prf:corollary}
+:label: Col:InjSurj:mn2
+
If $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ is surjective, then $m\geq n$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Col:InjSurj:mn2`
+:class: myproof
+
If $T$ is surjective, then its standard matrix $A$ has a pivot in every row. Consequently, the number of columns of $A$, which is $m$, must be greater than or equal to the number of rows of $A$, which is $n$.
::::::
@@ -430,11 +477,15 @@ A linear transformation $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ is called **bijectiv
Although we allow $m$ and $n$ to be different in the definition, it turns out that this cannot happen as the following proposition shows.
::::::{prf:proposition}
+:label: Prop:InjSurj:mequaln
+
If a linear transformation $T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ is bijective, then $m=n$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:mequaln`
+:class: myproof
+
Let $A$ be the standard matrix of $T$, which is an $n\times m$ matrix. By {prf:ref}`Prop:InjSurj:InjChars`, the number of pivots in $A$ is $m$. By {prf:ref}`Prop:InjSurj:SurjChars`, the number of pivots in $A$ is $n$. Hence the claim follows.
::::::
@@ -442,6 +493,8 @@ Let $A$ be the standard matrix of $T$, which is an $n\times m$ matrix. By {prf:r
If the domain and codomain are both $\mathbb{R}^{n}$ then something striking happens: injectivity, surjectivity, and bijectivity all become equivalent!
::::::{prf:proposition}
+:label: Prop:InjSurj:Equiv
+
Let $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear transformation. The following are equivalent:
<ol type = "i">
@@ -468,7 +521,9 @@ $T$ is bijective.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:InjSurj:Equiv`
+:class: myproof
+
It suffices to show that $T$ is injective if and only if it is surjective. Assume $T$ is injective. Then the standard matrix $A$ of $T$ has a pivot in each column. But as the number of columns and the number of rows are the same, there must be a pivot in each row, showing surjectivity of $T$.
Similarly, if $T$ is surjective, then the standard matrix $A$ of $T$ has a pivot in each column. Again, the number of columns is the same as the number of rows, so there is also a pivot in every row and we conclude that $T$ must be injective.
@@ -507,7 +562,8 @@ for any vector $\vect{v}$ in $\R^{n}$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:InjSurj:BijIsInv`
+:class: myproof
By {prf:ref}`Thm:MatrixInv:InvertibilityCharacterizations`, we know that $A$ is invertible if and only if the system $\vect{x}=\vect{b}$ has a unique solution for each $\vect{b}$ in $\R^{n}$. The existence of a solution is equivalent to surjectivity of $T$ and its uniqueness is equivalent to injectivity of $T$. This establishes the equivalence of [i.](#It:InjSurj:TBij) and [ii.](#It:InjSurj:AInv)
@@ -522,6 +578,7 @@ Then $TS(\vect{v})=AB\vect{v}=\vect{v}$ for any $\vect{v}$ in $\R^{n}$. The only
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a5fcbee0-9af1-4596-b3d4-05dcdf7640b3?id=92235
:label: grasple_exercise_3_5_1
:dropdown:
@@ -530,6 +587,7 @@ Then $TS(\vect{v})=AB\vect{v}=\vect{v}$ for any $\vect{v}$ in $\R^{n}$. The only
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ba7d83d3-c00a-4d8b-910e-84e6a12b28e4?id=92236
:label: grasple_exercise_3_5_2
:dropdown:
@@ -538,6 +596,7 @@ Then $TS(\vect{v})=AB\vect{v}=\vect{v}$ for any $\vect{v}$ in $\R^{n}$. The only
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a92c3a97-1bf0-47dc-a549-c1130d053e33?id=92237
:label: grasple_exercise_3_5_3
:dropdown:
@@ -546,6 +605,7 @@ Then $TS(\vect{v})=AB\vect{v}=\vect{v}$ for any $\vect{v}$ in $\R^{n}$. The only
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a18a8d11-11f0-4bce-96f9-39946d6543a9?id=92238
:label: grasple_exercise_3_5_4
:dropdown:
diff --git a/Chapter3/LUdecomp.md b/Chapter3/LUdecomp.md
index d169f99..5046035 100644
--- a/Chapter3/LUdecomp.md
+++ b/Chapter3/LUdecomp.md
@@ -1,47 +1,85 @@
<!-- :::{review}
::: -->
-(Sec:MatFactor)=
+(Sec:LUdecomp)=
-# Matrix Factorisation
+# The $LU$ decomposition
-When solving linear systems, it is very convenient to find an echelon form, so you can solve it using backward substitution as shown in {prf:ref}`Ex:LinSystems:I`.
+As we have seen, one way to solve a linear system is to row reduce it to echelon form and then use back substitution. See for instance {prf:ref}`Ex:LinSystems:I`.
+
+
+::::::{prf:example}
+:label: Ex:LUdecomp:FirstLU
Consider the system $A\mathbf{x} = \mathbf{b}$ where
$$
A =
\begin{bmatrix}
-1 & 1 & 1 \\
-0 & 2 & 6 \\
--1 & -1 & 3
+1 & 1 & -1 \\
+2 & 4 & -3 \\
+1 & -1 & -3
\end{bmatrix},
\qquad
\mathbf{b} =
\begin{bmatrix}
-3 \\ 2 \\1
+2 \\ 1 \\8
\end{bmatrix}.
$$
-In this section we will learn how to solve an $m\times n$ linear system $A\mathbf{x}=\mathbf{b}$ by decomposing (or factorising) a matrix into a product of two matrices. In the previous example we can express $A$ as
+::::::
+
+In this section we will learn how to solve an $m\times n$ linear system $A\mathbf{x}=\mathbf{b}$ by decomposing (or factorising) a matrix into a product of two 'special' matrices. In the previous example we can express $A$ as
$$
-A =
+A =
\begin{bmatrix}
1 & 0 & 0 \\
-0 & 1 & 0 \\
--1 & 0 & 1
+2 & 1 & 0 \\
+1 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
-1 & 1 & 1 \\
-0 & 2 & 6 \\
-0 & 0 & 4
-\end{bmatrix}.
+1 & 1 & -1 \\
+0 & 2 & -1 \\
+0 & 0 & 3
+\end{bmatrix} = L\,U.
+$$
+
+This is called an $LU$-*decomposition* of the matrix $A$. With this factorisation we can quickly solve the system in two steps, using both backward and forward substitution. In fact, the matrix $U$ is an echelon matrix equivalent to $A$.
+To solve the equation $LU\vect{x} = \vect{b}$ we can introduce the auxiliary vector $\vect{y}$ via
+
+$$
+ \vect{y} = U\vect{x}.
+$$
+
+We then first solve
+
+::::{math}
+:label: Eq:LUDecomp:Lyb
+
+ L\vect{y} = \vect{b}
+
+::::
+
+and after that we find $\mathbf{x}$ by solving
+
+$$
+ U\vect{x} = \tilde{\vect{y}},
+$$
+
+
+for the solution $ \tilde{\vect{y}}$ of system {eq}`Eq:LUDecomp:Lyb`. For this solution $ \tilde{\vect{x}}$ we then have
+
+$$
+ A \tilde{\vect{x}} = L\,U\, \tilde{\vect{x}} = L(U\tilde{\vect{x}})= L\tilde{\vect{y}} = \vect{b}.
$$
-When the factorisation is given, solving the system is quicker since we can solve it using backward and forward substitution. There is no need to look for an echelon form and then apply backward substitution.
-Considering the matrix above and the system
+
+::::::{prf:example}
+:label: Ex:LUdecomp:FirstLUcontinued
+
+The system $A\vect{x} = \vect{b}$ of {prf:ref}`Ex:LUdecomp:FirstLU` can be written as
$$
\begin{bmatrix}
@@ -58,7 +96,7 @@ $$
x_1\\ x_2 \\ x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\ 2 \\1
+2 \\ 1 \\8
\end{bmatrix},
$$
@@ -67,41 +105,49 @@ by setting $\mathbf{y} = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 6 \\ 0 & 0 & 4 \en
$$
\begin{bmatrix}
1 & 0 & 0 \\
-0 & 1 & 0 \\
--1 & 0 & 1
+2 & 1 & 0 \\
+1 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
y_1\\y_2\\y_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\ 2 \\1
-\end{bmatrix}.
+2 \\ 1 \\ 8
+\end{bmatrix}, \quad \text{i.e.,}\,\,
+\left\{\begin{array}{ccccccc}
+ y_1 & & & & &=& 2 \\
+ 2y_1 &+& y_2 & & &=& 1 \\
+ y_1 &-& y_2 &+& y_3 &=& 8.
+ \end{array} \right.
+
$$
With forward substitution we can find the solution:
-$$
-\begin{bmatrix}
+$$
+ y_1 = 2 \quad \Longrightarrow \quad y_2 = -3 \quad \Longrightarrow \quad y_2 = 3 \quad
+ \Longrightarrow \quad \begin{bmatrix}
y_1\\y_2\\y_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\ 2 \\ 4
-\end{bmatrix},
+2 \\ -3 \\ 3
+\end{bmatrix}
$$
+
Then, solving
$$
\begin{bmatrix}
-1 & 1 & 1 \\
-0 & 2 & 6 \\
-0 & 0 & 4
-\end{bmatrix}
+1 & 1 & -1 \\
+0 & 2 & -1 \\
+0 & 0 & -3
+\end{bmatrix}
\begin{bmatrix}
x_1\\ x_2 \\ x_3
\end{bmatrix} =
\begin{bmatrix}
-3 \\ 2 \\ 4
+2 \\ -3 \\ 3
\end{bmatrix},
$$
@@ -110,279 +156,51 @@ we obtain
$$
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
-\end{bmatrix}
+\end{bmatrix}
+=
\begin{bmatrix}
-4 \\ -2 \\ 1
+3\\ -2 \\ -1
\end{bmatrix},
$$
which is the solution to our original system. Since the factorisation was given, we did not have to solve the system from scratch by trying to find an echelon form.
-There are several methods for factorising matrices. The factorisations that we will see in this section use **direct methods**. Direct methods are methods that produce the exact solution in a finite number of steps.
-
-There are special cases where solving linear systems can be done quickly. These cases involve triangular or trapezoidal matrices (we will discuss these matrices in the next section). In general, when the matrix associated with an $m\times n$ linear system is a trapezoidal matrix we can use backward or forward substitution for solving them. Remember the echelon forms from {numref}`Subsec:LinSystems:RowReduction`? Echelon forms are trapezoidal matrices.
-
-The most common factorisation methods make use of this kind of matrices. This is why we will first introduce the idea of a trapezoidal and triangular matrix and then discuss the corresponding factorisation methods and their applications.
-
-## Trapezoidal and Triangular Matrices
-
-At this point one probably figured out that when we find an echelon form for an $m\times n$ matrix, that echelon form is actually a trapezoidal matrix. However, to talk about trapezoidal matrices, first we need to revisit the concept of "main diagonal" of a matrix introduced in {prf:ref}`Def:MatrixOps:MainDiagonal`. We will extend this concept to non-square matrices.
-
-:::{prf:definition} Main diagonal of a matrix
-
-For an $m\times n$ matrix $A$, we call the elements $a_{ii}$ the **diagonal elements** of $A$. The (ordered) set of the diagonal elements is called the **main diagonal** of $A$.
-
-:::
-
-Now we are ready to define the objects that we will use during this section.
-
-::::::{prf:definition} Trapezoidal and Triangular Matrices
-
-<ul>
-<li>
-
-We say that an $m\times n$ matrix $U$ is an **upper trapezoidal matrix** if all the entries below the main diagonal are zero. In other words, an upper trapezoidal matrix that has the form
-
-$$
-U=
-\begin{bmatrix}
-\blacksquare & \ast & \cdots & \cdots & \ast & \cdots & \cdots & \ast \\
-0 & \blacksquare & \ddots & &\vdots & & &\vdots \\
-\vdots & \ddots & \ddots & \ddots & \vdots & & & \vdots\\
-0 & \cdots & 0 & \blacksquare & \ast & \cdots & \cdots & \ast
-\end{bmatrix} \quad \text{for }\,\, m<n,
-$$
-
-$$
-U=
-\begin{bmatrix}
-\blacksquare & \ast & \cdots & \ast \\
-0 & \blacksquare & \ddots &\vdots \\
-\vdots & \ddots & \ddots & \ast\\
-0 & \cdots & 0 & \blacksquare \\
-0 & \cdots & \cdots & 0 \\
-\vdots & & & \vdots \\
-\vdots & & & \vdots \\
-0 & \cdots & \cdots & 0 \\
-\end{bmatrix} \quad \text{for }\,\, m>n,
-$$
-
-where the $\blacksquare$ represent any real number in the main diagonal, and the $\ast$ represent any real number above the main diagonal. If the matrix is a square matrix, then it is an **upper triangular** matrix.
-
-</li>
-<li>
-
-We say that an $m\times n$ matrix $L$ is a **lower trapezoidal matrix** if all the entries above the main diagonal are zero. I.e., a lower trapezoidal matrix that has the form
-
-$$
-L=
-\begin{bmatrix}
-\blacksquare & 0 & \cdots & \cdots & 0 & \cdots & \cdots &0\\
-\ast & \blacksquare & \ddots & &\vdots & & & \vdots\\
-\vdots & \ddots & \ddots & \ddots & \vdots & & & \vdots \\
-\ast & \cdots & \ast & \blacksquare & 0 & \cdots & \cdots & 0
-\end{bmatrix} \quad \text{for }\,\, m<n,
-$$
-
-$$
-L=
-\begin{bmatrix}
-\blacksquare & 0 & \cdots & 0 \\
-\ast & \blacksquare & \ddots &\vdots \\
-\vdots & \ddots & \ddots & 0\\
-\ast & \cdots & \ast & \blacksquare \\
-\ast & \cdots & \cdots & \ast \\
-\vdots & & & \vdots \\
-\vdots & & & \vdots \\
-\ast & \cdots & \cdots & \ast \\
-\end{bmatrix} \quad \text{for }\,\, m>n,
-$$
-
-where the $\blacksquare$ represent any real number in the main diagonal, and the $\ast$ represent any real number below the main diagonal. If the matrix is a square matrix, then it is a **lower triangular** matrix.
-
-</li>
-</ul>
-
-::::::
-
-Observe that, when all $\ast$ are zero and the matrix is square, then the matrix is a diagonal matrix.
-
-::::::{prf:example}
-
-Below there are examples of trapezoidal matrices.
-
-:::{latexlist}
-
-\item The matrix
-
-$$
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & -9 & 0 \\
-0 & 0 & 11
-\end{bmatrix}
-$$
-
-is a $3\times 3$ both upper and lower triangular matrix and, therefore, it is a diagonal matrix.
-
-\item The matrix
-
-$$
-\begin{bmatrix}
-2 & 0 & 0 & 0 \\
-4 & 6 & 0 & 0 \\
-8 & 10 & 12 & 0 \\
-14 & 16 & 18 & 20
-\end{bmatrix}
-$$
-
-is a $4\times 4$ lower triangular matrix.
-
-\item The matrix
-
-$$
-\begin{bmatrix}
-1 & 0 & 0 & 3 \\
-0 & -3 & 0 & 0 \\
-0 & 0 & 2 & 0 \\
-0 & 0 & 0 & -4 \\
-0 & 0 & 0 & 0
-\end{bmatrix}
-$$
-
-is a $5\times 4$ upper trapezoidal matrix.
-
-:::
-
-::::::
-
-Trapezoidal matrices have nice properties with respect to their sum and their product.
-
-::::::{prf:proposition} Properties of trapezoidal matrices
-:label: prop:PropertiesTriangularMatrices
-
-Let $A$ and $B$ be two upper trapezoidal matrices. Then the following properties hold:
-
-:::{latexlist}
-:enumerated: true
-:type: i
-
-\item $A+B$ is an upper trapezoidal matrix (whenever the sum makes sense).
-\label{Item:prop:PropertiesTriangularMatrices_sum}
-
-\item $AB$ is an upper trapezoidal matrix (whenever the product makes sense).
-\label{Item:prop:PropertiesTriangularMatrices_product}
-
-:::
-
::::::
-::::{prf:proposition}
-:label: Prop:LUDecomp:PropertiesTriangularMatricesInverse
-
-If $A$ is an invertible upper triangular matrix, then $A^{-1}$ is upper triangular matrix.
-
-::::
-
-The proof of {prf:ref}`Prop:LUDecomp:PropertiesTriangularMatricesInverse` is technical and it involves computations. We leave it as an exercise (see {numref}`Exc:LUdecomp:Theory1`).
-
-So the reader may skip this proof and convince themselves that the properties hold true by looking at the following example.
-
-::::::{prf:example}
-Consider the matrices
-
-$$
-A=
-\begin{bmatrix}
-1 & 2 & 3 \\
-0 & 4 & 5 \\
-0 & 0 & 6
-\end{bmatrix}
-, \qquad
-B =
-\begin{bmatrix}
-1 & 3 & 5 \\
-0 & 7 & 9 \\
-0 & 0 & -11
-\end{bmatrix}
-.
-$$
-
-We can easily see that the first two properties of the proposition hold.
-
-$$
-A+B =
-\begin{bmatrix}
-2 & 5 & 8 \\
-0 & 11 & 14 \\
-0 & 0 & -5
-\end{bmatrix}
-,
-$$
-
-$$
-AB =
-\begin{bmatrix}
-1 & 17 & -10 \\
-0 & 28 & -19 \\
-0 & 0 & -66
-\end{bmatrix}
-.
-$$
-
-For the third one, we have that
-
-$$
-A^{-1} =
-\begin{bmatrix}
-1 & -{1}/{2} & -{1}/{12} \\
-0 & {1}/{4} & -{5}/{24} \\
-0 & 0 & {1}/{6}
-\end{bmatrix}
-,
-$$
-which is upper triangular. It is left to the reader to check that $AA^{-1}=I$.
+Here is one to try for yourself.
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/b9a339dc-721f-4c85-aa77-3455269a3b7a?id=105809
+:label: grasple_exercise_3_6_1T
+:dropdown:
+:description: To solve a system $LU\vect{x} = \vect{b}$.
::::::
-::::::{prf:proof} (of {prf:ref}`prop:PropertiesTriangularMatrices`)
-
-The proof of {itemref}`Item:prop:PropertiesTriangularMatrices_sum` follows from the fact that matrices are added component-wise. Therefore, the entries below the main diagonal of $A+B$ will be zero since both $A$ and $B$ have zero entries below the main diagonal.
-To prove {itemref}`Item:prop:PropertiesTriangularMatrices_product` we fix $j < n$ and compute $AB_{(j+k)j}$ for $k=1,2,\dots, m-j$. In other words, we compute the entries below the main diagonal in column $j$:
+There are several methods for factorising matrices. The factorisations that we will see in this section use **direct methods**. Direct methods are methods that produce the exact factorisations in a finite number of steps.
-$$
-AB_{(j+k)j} = \sum_{l=1}^{n} a_{(j+k)l}b_{lj}.
-$$
+%There are special cases where solving linear systems can be done quickly. These cases involve %triangular or trapezoidal matrices (we will discuss these matrices in the next section). In general, %when the matrix associated with an $m\times n$ linear system is a trapezoidal matrix we can use %backward or forward substitution for solving them. Remember the echelon forms from {numref}%`Subsec:LinSystems:RowReduction`? Echelon forms are trapezoidal matrices.
-If we define $A = \begin{bmatrix}\mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n \end{bmatrix}$ an $m\times n$ upper trapezoidal matrix, and $B=\begin{bmatrix}\mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_p \end{bmatrix}$ and $n\times p$ upper trapezoidal matrix, then we can understand the previous sum as the the result of the dot product of the vectors corresponding to $\mathbf{a}_{j+k}^T$ and $\mathbf{b}_j$.
+%The most common factorisation methods make use of this kind of matrices. This is why we will first %introduce the idea of a trapezoidal and triangular matrix and then discuss the corresponding %factorisation methods and their applications.
-We observe that for $l\le j$ the components $a_{(j+k)l}$ are zero, and for $l>j$ the components $b_{lj}0$ are zero. Therefore,
-
-$$
-\begin{align*}
-AB_{(j+k)j} &= \sum_{l=1}^{n} a_{(j+k)l}b_{lj} \\
-&= \sum_{l=1}^{j} 0 b_{lj} + \sum_{l=j+1}^n a_{(j+k)j} 0 \\
-&= 0.
-\end{align*}
-$$
+In the next subsection we will address the questions of whether an $LU$ decomposition always exists and if so, how to construct it. With an extra condition on $L$ the decomposition, if it exists, will appear to be unique. In the case where an $LU$ decomposition does not exist we can instead consider the slightly more general $PLU$ decomposition. In the remainder of the section we will consider the generalization to non-square matrices and we will analyze to which extent the $(P)LU$ decomposition gives an efficiency boost.
-::::::
-## LU Decomposition
+## $LU$ decomposition of a square matrix
-As mentioned at the begining of this section, one of the ways to solve a linear system is to factorise the matrix of coefficients by "splitting" it into a product of two trapezoidal matrices. For the following factorisation, we will name these matrices $L$ and $U$, and they will have the property that $L$ is lower triangular and $U$ is upper trapezoidal. As you probably have guessed, we call this factorisation an $LU$ decomposition.
::::::{prf:definition}
-Let $A$ be an $m\times n$ matrix. We call a $LU$
-decomposition of $A$ all decompositions of the type
+:label: Def:LUdecomp:DefinitionLU
+
+Let $A$ be an $n\times n$ matrix. An **$LU$ decomposition** of $A$ is a factorization of the type
$$
A=LU
$$
-with $L$ being an $m\times m$ lower triangular matrix and $U$ an $m\times n$ upper trapezoidal matrix, both with real coefficients with the form
+where $L$ is an $n\times n$ lower triangular matrix with $1$s on the diagonal, and $U$ an $n\times n$ upper triangular matrix. So,
$$
L=
@@ -391,46 +209,32 @@ L=
l_{21} & 1 \\
l_{31} & l_{32} & \ddots \\
\vdots & \vdots & \ddots & \ddots\\
-l_{m1} & l_{m2} & \cdots & l_{m(m-1)} & 1
+l_{n1} & l_{n2} & \cdots & l_{n(n-1)} & 1
\end{bmatrix}
,
\quad
U=
\begin{bmatrix}
u_{11} & u_{12} & u_{13} & \cdots &u_{1n} \\
-&
-u_{22}
-&
-u_{23} & \cdots & u_{2n} \\
-& & u_{33} & \cdots & u_{3n} \\
-& & &
-\ddots
-&
-\vdots \\
-& & & & u_{mn}
-\end{bmatrix}, \quad m<n
-,
+ & u_{22} & u_{23} & \cdots & u_{2n} \\
+ & & u_{33} & \cdots & u_{3n} \\
+ & & &
+\ddots & \vdots \\
+& & & & u_{nn}
+\end{bmatrix}.
$$
-or,
+For convenience we have used blanks for zeros.
+::::::
-$$
-U = \begin{bmatrix}
-u_{11} & \cdots & \cdots & u_{1n}\\
- & \ddots & & \vdots \\
- & & \ddots & \vdots \\
- & & & u_{nn} \\
- & \\
- & \\
- &
-\end{bmatrix},\quad m>n.
-$$
+The next proposition captures the main interest of the $LU$ decomposition, as is already illustrated in
+{prf:ref}`Ex:LUdecomp:FirstLUcontinued`
-To not clutter and to emphasise the triangular/trapezoidal matrices, we use empty spaces instead of zeros.
-::::::
+::::::{prf:proposition}
+:label: Prop:LUdecomp:Usefulness
-Suppose that $A=LU$ so that a linear system of equations $A\mathbf{x}=\mathbf{b}$ can be written as $LU\mathbf{x}=\mathbf{b}$. Then, by setting $\mathbf{y} = U\mathbf{x}$, we can solve the linear system in two steps:
+Suppose that $A=LU$, so that a linear system of equations $A\mathbf{x}=\mathbf{b}$ can be written as $LU\mathbf{x}=\mathbf{b}$. Then, by setting $\mathbf{y} = U\mathbf{x}$, we can solve the linear system in two steps.
:::{latexlist}
:enumerated: true
@@ -441,13 +245,23 @@ Suppose that $A=LU$ so that a linear system of equations $A\mathbf{x}=\mathbf{b}
:::
-At this point the reader may think that such a decomposition involves a lot of work. However, it takes (roughly) the same number of operations as for finding an echelon form (see {numref}`Subsec:LinSystems:RowReduction`). In addition, this method has the advantage that once the decomposition of a matrix $A$ has been found, then it is faster to solve multiple systems involving the matrix $A$. And why is that? Well, if one thinks about it, once we have an $LU$ decomposition of $A$ then we just need to solve linear systems that involve trapezoidal/triangular matrices, which are "faster" to solve than solving $A\mathbf{x}=\mathbf{b}$. We will formalise this concept below.
+The solution of the second system is then a solution of the system $A\mathbf{x}=\mathbf{b}$.
+
+::::::
+
+The next proposition tells us which matrices do have an $LU$ decomposition.
+More importantly, it shows how such a decomposition can be found, which is in fact a row reduction that keeps track of the row operations involved.
-The $LU$ decomposition is a particular case of the more general algorithm that we find in {numref}`Sec:PLUDecomp`
+::::::{prf:proposition}
+:label: Prop:LUdecomp:Existence
-The idea behind an $LU$ decomposition is to find an echelon form of the matrix of coefficients without exchanging rows, while keeping track of those changes. Let's see how to find an $LU$ decomposition with a complete example using an square matrix of coefficients. Along the rest of this subsection, we will assume that we perform **no row scaling when preforming row operations**.
+A matrix $A$ can be written as $A = LU$, with $L$ and $U$ as described in {prf:ref}`Def:LUdecomp:DefinitionLU` if and only it can be row reduced to an echelon matrix with only additions of multiples of rows to rows below it. We will call this a *top-down row reduction*.
+::::::
+
+Instead of a giving a formal proof, we will illustrate matters first with an example.
::::::{prf:example}
+:label: Ex:LUdecomp:SecondLU
We consider the matrix
$$
@@ -460,7 +274,7 @@ A=
.
$$
-To find an $LU$ decomposition we will find an echelon form performing row operations without exchanging rows. In addition, we will keep track of the changes we perform.
+To find an $LU$ decomposition we will works towards an echelon form in the top-down direction. In addition, we will keep track of the actions we perform.
For the first step,
$$
@@ -469,30 +283,38 @@ $$
[R_1] \\
{[R_2-2/3R_1]} \\
{[R_3-1/3R_1]} \\
-\end{array}\sim
+\end{array}
+\quad\sim\quad
\begin{bmatrix}
3 & 1 & -2 \\
0 & {10}/{3} & {7}/{3} \\
0 & {5}/{3} & {5}/{3}
-\end{bmatrix}
+\end{bmatrix}.
$$
-The numbers $2/3$ and $1/3$ are called multipliers.
-Observe that the row operations can be understood as a product of matrices. Let's call
+The numbers $2/3$ and $1/3$ are called *multipliers*.
+
+A second row reduction step, involving the multiplier $1/2$, leads to an echelon/upper triangular matrix.
$$
-F^{(1)} =
+
+\left[\begin{array}{rrr}3 & 1 & -2\\0 &{10}/{3}&{7}/{3}\\0 &{5}/{3}&{5}/{3}\end{array} \right] \begin{array}{l}
+[R_1] \\
+{[R_2]} \\
+{[R_3-1/2R_2]} \\
+\end{array}\sim
\begin{bmatrix}
-1 & 0 & 0 \\
--2/3 & 1 & 0 \\
--1/3 & 0 & 1
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & 0 & {1}/{2}
\end{bmatrix}
+.
$$
-Then,
+We can effectuate these row operations by matrix multiplications.
$$
-F^{(1)} A =
+ F_1A =
\begin{bmatrix}
1 & 0 & 0 \\
-2/3 & 1 & 0 \\
@@ -509,762 +331,1173 @@ F^{(1)} A =
0 & {10}/{3} & {7}/{3} \\
0 & {5}/{3} & {5}/{3}
\end{bmatrix}
+= A_2.
$$
-Now,
+In the same way the second row reduction step can be described by
-$$
-\left[\begin{array}{rrr}3 & 1 & -2\\0 &{10}/{3}&{7}/{3}\\0 &{5}/{3}&{5}/{3}\end{array} \right] \begin{array}{l}
-[R_1] \\
-{[R_2]} \\
-{[R_3-1/2R_2]} \\
-\end{array}\sim
+$$
+ F_2A_2 =
+\begin{bmatrix}
+1 & 0 & 0 \\
+0 & 1 & 0 \\
+0 & -1/2 & 1
+\end{bmatrix}
\begin{bmatrix}
3 & 1 & -2 \\
0 & {10}/{3} & {7}/{3} \\
-0 & 0 & {1}/{2}
-\end{bmatrix}
-.
-$$
-
-Since an echelon form of an $n\times n$ matrix is an upper triangular matrix, we define
-
-$$
-U=
+0 & {5}/{3} & {5}/{3}
+\end{bmatrix} =
\begin{bmatrix}
3 & 1 & -2 \\
0 & {10}/{3} & {7}/{3} \\
0 & 0 & {1}/{2}
\end{bmatrix}
-.
+= A_3.
+
$$
-To find $L$, we observe that the last row operation that we performed can be written as a product of matrices:
+The echelon matrix $A_3$ can act as our upper triangular matrix $U$, and the above computations
+can be summarized as
-\begin{align*}
-U &= F^{(2)}F^{(1)}A =
-\begin{bmatrix}
+$$
+ U = A_3 = F_2F_1A = FA = \begin{bmatrix}
1 & 0 & 0 \\
-0 & 1 & 0 \\
+-2/3 & 1 & 0 \\
0 & -1/2 & 1
\end{bmatrix}
-\begin{bmatrix}
-1 & 0 & 0 \\
--2/3 & 1 & 0 \\
--1/3 & 0 & 1
-\end{bmatrix}
-\begin{bmatrix}
-3 & 1 & -2 \\
-2 & 4 & 1 \\
-1 & 2 & 1
-\end{bmatrix}
-\\
-&=
-\begin{bmatrix}
-1 & 0 & 0 \\
--\frac{2}{3} & 1 & 0 \\
-0 & -\frac{1}{2} & 1
-\end{bmatrix}
-A\\
-&= FA.
-\end{align*}
+A,
+$$
+where $F = F_2F_1$.
-Since $F=F^{(2)}F^{(1)}$ is non-singular and lower triangular, we can compute its inverse,
+We conclude that
$$
-F^{-1} =
-\begin{bmatrix}
+ A = F^{-1} U = \begin{bmatrix}
1 & 0 & 0 \\
-{2}/{3} & 1 & 0 \\
-{1}/{3} & {1}/{2} & 1
+2/3 & 1 & 0 \\
+1/3 & 1/2 & 1
+\end{bmatrix}
+ \begin{bmatrix}
+ 3 & 1 & -2 \\
+ 0 & {10}/{3} & {7}/{3} \\
+ 0 & 0 & {1}/{2}
\end{bmatrix}
-,
+
$$
-which is also lower triangular (we knew it from {prf:ref}`prop:PropertiesTriangularMatrices`). By writing $F^{-1}U=A$, we can take $L=F^{-1}$.
+which is indeed a product of a lower triangular matrix $L$ (with 1's on it diagonal) and an upper triangular matrix $U$.
::::::
-::::::{prf:remark}
-Pay attention!
+The above procedure shows how in principle an $LU$ decomposition can be computed. However, writing down the explicit elementary like matrices $F_i$, computing their product $F$, and finding the inverse of $F$ is unnecessary. Look at the matrix $L = F^{-1}$ we found! The entries below the diagonal are exactly the numbers $2/3$, $1/3$ and $1/2$ we called the *multipliers*.
+The following algorithm describes this 'shortcut' to find an $LU$ decomposition.
-In practice, we do not need to compute the product of the matrices $F^{(k)}$, and we do not need to find any inverse. By looking at the previous example, one can see that the matrix $L$ has ones in its diagonal and the entries below the main diagonal have, in each column, the corresponding multipliers. This is true in the general case (as long as one does not need to exchange rows). We will see all the details for the general case in {prf:ref}`thm:existence_and_uniqueness_LU` below.
+::::::{prf:algorithm}
+:label: Alg:LUdecomp:LUalgorithm
-::::::
+Suppose the $n\times n$ matrix $A$ can be row reduced top-down to the echelon matrix $U$. If the numbers $m_{jk}$ denote the multiples of the $k$th row that are subtracted from the rows below it in the $k$th step (so $1 \leq k < j \leq n$), then
-The following example shows how we implement the $LU$ decomposition in practice. Remember that no row exchanges are needed.
+$$
+ A = LU, \quad \text{for} \,\,
+ L = \begin{bmatrix}
+1\\
+m_{21} & 1 \\
+m_{31} & m_{32} & \ddots \\
+\vdots & \vdots & \ddots & \ddots\\
+m_{n1} & m_{n2} & \cdots & m_{n,n-1} & 1
+\end{bmatrix}.
+$$
-::::::{prf:example}
-Find an $LU$ decomposition of the matrix
-$$
-A=
-\begin{bmatrix}
-5 & 5 & 5 \\
-3 & 4 & 1 \\
-2 & 1 & 3
-\end{bmatrix} .
-$$
+::::::
+
+
+Another look at {prf:ref}`Ex:LUdecomp:SecondLU` explains why the detour via the matrices $F_i$ and the inverse of their product can be skipped, as is expressed in the above algorithm.
+
+
+::::::{prf:example}
+:label: Ex:LUdecomp:SecondLUSecondLook
+
+For the matrix $A= \begin{bmatrix}
+ 3 & 1 & -2 \\
+ 2 & 4 & 1 \\
+ 1 & 2 & 1
+ \end{bmatrix}$.
+
+
+The 'trick' is not to work with the matrices $F_1$ and $F_2$, but with their inverses.
-We will proceed using row operations without exchanging any rows. At the same time, we will construct the matrix $L$ using the multipliers. Our matrix $L$ will have the form
+The first row reduction step
$$
-L=
+ F_1A =
\begin{bmatrix}
1 & 0 & 0 \\
-\ast & 1 & 0 \\
-\ast & \ast & 1
-\end{bmatrix} .
+-2/3 & 1 & 0 \\
+-1/3 & 0 & 1
+\end{bmatrix}
+\begin{bmatrix}
+3 & 1 & -2 \\
+2 & 4 & 1 \\
+1 & 2 & 1
+\end{bmatrix}
+=
+\begin{bmatrix}
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & {5}/{3} & {5}/{3}
+\end{bmatrix}
+= A_2.
$$
-The changes we make are the following:
+can be rewritten as
-\begin{align*}
-\left[\begin{array}{rrr}5 & 5 & 5\\3 & 4 & 1\\2 & 1 & 3\end{array} \right] \begin{array}{l}
-[R_1] \\
-{[R_2-\mathbf{3/5}R_1]} \\
-{[R_3-\mathbf{2/5}R_1]} \\
-\end{array} &\sim
+$$
+ A = F_1^{-1}A_2 = L_1A_2 =
+\begin{bmatrix}
+1 & 0 & 0 \\
+2/3 & 1 & 0 \\
+1/3 & 0 & 1
+\end{bmatrix}
\begin{bmatrix}
-5 & 5 & 5 \\
-0 & 1 & -2 \\
-0 & -1 & 1
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & {5}/{3} & {5}/{3}
\end{bmatrix}
-\\
-&\longrightarrow
-L =
+$$
+
+
+Likewise the second row reduction step
+
+$$
+ F_2A_2 =
\begin{bmatrix}
1 & 0 & 0 \\
-\mathbf{3/5} & 1 & 0 \\
-\mathbf{2/5} & \ast & 1
-\end{bmatrix} , \\
-\left[\begin{array}{rrr}5 & 5 & 5\\0 & 1 & -2\\0 & -1 & 1\end{array} \right] \begin{array}{l}
-[R_1] \\
-{[R_2]} \\
-{[R_3-(\mathbf{-1})R_2]} \\
-\end{array}
-&\sim
+0 & 1 & 0 \\
+0 & -1/2 & 1
+\end{bmatrix}
\begin{bmatrix}
-5 & 5 & 5 \\
-0 & 1 & -2 \\
-0 & 0 & -1
-\end{bmatrix}\\
-&\longrightarrow
-L =
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & {5}/{3} & {5}/{3}
+\end{bmatrix} =
\begin{bmatrix}
-1 & 0 & 0 \\
-3/5 & 1 & 0 \\
-2/5 & \mathbf{-1} & 1
-\end{bmatrix} .
-\end{align*}
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & 0 & {1}/{2}
+\end{bmatrix}
+$$
+
+can be can be represented as
-Therefore, an $LU$ decomposition of $A$ is
$$
-L=
+A_2 = F_2^{-1}A_3 = L_2U =
\begin{bmatrix}
1 & 0 & 0 \\
-{3}/{5} & 1 & 0 \\
-{2}/{5} & -1 & 1
+0 & 1 & 0 \\
+0 & 1/2 & 1
\end{bmatrix}
-,\qquad
-U=
\begin{bmatrix}
-5 & 5 & 5 \\
-0 & 1 & -2 \\
-0 & 0 & -1
-\end{bmatrix}
-.
+3 & 1 & -2 \\
+0 & {10}/{3} & {7}/{3} \\
+0 & {5}/{3} & {5}/{3}
+\end{bmatrix}.
+
$$
-::::::
+Combining the two equations gives
-At this point, there are important questions that we still need to address: when does a matrix have an $LU$ decomposition? Can a matrix have more than one $LU$ decomposition? It would be convenient if there is an $LU$ decomposition, that such decomposition is unique.
+$$
+ A = L_2L_1U = LU =
+ \begin{bmatrix}
+ 1 & 0 & 0 \\
+ 2/3 & 1 & 0 \\
+ 1/3 & 1/2 & 1
+ \end{bmatrix}
+ \begin{bmatrix}
+ 3 & 1 & -2 \\
+ 0 & {10}/{3} & {7}/{3} \\
+ 0 & 0 & {1}/{2}
+ \end{bmatrix}.
+$$
-The case when there is a unique $LU$ decomposition is very special, as the following result indicates.
+We see the multipliers nicely fall into place!
-::::::{prf:theorem}
-:label: thm:existence_and_uniqueness_LU
+::::::
-Suppose that $A$ is an
-$m\times n$ matrix with real entries. Then
+Here is an example where we apply the algorithm applied without further ado to a $4 \times 4$ matrix.
+
+::::::{prf:example}
+:label: Ex:LUdecomp:LUviaAlgorithm
+We use the $LU$-algorithm to the matrix $A = \begin{bmatrix}
+ 2 & 1 & -2 & 3 \\
+ 2 & -3 & -4 & 7 \\
+ -4 & 0 & 2 & -5 \\
+ -6 & -1 & 8 & -8
+ \end{bmatrix}.$
+
+We row reduce the matrix $A$ top-down, diligently registering the multipliers.
+Recall that the multiplier $m_{jk}$ is defined as the multiple of row $k$ that is *subtracted* from row $j$ in the $k$-th step of the reduction process.
+
+Here he goes:
+
+$$
+\left[\begin{array}{rrrr}
+ 2 & 1 & -2 & 3 \\
+ 2 & -3 & -4 & 7 \\
+ -4 & 0 & 2 & -5 \\
+ 6 & 1 & -8 & 8\end{array} \right] \begin{array}{l}
+[R_1] \\
+{[R_2-\class{red}{1}R_1]} \\
+{[R_3-\class{red}{(-2)}R_1]} \\
+{[R_4-\class{red}{3}R_1]} \\
+\end{array}
+\sim
+\left[\begin{array}{rrrr}
+ 2 & 1 & -2 & 3 \\
+ 0 & -4 & -2 & 4 \\
+ 0 & 2 & 2 & 1 \\
+ 0 & -2 & -2 &-1\end{array} \right] \begin{array}{l}
+[R_1] \\
+[R_2] \\
+{[R_3-\class{blue}{(-1/2)}R_2]} \\
+{[R_4-\class{blue}{1/2}R_2]} \\
+\end{array}
+$$
-<ol type = "i">
-<li id="Item:thm:existence_and_uniqueness_LU:existence">
+$$
+\sim
+ \left[\begin{array}{rrrr}
+ 2 & 1 & -2 & 3 \\
+ 0 & -4 & -2 & 4 \\
+ 0 & 0 & -3 & 3 \\
+ 0 & 0 & -1 & -3
+ \end{array} \right] \begin{array}{l}
+[R_1] \\
+[R_2] \\
+{[R_3]} \\
+{[R_4-\class{green}{1/3}R_2]} \\
+\end{array}
+\,\,=\,\,
+ \left[\begin{array}{rrrr}
+ 2 & 1 & -2 & 3 \\
+ 0 & -4 & -2 & 4 \\
+ 0 & 0 & -3 & 3 \\
+ 0 & 0 & 0 & -4
+ \end{array} \right]
+\,\,\sim\,\, U.
+$$
-If we can find an echelon form for the matrix $A$ without exchanging rows, then there exists an $LU$ decomposition of $A$, and it has the form
+Putting every multiplier in its right place gives
$$
-L=
-\begin{bmatrix}
-1 \\
-m_{21} & 1 \\
-\vdots & m_{32} &\ddots \\
-\vdots & \vdots &\ddots & \ddots & \\
-m_{m1} & m_{m2} & \cdots & m_{m(m-1)} & 1
-\end{bmatrix}
-, \quad
-U=
-\begin{bmatrix}
-a^{(0)}_{11} & a^{(0)}_{12} & \cdots & a^{(0)}_{1n}\\
-& a^{(1)}_{22} & \cdots & a^{(1)}_{2n} \\
-& & \ddots & \vdots \\
-& & &a^{(n-1)}_{nn} \\
-\\\\
-\end{bmatrix}\quad (m>n),
+ L = \left[\begin{array}{rrrr}
+ 1 & 0 & 0 & 0 \\
+ \class{red}{1} & 1 & 0 & 0 \\
+ \class{red}{-2} & \class{blue}{-1/2} & 1 & 0 \\
+ \class{red}{3} & \class{blue}{1/2} & \class{green}{1/3} & 1
+ \end{array} \right].
$$
-or
+::::::
+
+
+::::::{prf:example}
+:label: Ex:LUdecomp:NoLU
+
+For the matrix $A= \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 4 & 8 & 6 \\
+ 2 & 5 & 4
+ \end{bmatrix}$
+the procedure breaks down at the second step.
+
+$$
+ \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 4 & 8 & 6 \\
+ 2 & 5 & 7
+ \end{bmatrix}
+ \begin{array}{l}
+ [R_1] \\
+ {[R_2-4R_1]} \\
+ {[R_3-2R_1]}
+ \end{array} \quad \sim \quad
+ \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 0 & 0 & 2 \\
+ 0 & 1 & 5
+ \end{bmatrix}.
+$$
+The next step towards an echelon matrix needs a row swap. But then the lower triangular structure of $L$ will be broken. In {numref}`Subsection %s <Subsec:LUdecomp:PLUdecomp>` we will study what we can do in such a situation.
+
+::::::
+
+People that are not satisfied with two examples to show
+that {prf:ref}`Alg:LUdecomp:LUalgorithm` works can have a look at the following (informal) proof of
+{prf:ref}`Prop:LUdecomp:Existence`. It will also prove the statement that the existence of a $LU$ decomposition of a matrix $A$ implies that the matrix $A$ can be row reduced top-down to a matrix in echelon form.
+
+
+:::{admonition} Proof of {prf:ref}`Prop:LUdecomp:Existence`
+:class: myproof, dropdown
+
+Suppose $A$ is an $n\times n$ matrix that can be row reduced to an upper triangular (= echelon) matrix $U$. We can row reduce $A$ from top to bottom, where we use the same form as in {prf:ref}`Ex:LUdecomp:SecondLUSecondLook`.
+For instance the first two steps are
+
$$
-\begin{bmatrix}
-a^{(0)}_{11} & a^{(0)}_{12} & \cdots & a^{(0)}_{1m} &\cdots &a_{1n}^{(0)}\\
-& a^{(1)}_{22} & \cdots & a^{(1)}_{2m} & \cdots & a_{2n}^{(1)}\\
-& & \ddots & \vdots & & \vdots\\
-& & &a^{(m-1)}_{mm} & \cdots & a_{mn}^{(m-1)}\\
-\end{bmatrix}\quad (m<n),
+ A = \begin{bmatrix}
+ a_{11} & a_{12}& a_{13} & \ldots& a_{1n} \\
+ a_{21} & a_{22}& a_{23} & \ldots& a_{2n} \\
+ a_{31} & a_{32}& a_{33} & \ldots& a_{3n} \\
+ \vdots & \vdots& \vdots& & \vdots \\
+ a_{n1} & a_{n2}& a_{n3}& \ldots& a_{nn}
+ \end{bmatrix} = L_1A_1
+ =
+ \begin{bmatrix}
+ 1 & \\
+ m_{21} & 1 \\
+ m_{31} & 0 & 1 \\
+ \vdots & \vdots& & \ddots \\
+ m_{n1} & 0 & \ldots& & 1
+ \end{bmatrix}A_1
$$
-where the values $m_{ij}$ are the corresponding multiplers and the superscripts in $a_{ij}^{(k)}$ indicate that the value has been obtained after performing $k$ row operations.
+and
-</li>
-<li id="Item:thm:existence_and_uniqueness_LU:uniqueness">
+$$
+ A_1 = L_2A_2 =
+ \begin{bmatrix}
+ 1 & \\
+ 0 & 1 & \\
+ 0 &m_{32} & 1 \\
+ \vdots & \vdots& & \ddots \\
+ 0 & m_{n2} & 0 & \ldots& & 1
+ \end{bmatrix}A_2,
+$$
-If $A$ is an $n\times n$ non-singular matrix, then the $LU$ decomposition is unique.
+where $A_2$ will be of the form
-</li>
-</ol>
+$$
+ A_2 = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} \\
+ 0 & \tilde{a}_{22} & \tilde{a}_{23} & \ldots &\tilde{a}_{2n} \\
+ 0 & 0 & \tilde{a}_{33} & \ldots &\tilde{a}_{3n} \\
+ \vdots & \vdots & \vdots & & \vdots \\
+ 0 & 0 & \tilde{a}_{n3} & \ldots &\tilde{a}_{nn} \\
+ \end{bmatrix}.
+$$
-::::::
+We thus end up with a product
-::::::{prf:proof} Proof of {prf:ref}`thm:existence_and_uniqueness_LU`
+$$
+ A = L_{n-1}L_{n-2}\cdots L_2L_1\,U = L\,U,
+$$
+
+where each matrix $L_k$ is a matrix containing the multipliers of the $k$th step in de proces.
-We start with the proof of <a href="#Item:thm:existence_and_uniqueness_LU:uniqueness">ii. </a>. We will
-proceed by contradiction. Suppose that we have two $LU$ decompositions of $A$. That is, $A= L_1U_1$ and $A=L_2U_2$. Then
+The crucial thing is that in the product $L_{n-1}L_{n-2}\cdots L_2L_1$ in this order the multipliers do not 'interact'. For instance, for $n = 4$, we would get
$$
-L_1U_1=L_2U_2.
+ \begin{array}{l} L =
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ m_{21} & 1 & 0 & 0 \\
+ m_{31} & 0 & 1 & 0 \\
+ m_{41} & 0 & 0 & 1
+ \end{bmatrix}
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & m_{32} & 1 & 0 \\
+ 0 & m_{42} & 0 & 1
+ \end{bmatrix}
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 0 \\
+ 0 & 0 & m_{43} & 1
+ \end{bmatrix} = \\
+ =
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ m_{21} & 1 & 0 & 0 \\
+ m_{31} & 0 & 1 & 0 \\
+ m_{41} & 0 & 0 & 1
+ \end{bmatrix} =
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & m_{32} & 1 & 0 \\
+ 0 & m_{42} & m_{43} & 1
+ \end{bmatrix} =
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\
+ m_{21} & 1 & 0 & 0 \\
+ m_{31} & m_{32} & 1 & 0 \\
+ m_{41} & m_{42} & m_{43} & 1
+ \end{bmatrix}.
+ \end{array}
$$
-Since $A$ is non-singular, $L_{i}$ and $U_{i}$ are non-singular for $i=1,2$. Multiplying both sides of the previous identity by $L^{-1}_{2}$ to the left, and by $U^{-1}_{1}$ to the right we obtain:
+The pattern is clear, we skip the technical details to prove it for $n\times n$ matrices.
-:::{math}
-:label: eq:lowerequalupper
-L*{2}^{-1}L*{1}=U*{2}U^{-1}*{1}.
+To prove the converse, assume that $A$ has an $LU$ decomposition, i.e.,
-:::
-Now we can visualise equation {eq}`eq:lowerequalupper` like this
+$$
+ A = LU =
+ \left[\begin{array}{rrrrr}
+ 1 & & & \\
+ \ell_{11} & 1 & & \\
+ \ell_{21} & \ell_{32} & 1 & \\
+ \vdots & \vdots & \vdots & \ddots & \\
+ \ell_{n1} & \ell_{n2}& \ell_{n3} & \cdots & 1 \\
+ \end{array} \right] \begin{bmatrix}
+u_{11} & u_{12} & u_{13} & \cdots &u_{1n} \\
+ & u_{22} & u_{23} & \cdots & u_{2n} \\
+ & & u_{33} & \cdots & u_{3n} \\
+ & & &
+\ddots & \vdots \\
+& & & & u_{nn}
+\end{bmatrix}.
+$$
+
+By {prf:ref}`Prop:LUdecomp:L-properties` the inverse of $L$ has the same structure as $L$, i.e.,
$$
-\begin{bmatrix}
-1 \\
-\ast & 1 \\
-\vdots & \ast & \ddots \\
-\vdots & \vdots & \ddots &\ddots \\
-\ast & \ast & \cdots & \ast& 1
-\end{bmatrix}
- =
-\begin{bmatrix}
-\blacksquare & \ast & \cdots & \cdots & \ast \\
-& \blacksquare & \ast & \cdots & \ast \\
-& & \ddots & \ddots & \vdots \\
-& & & \ddots & \ast \\
-& & & & \blacksquare
-\end{bmatrix}
-.
+\left[\begin{array}{rrrrr}
+ 1 & & & \\
+ \ell^{\ast}_{11} & 1 & & \\
+ \ell^{\ast}_{21} & \ell^{\ast}_{32} & 1 & \\
+ \vdots & \vdots & \vdots & \ddots & \\
+ \ell^{\ast}_{n1} & \ell^{\ast}_{n2}& \ell^{\ast}_{n3} & \cdots & 1 \\
+ \end{array} \right].
+$$
+
+As in the proof of {prf:ref}`Prop:LUdecomp:Existence` $L^{-1}$ can be factorized as
+
+$$
+ L_2L_3\cdots L_{n} =
+ \left[\begin{array}{rrrrr}
+ 1 & & & \\
+ \ell^{\ast}_{11} & 1 & & \\
+ \ell^{\ast}_{21} & 0 & 1 & \\
+ \vdots & \vdots & \vdots & \ddots & \\
+ \ell^{\ast}_{n1} & 0 & 0 & \cdots & 1 \\
+ \end{array} \right]
+ \left[\begin{array}{rrrrr}
+ 1 & & & \\
+ 0 & 1 & & \\
+ 0 & \ell^{\ast}_{32} & 1 & \\
+ \vdots & \vdots & \vdots & \ddots & \\
+ 0 & \ell^{\ast}_{n2}& 0 & \cdots & 1 \\
+ \end{array} \right]
+ \cdots
+ \left[\begin{array}{rrrrr}
+ 1 & & & \\
+ 0 & 1 & & \\
+ 0 & 0 & 1 & \\
+ \vdots & \vdots & \vdots & \ddots & \\
+ 0 & 0 & \cdots & \ell^{\ast}_{n-1,n} & 1 \\
+ \end{array} \right]
$$
-For this identity to hold true, we need all $\blacksquare$ to be one and all $\ast$ to be zero. In other words,
+Pre-multiplication of a matrix $M$ with one of the matrices $L_k$ amounts to adding multiples of the $k$th row of $M$ to the lower rows. So the product $L_2L_3\cdots L_n$ is a series of top-down operations,
+and
$$
-L^{-1}_{2}L_1 = U_{2}U^{-1}_{1} = I,
+ L_2L_3\cdots L_n\,s A = U
$$
-which leads us to $L^{-1}_2=L^{-1}_1$ so $L_2=L_1$, and
-$U^{-1}_2=U^{-1}_1$ so $U_2=U_1$. This is a contradiction with the statement that $A$ had two different $LU$ decompositions.
+amounts to a top-down row reduction of $A$ to the upper triangular matrix $U$.
+
+::::::
+
+At the end of this section we will analyze whether an $LU$ decomposition if it exists, is unique. To answer that question we need the following properties of lower as well as upper triangular matrices.
+
+
+::::::{prf:proposition}
+:label: Prop:LUdecomp:L-properties
+
+Suppose $A$ and $B$ are lower triangular matrices with $1$s on their diagonals.
+Then the following properties hold.
+
+::::{latexlist}
+:enumerated: true
+:type: i
+
+\item $AB$ is also a lower triangular matrix with $1$s on its diagonal.
+
+\item $A^{-1}$ is also a lower triangular matrix with $1$s on its diagonal.
+
+::::
+
+In both properties 'lower' can be replaced by 'upper'.
+
+
+::::::
+
+
+
+There are several ways to prove {prf:ref}`Prop:LUdecomp:L-properties`.
+The best would be to think of a proof yourself, but you can also have a peek at the exposition below.
+
+::::::{admonition} Proof of {prf:ref}`Prop:LUdecomp:L-properties`
+:class: myproof, dropdown
+
+:::{latexlist}
+:enumerated: true
+:type: i
-To prove <a href="#Item:thm:existence_and_uniqueness_LU:existence">i. </a>, suppose that we can find an echelon form of $A$ without exchanging rows. For simplicity on the notation, we will prove the case of a squared matrix but the reader can easily adapt the proof for the non-suqared case.
+\item
+Suppose $A$ and $B$ are lower triangular matrices with $1$s on their diagonals.
-We can write the $k$-th step as a product of matrices:
+One way to prove that $AB$ also has these properties is to use the column-row expansion of the product.
+(Cf. {numref}`Exc:MatrixOps:ColumnRowExpansion`.) <br>
+Let $\vect{a}_k$ be the $k$th column of $A$, and $\vect{b}^{(k)}$ the $k$th row of $B$.
+Then
$$
-\begin{align*}
-&
-\begin{bmatrix}
-1 \\
-& \ddots \\
-& & 1 \\
-& & -m_{(k+1)k} & 1 \\
-& & \vdots & & \ddots \\
-& & -m_{nk} & & & 1
-\end{bmatrix}
-\begin{bmatrix}
-a^{(0)}_{11} & \cdots & a^{(0)}_{1k} & a^{(0)}_{1(k+1)} & \cdots & a^{(0)}_{1n} \\
-& \ddots & \vdots & \vdots & & \vdots \\
-& & a^{(k-1)}_{kk} & a^{(k-1)}_{k(k+1)} & \cdots & a^{(k-1)}_{kn} \\
-& & a^{(k-1)}_{(k+1)k} & a^{(k-1)}_{(k+1)(k+1)} & \cdots & a^{(k-1)}_{(k+1)n} \\
-& & \vdots & \vdots & & \vdots \\
-& & a^{(k-1)}_{nk} & a^{(k-1)}_{n(k+1)} & \cdots & a^{(k-1)}_{nn}
-\end{bmatrix}
-\\
-=&
-\begin{bmatrix}
-a^{(0)}_{11} & \cdots & a^{(0)}_{1k} & a^{(0)}_{1(k+1)} & \cdots & a^{(0)}_{1n} \\
-& \ddots & \vdots & \vdots & & \vdots \\
-& & a^{(k-1)}_{kk} & a^{(k-1)}_{k(k+1)} & \cdots & a^{(k-1)}_{kn} \\
-& & 0 & a^{(k)}_{(k+1)(k+1)} & \cdots & a^{(k)}_{(k+1)n} \\
-& & \vdots & \vdots & & \vdots \\
-& & 0 & a^{(k)}_{n(k+1)} & \cdots & a^{(k)}_{nn}
-\end{bmatrix}
-.
-\end{align*}
+ \vect{a}_k = \begin{bmatrix}0 \\ \vdots \\ 0 \\ a_{kk} \\ \vdots \\ a_{nk} \end{bmatrix} \quad
+ \text{and} \quad
+ \vect{b}_{(k)} = \begin{bmatrix}b_{k1} & \cdots & b_{kk} & 0 & \cdots & 0 \end{bmatrix},
$$
-We denote the previous product by $A^{(k)}=F^{(k)}A^{(k-1)}$. Since finding an echelon form is an iterative process, we can find $A^{(n)}$ by multiplying $A^{(0)}$ by the corresponding $F^{(k)}$ matrices to the left:
+where moreover $a_{kk} = b_{kk} = 1$. So the $k$th term in the column-row expansion of $AB$ becomes
$$
-A^{(n)} = F^{(n-1)}F^{(n-2)}\cdots F^{(3)}F^{(2)}F^{(1)}A^{(0)}
-$$
+\begin{array}{rcl}
+ \vect{a}_k\vect{b}^{(k)} &=&
+ \begin{bmatrix}b_{k1}\vect{a}_k & b_{k2}\vect{a}_k & \cdots & b_{kk}\vect{a}_k & \vect{0} & \cdots & \vect{0}\end{bmatrix} \\
+ &=&
+ \begin{bmatrix}
+ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
+ \vdots & \vdots & & \vdots & \vdots & & \vdots \\
+ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
+ a_{kk}b_{k1} & a_{kk}b_{k2} & \cdots & a_{kk}b_{kk} & 0 & \cdots & 0 \\
+ \vdots & \vdots & & \vdots & \vdots & & \vdots \\
+ a_{nk}b_{k1} & a_{nk}b_{k2} & \cdots & a_{nk}b_{kk} & 0 & \cdots & 0
+ \end{bmatrix}
+ \end{array}.
+$$
+
+This is a lower triangular matrix, with a $1$ on position $(k,k)$ and for the rest $0$s on the diagonal.
+Adding these $n$ column-row products gives a matrix of the required form.
+
+\item
+Next suppose that $A$ is an upper triangular matrix with $1$s on the diagonal.
+If we apply the algorithm of {prf:ref}`Prop:MatrixInv:Algorithm` to find the inverse using the augmented matrix
+
+$$
+ [\,A | I \,] =
+ \left[\begin{array}{ccccc|ccccc}
+ 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\
+ a_{21} & 1 & 0 & \cdots & 0 & 0 & 1 & 0 & \cdots & 0 \\
+ a_{31} & a_{32} & 1 & \cdots & 0 & 0 & 0 & 1 & \cdots & 0 \\
+ \vdots &\vdots & & \ddots & 0 & \vdots & \vdots & & \ddots & 0 \\
+ a_{n1} &a_{n2} & a_{n3} & \cdots & 1 & 0 & 0 & 0 & \cdots & 1
+ \end{array} \right],
+$$
+to row reduce $A$ to $I$ we only need to subtract multiples of rows from rows below it. Thus in the matrix to the right of the bar, where the identity matrix is transformed into the inverse matrix of $A$, the triangular structure with $1$s on the diagonal remains.
+
+:::
-Since $A^{(n)}$ is an upper triangular matrix, we can take it as $U$. Since all matrices $F^{(k)}$ are invertible, we can recover $A^{(0)}$ multiplying $U$ to the left by the corresponding inverse matrices:
+::::::
+
+The uniqueness issue is not of major importance, but for completeness' sake below you can open a possible argument to prove it.
+
+::::::{admonition} Proof of the uniqueness of the $LU$-decomposition of an invertible matrix.
+:class: myproof, dropdown
+
+Suppose $A$ is an invertible matrix with two $LU$ decompositions
$$
-A=A^{(0)}={F^{(1)}}^{-1}{F^{(2)}}^{-1}\cdots{F^{(n-2)}}^{-1}{F^{(n-1)}}^{-1}A^{(n)}
+ A = L_1U_1, \quad A = L_2U_2.
$$
-So we can define $L={F^{(1)}}^{-1}{F^{(2)}}^{-1}\cdots{F^{(n-2)}}^{-1}{F^{(n-1)}}^{-1}$.
-We just need to see that $L$ is, indeed, lower triangular with ones on the diagonal, and that it has the multipliers $m_{ij}$ in the entries below it. To see that, we write $F^{(k)} = I - M_k E_k$ where
+We then have to show that
$$
-M_k =
-\begin{bmatrix}
-0\\ \vdots\\ 0\\m_{(k+1)k}\\m_{(k+2),k}\\ \vdots \\ m_{nk}
-\end{bmatrix}
-,\qquad E_k =
-\begin{bmatrix}
-0 & \cdots & 0 & \stackrel{\text{$k$-th}}{1} & 0 & \cdots & 0
-\end{bmatrix}
-.
+ L_1=L_2 \quad \text{and} \quad U_1 = U_2.
$$
-Observe that the matrix $I-M_kE_k$ is the matrix
+Since $A$ is invertible, the matrices $L_1, L_2, U_1, U_2$ are also invertible. <BR>
+From
$$
-\begin{bmatrix}
-1 & & & & & \\
- & \ddots & & & & \\
- & & 1 & & & \\
- & & -m_{(k+1)k} & \\
- & & -m_{(k+2)k} & \ddots & \\
- & & \vdots & & & \ddots \\
- & & -m_{nk} & & & & 1
-\end{bmatrix},
+L_1U_1 = L_2U_2
$$
-and that $E_iM_j = 0$ for $i \le j$.
+it follows that
-Now, we take the following product:
+::::{math}
+:label: Eq:LUdecomp:EqualityL1L2U1U2
-\begin{align*}
-(I-M_kE_k)(I+M_kE_k) &= I - M_kE_k + M_kE_k -M_kE_kM_kE_k \\
-&= I - M_kE_k M_kE_k \\
-&= I - M_k(E_k M_k)E_k \\
-&= I - M_k 0 E_k \\
-&= I,
-\end{align*}
+ L_2^{-1}L_1 = U_2U_1^{-1}.
+::::
-where the zero in the second from the last identity is the zero matrix, we deduce that ${F^{(k)}}^{-1} = I + M_k E_k$.
-Then,
+From {prf:ref}`Prop:LUdecomp:L-properties` we know that $L_2^{-1}$ is a lower triangular matrix with $1$s on the diagonal,
+and that the product $L_2^{-1}L_1$ is also of this form. At the same time the product $U_2U_1^{-1}$ must be an upper triangular matrix with $1$s on its diagonal. Then {eq}`Eq:LUdecomp:EqualityL1L2U1U2` implies that
$$
-\begin{align*}
-L &= {F^{(1)}}^{-1}{F^{(2)}}^{-1}\cdots{F^{(n-1)}}^{-1}{F^{(n-1)}}^{-1}\\
-&= \prod_{k=1}^{n-1} (I+M_kE_k) \\
-&= I + \sum_{k=1}^{n-1} M_kE_k,
-\end{align*}
+ L_2^{-1}L_1 = U_2U_1^{-1} = I,
$$
-where we have used that fact that, for $i < j$ $E_iM_j=0$.
-Thus,
+from which the identities
$$
-\begin{align*}
-L &= I + M_1E_1 + \cdots M_{n-1}E_{n-1} \\
-&=
-\begin{bmatrix}
-1 \\
-m_{21} & 1 \\
-\vdots & m_{32} &\ddots \\
-\vdots & \vdots &\ddots & \ddots & \\
-m_{n1} & m_{n2} & \cdots & m_{n(n-1)} & 1
-\end{bmatrix}
-.
-\end{align*}
+ L_1 = L_2, \quad U_1 = U_2
$$
+immediately follow.
+
::::::
-(Sec:PLUDecomp)=
+Lastly we give an example to show that the uniqueness may fail for a singular matrix.
+
+::::::{prf:example}
+:label: Eq:LUdecomp:NonUniqueLU
+
+For the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{bmatrix}$ it holds that
+
+$$
+ A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & a & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},
+$$
+
+where the parameter $a$ is free to choose.
+::::::
-## PLU Decomposition
+(Subsec:LUdecomp:PLUdecomp)=
-The LU decomposition is quite limiting in the sense that it requires to be able to find an echelon form of the matrix of coefficients without exchanging any rows in the process. Clearly, it is not always possible to find an echelon form of a matrix without exchanging rows. Let's see what happens when we encounter this situation.
+## Generalization to non-square matrices and $PLU$ decomposition
-::::{prf:example}
+In this section we describe what can be said regarding $LU$ decompositions for non-square matrices,
+and present a 'workaround' for matrices for which there is no top-down row reduction to echelon form.
-We can try to find an $LU$ decomposition of the matrix
+We start with an example to show that not much changes if we apply the algorithm of {prf:ref}`Alg:LUdecomp:LUalgorithm` to non-square matrices.
+
+::::::{prf:example}
+:label: Ex:LUdecomp:Nonsquare
+
+The matrix $A$ is given by
$$
-A = \begin{bmatrix}
-0 & 1 & 2 \\
-3 & 4 & 5 \\
-6 & 7 & 8
+A =
+\begin{bmatrix}
+1 & 3 & 1 & -1 \\
+-1 & 1 & 1 & 2 \\
+ 2 & -1 & -1 & 3
\end{bmatrix}.
$$
-In this case, we can not find an echelon form for this matrix without exchanging rows. Let's perform the following row operations:
+We can row reduce $A$ top-down to an echelon matrix:
-\begin{align*}
-&\begin{bmatrix}
-0 & 1 & 2 \\
-3 & 4 & 5 \\
-6 & 7 & 8
+$$
+\begin{array}{rcl}
+\begin{bmatrix}
+1 & 3 & 1 & -1 \\
+-1 & 1 & 1 & 2 \\
+ 2 & -2 & -1 & 3
\end{bmatrix}
\begin{array}{l}
[R_1] \\
-{[R_2]}\\
-{[R_3-2 R_2]}
-\end{array} &\sim &
+{[R_2-\class{blue}{(-1)}R_1]} \\
+{[R_3-\class{blue}{2}R_1]} \\
+\end{array}
+& \sim &
\begin{bmatrix}
-0 & 1 & 2 \\
-3 & 4 & 5 \\
-0 & -1 & -2
-\end{bmatrix}
-\begin{array}{l}
-[R_1 \leftrightarrow R_2] \\
-{[R_2 \leftrightarrow R_1]} \\
-{[R_3]} \\
-\end{array} \\
-&\begin{bmatrix}
-3 & 4 & 5 \\
-0 & 1 & 2 \\
-0 & -1 & -2
+1 & 3 & 1 & -1 \\
+0 & 4 & 2 & 1 \\
+0 & -8 & -3 & 5
\end{bmatrix}
\begin{array}{l}
-[R_1]\\
+[R_1] \\
{[R_2]} \\
-{[R_3 + R_2]}
-\end{array} &\sim &
+{[R_3-\class{blue}{(-2)}R_2]} \\
+\end{array}\\
+&\sim&
\begin{bmatrix}
-3 & 4 & 5 \\
-0 & 1 & 2 \\
-0 & 0 & 0
-\end{bmatrix}
-\end{align*}
+1 & 3 & 1 & -1 \\
+0 & 4 & 2 & 1 \\
+0 & 0 & 1 & 7
+\end{bmatrix}.
+\end{array}
+$$
-Observe that we obtained an upper triangular matrix $U$ (echelon form). However, if we consider the changes we performed to the matrix $A$, we will see that
+We then have
$$
+ A = LU = \begin{bmatrix}
+1 & 0 & 0 \\ \class{blue}{-1} & 1 & 0 \\ \class{blue}{2} & \class{blue}{-2} & 1 \end{bmatrix}
\begin{bmatrix}
-1 & 0 & 0\\
-0 & 1 & 0 \\
-0 & 1 & 1
-\end{bmatrix}
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
-0 & -2 & 1
-\end{bmatrix}
-=
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-1 & -2 & 1
-\end{bmatrix}
+1 & 3 & 1 & -1 \\
+0 & 4 & 2 & 1 \\
+0 & 0 & 1 & 7
+\end{bmatrix}.
$$
-is not a lower triangular matrix.
+If we want to solve an equation like
-:::::
+$$
+ A\vect{x}= \vect{b},
+$$
-To find a lower triangular matrix we observe the following fact: subtracting $2$ times the second row to the third and then exchanging the first and second rows is equivalent to first exchanging the first and second rows and then subtracting $2$ times the first row to the third.
+we can use the same two-step procedure as in {prf:ref}`Ex:LUdecomp:FirstLUcontinued`.
-::::{prf:example}
+For instance, to solve the system
$$
+ LU\vect{x} = \begin{bmatrix}
+1 & 0 & 0 \\ -1 & 1 & 0 \\ 2 & -2 & 1 \end{bmatrix}
\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
-0 & -2 & 1
-\end{bmatrix}=
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
--2 & 0 & 1
-\end{bmatrix}
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}.
+1 & 3 & 1 & -1 \\
+0 & 4 & 2 & 1 \\
+0 & 0 & 1 & 7
+\end{bmatrix}\vect{x} = \begin{bmatrix}2 \\ -2 \\ 5 \end{bmatrix}
$$
+we can first solve
-::::
+$$
+ L\vect{y} = \begin{bmatrix}
+1 & 0 & 0 \\ -1 & 1 & 0 \\ 2 & -2 & 1 \end{bmatrix}\vect{y} =
+ \begin{bmatrix}2 \\ -2 \\ 5 \end{bmatrix}.
+$$
-So we found an alternative order of row operations that brought us to the same echelon form. The key idea is to **perform all the row exchanges first** and **then add multiples of one row to another** to obtain an echelon form. Let's see the results:
+Using forward substitution we find the solution
-:::{math}
-:label: eq:LUDecomp:almostPLU
+$$
+ \tilde{\vect{y}} = \begin{bmatrix}2 \\ 0 \\ 1 \end{bmatrix}
+$$
-\begin{array}{rcl}
-\begin{bmatrix}
-1 & 0 & 0\\
-0 & 1 & 0 \\
-0 & 1 & 1
-\end{bmatrix}
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
--2 & 0 & 1
-\end{bmatrix}
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}A &= \begin{bmatrix}
-3 & 4 & 5 \\
-0 & 1 & 2 \\
-0 & 0 & 0
-\end{bmatrix} \\\\
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
--2 & 1 & 1
-\end{bmatrix}
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}A &=
-\begin{bmatrix}
-3 & 4 & 5 \\
-0 & 1 & 2 \\
-0 & 0 & 0
-\end{bmatrix}
-\end{array}
+and then the system
-:::
+$$
+ U\vect{x} = \begin{bmatrix}
+1 & 3 & 1 & -1 \\
+0 & 4 & 2 & 1 \\
+0 & 0 & 1 & 7\end{bmatrix}\vect{x} =
+ \begin{bmatrix}2 \\ 0 \\ 1 \end{bmatrix}
+$$
-Now we see that on the left-hand side of {eq}`eq:LUDecomp:almostPLU` we obtain a lower triangular matrix, for which we can find an inverse to obtain:
+gives the solution(s)
$$
-\begin{bmatrix}
-0 & 1 & 0 \\
-1 & 0 & 0 \\
-0 & 0 & 1
-\end{bmatrix}A =
-\begin{bmatrix}
-1 & 0 & 0 \\
-0 & 1 & 0 \\
-2 & -1 & 1
-\end{bmatrix}
-\begin{bmatrix}
-3 & 4 & 5 \\
-0 & 1 & 2 \\
-0 & 0 & 0
-\end{bmatrix}
+ \vect{x} = \begin{bmatrix}
+ \tfrac52 -\tfrac{7}{4}x_4\\
+ -\tfrac12 +\tfrac{13}{4}x_4\\
+ 1 - 7x_4\\
+ x_4
+ \end{bmatrix}, \quad x_4 \,\, \text{free.}
$$
-This is not an $LU$ decomposition, but almost! This is what we call a **PLU** decomposition. The following result summarizes it:
+::::::
-:::::{prf:theorem} Existence of a PLU Decomposition
-:label: LUDecomp:existencePLU
-Suppose that $A$ is an $m\times n$ matrix with real coefficients. Then there exist matrices $P$, $L$ and $U$ such that
+The generalization of {prf:ref}`Prop:LUdecomp:Existence` to non-square matrices is captured in the next proposition.
-$$ PA = LU, $$
+::::::{prf:proposition}
+:label: Prop:LUdecomp:ExistenceNonsquare
-where $P$ is an $m\times m$ matrix that exchanges the rows of $A$, $L$ is an $m\times m$ lower triangular matrix obtained inverting the product matrices that perform row operations to that only add multiples of a row to other rows in the matrix $PA$, and the matrix $U$ is an $m\times n$ upper-trapezoidal matrix (echelon form) that is obtained after performing the corresponding row operations to $PA$.
+An $m\times n$ matrix $A$, with $m \leq n$ can be written as $A = LU$, with $L$ an $m\times m$ upper triangular matrix with $1$s on its diagonal and $U$ an echelon matrix if and only if $A$ can be row reduced top-down to the echelon matrix $U$. <BR>
+Moreover, if the rows of $A$ are linearly independent the matrices $L$ and $U$ are unique.
-:::::
+::::::
-## Application of the (P)LU Decomposition
+A similar proposition holds for $m \times n$ matrices with $m > n$. However, in this case the systems
+$A\vect{x} = \vect{b}$ are inconsistent for most vectors $\vect{b}$, and then other techniques come into play (e.g., see {numref}`SubSec:LeastSquares:LS-solutions`).
-One way to measure the performance of an algorithm is counting the number of arithmetic operations [^flopnote] that are necessary for solving a problem. By arithmetic operations we will take into account sums, products, multiplications and divisions. Suppose that we want to solve the linear system $A\mathbf{x}=\mathbf{b}$ by taking the augmented matrix $[ A | \mathbf{b}]$, finding an echelon form with the same solution set, and then using backward substitution.
+In the remainder of this subsection we address the next best thing we can do in case a matrix does not have an $LU$ decomposition.
-In the worst-case scenario, for a $3\times 3$ matrix $A$, ($3\times 4$ augmented matrix), we need the following number of arithmetic operations:
-<ul>
-<li>
+::::::{prf:example}
+:label: Ex:LUdecomp:NoLUcontinued
-To convert the components $a_{21}$ and $a_{31}$ to a zero value, we need to compute two $m_{ij}$(two divisions), then we need to multiply each component of the first row, starting at $a_{12}$, by each $m_{ij}$(this happens twice, so $2\times 3=6$ products) and then we need to subtract the resulting values to the corresponding components in each row ($2\times 3=6$ subtractions). Therefore, we need a total of $14$ arithmetic operations(8 products/divisions and 6 additions/subtractions).
+Let us return to {prf:ref}`Ex:LUdecomp:NoLU` where the matrix $A$ cannot be written as $LU$.
+We copy from there
-</li>
-<li>
+$$
+ A \,=\, \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 4 & 8 & 6 \\
+ 2 & 5 & 7
+ \end{bmatrix}
+ \begin{array}{l}
+ [R_1] \\
+ {[R_2-4R_1]} \\
+ {[R_3-2R_1]}
+ \end{array} \quad \sim \quad
+ \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 0 & 0 & 2 \\
+ 0 & 1 & 5
+ \end{bmatrix}.
+$$
-To convert the component $a_{32}$ to a zero value, we need to compute one $m_{ij}$(one division), then we need to multiply each component of the second row starting at $a_{23}$(this is 2 products), and then we need to subtract the resulting values to the corresponding components in the third row (2 subtractions). This totals 5 arithmetic operations.
+We need a row swap to bring the last matrix to echelon form. The key idea is to **perform all the row exchanges first** and **then add multiples of rows (top-down) to other rows** to obtain an echelon form. Here the one row exchange is captured by the matrix
-</li>
-</ul>
+$$
+ P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}.
+$$
-So just to bring the augmented matrix to an echelon form requires 19 arithmetic operations (11 multiplications/divisions and 8 additions/subtractions).
-Now, to solve the system we use backward substitution:
+We get
-<ul>
-<li>
+$$
+ PA \,=\, \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 2 & 5 & 7 \\
+ 4 & 8 & 6
+ \end{bmatrix} \,\, = \,\,
+ \begin{bmatrix}
+ 1 & 0 & 0 \\
+ 2 & 1 & 0 \\
+ 4 & 0 & 1
+ \end{bmatrix}\,
+ \begin{bmatrix}
+ 1 & 2 & 1 \\
+ 0 & 1 & 5 \\
+ 0 & 0 & 2
+ \end{bmatrix}.
+$$
-To find $x_3$ requires one division.
-</li>
-<li>
+This is what we will call a *PLU* decomposition.
-To find $x_2$ requires (one multiplication, one subtraction, and one division).
+Note that first subtracting row 1 four times from row 2 and two times from row 3, followed by exchanging row 2 and row 3 leads to the same echelon matrix as first swapping row 2 and 3 and then subtracting row 1 two times from row 2 and four times from row 3.
-</li>
-<li>
+::::::
-To find $x_1$ requires (two multiplications, two subtractions, and one division).
-</li>
-</ul>
+::::::{prf:definition}
+:label: Def:LUdecomp:PermutationMatrix
-This means that solving a linear system with a $3\times 3$ matrix of coefficients in echelon form requires 9 arithmetic operations.
+A **permutation matrix** is an $n \times n$ matrix $P$ with only entries $0$ and $1$ in such a way that each row and each column contain exactly one $1$.
-So in total, we needed 28 arithmetic operations.
+::::::
-Supposing that $A=LU$ and that $L$ and $U$ are given, then, we solve first $L\mathbf{y}=\mathbf{b}$ and then $U\mathbf{x}=\mathbf{y}$.
+::::::{prf:example}
-<ul>
-<li>
+Two $4\times 4$ permutation matrices are
-For $L\mathbf{y}=\mathbf{b}$ we use forward substitution. Since the elements in the main diagonal are ones, then we have that we need no operations to determine $y_1$, we need one subtraction and one division for $y_2$, and two subtractions and one division for $y_3$. This totals 6 arithmetic operations.
+::::{math}
+:label: Eq:LUdecomp:P1andP2
-</li>
-<li>
+ P_1 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0
+ \end{bmatrix} \quad \text{and} \quad
+ P_2 = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0
+ \end{bmatrix}
+::::
-To solve $U\mathbf{x}=\mathbf{y}$ we use backward substitution, and we have just seen that it requires 9 arithmetic operations.
+Note that for an arbitrary $4 \times 4$ matrix $A$ we have
-</li>
-</ul>
+$$
+ P_1 A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & \class{blue}{1} \\ 0 & 0 & 1 & 0
+ \end{bmatrix}
+ \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\
+ a_{31} & a_{32} & a_{33} & a_{34} \\
+ \class{blue}{a_{41}} & \class{blue}{a_{42}} &
+ \class{blue}{a_{43}} & \class{blue}{a_{44}}
+ \end{bmatrix} =
+ \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\
+ \class{blue}{a_{41}} & \class{blue}{a_{42}} &
+ \class{blue}{a_{43}} & \class{blue}{a_{44}} \\
+ a_{31} & a_{32} & a_{33} & a_{34}
+ \end{bmatrix}
+$$
-So when the matrix $A$ is already $LU$ factorised, the number of operations required to solve the system is significantly lower.
+and
-Suppose now that $A$ is a non-singular $n\times n$ matrix. Let's count the number of operations in a general case, and we can see how advantageous it is to factorise our matrix $A$.
+$$
+ P_2 A = \begin{bmatrix} 0 & \class{blue}{1} & 0 & 0 \\ 0 & 0 & \class{red}{1} & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0
+ \end{bmatrix}
+ \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
+ \class{blue}{a_{21}} & \class{blue}{a_{22}} &
+ \class{blue}{a_{23}} & \class{blue}{a_{24}} \\
+ \class{red}{a_{31}} & \class{red}{a_{32}} &
+ \class{red}{a_{33}} & \class{red}{a_{34}} \\
+ a_{41} & a_{42} & a_{43} & a_{44}
+ \end{bmatrix} =
+ \begin{bmatrix} \class{blue}{a_{21}} & \class{blue}{a_{22}} &
+ \class{blue}{a_{23}} & \class{blue}{a_{24}} \\
+ \class{red}{a_{31}} & \class{red}{a_{32}} &
+ \class{red}{a_{33}} & \class{red}{a_{34}} \\
+ a_{41} & a_{42} & a_{43} & a_{44} \\a_{11} & a_{12} & a_{13} & a_{14}
+ \end{bmatrix}
+$$
-<ul>
-<li>
+In general (pre-)multiplication of a matrix $A$ with any permutation matrix reorders the rows of $A$.
-For solving the system directly, for each column $j$ we need to compute the $m_{ij}$($(n-1)$ divisions), then multiply $m_{ij}$ by all the components of row $i$ except for the first $i$, so that is $(n-i)(n-i+1)$ products, and then we need to perform the same number of subtractions. This totals $(n-i)(n-i+2)$ products/divisions and $(n-i)(n-i+2)$ additions/subtractions. Now we need to add these values for all the columns.
+::::::
+
+
+The following properties are rather obvious.
+
+
+::::::{prf:proposition}
+:label: Prop:LUdecomp:PermutationMatrices
+
+:::{latexlist}
+:enumerated: true
+:type: i
+
+\item The product of two $n\times n$ permutation matrices is again a permutation matrix.
+\label{Item:Prop:LUdecomp:PermutationMatrices_Product}
+
+\item The inverse of a permutation matrix is its transpose. Thus, $P^{-1} = P^T$.
+\label{Item:Prop:LUdecomp:PermutationMatrices__Inverse}
+
+:::
+
+::::::
+
+::::::{prf:example}
+
+For the matrices $P_1$ and $P_2$ from {eq}`Eq:LUdecomp:P1andP2` we have
-<ul>
-<li>
+$$
+ P_1P_2 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0
+ \end{bmatrix}
+ \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0
+ \end{bmatrix}
+ =
+ \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1
+ \end{bmatrix}
+$$
-For the products/divisions we have:
+and
$$
-\begin{align*}
-\sum_{i=1}^{n-1}(n-i)(n-i+2) &= \sum_{i+1}^{n-1} i^{2} - 2 i n + n^{2} - 2 i + 2 n \\
-&=\sum_{i=1}^{n-1}(n^2-2ni+i^2)+\sum_{i=1}^{n-1}(2n-2i)\\
-&=\sum_{i=1}^{n-1}(n-i)^2 + 2\sum_{i=1}^{n-1}(n-i) \\
-&=\sum_{i=1}^{n-1}i^2 +2\sum_{i=1}^{n-1}i \\
-&\frac{n(n-1)(2n-1)}{6}+2\frac{n(n-1)}{2} \\
-&\frac{2n^3+3n^2-5n}{6}.
-\end{align*}
+ P_2^TP_2 = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0
+ \end{bmatrix}
+ \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0
+ \end{bmatrix}
+ =
+ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1
+ \end{bmatrix}.
$$
-</li>
-<li>
+::::::
+
+::::::{admonition} Proof of {prf:ref}`Prop:LUdecomp:PermutationMatrices`
+:class: myproof, dropdown
+
+:::{latexlist}
+:enumerated: true
+:type: i
+
+\item The product of two $n\times n$ permutation matrices is again a permutation matrix.
+
+Suppose $P_1$ and $P_2$ are permutation matrices. With the product $P_1P_2$ the rows of $P_2$ are reordered, and that leaves the properties a $1$ in each row, a $1$ in each column, all other entries equal to $0$, intact.
+
+
+\item The inverse of a permutation matrix is its transpose. Thus, $P^{-1} = P^T$.
+
+In a product $A^TA$, the entry on position $(i,j)$ is the inner product of the $i$th column of $A$
+with the $j$th column of $A$. (Cf., {numref}`Exc:MatrixOps:InterpretATB`.) Since the $n$ columns of $A$ are the $n$ columns $\vect{e}_j$ of the identity matrix (in some order), and
+
+$$
+ \vect{e}_i\ip\vect{e}_j = \left\{\begin{array}{l}
+ 1, \,\,\text{if}\,\, i = j\\
+ 0, \,\,\text{if}\,\, i \neq j,
+ \end{array}
+ \right.
+$$
+
+the result immediately follows.
+
+:::
+
+::::::
+
+::::::{prf:theorem} Existence of a $PLU$ Decomposition
+:label: LUDecomp:existencePLU
+
+Suppose that $A$ is an $m\times n$ matrix with real coefficients, and let $m \leq n$. Then there exist a permutation matrix $P$, an upper triangular matrix $L$ and an echelon matrix $U$ such that
+
+$$
+ PA = LU.
+$$
+
+::::::
+
+::::::{prf:remark}
-For the additions/subtractions we have:
+As was mentioned before, the key idea is to perform the row exchanges first. These can be put together in the permutation matrix $P$. The algorithm to actually find the $LU$ decomposition of $PA$ without doing the whole row reduction process for $PA$ all over again is rather intricate, and in our view belongs to a course of numerical linear algebra. There it will be explained that it may also be preferable to work towards a $PLU$ decomposition instead of an $LU$ decomposition in cases where theoretically it is not absolutely necessary. For numerical reasons, having to do with finite accuracy when representing real numbers in computers, it may be better to choose the pivots in another order than just top-down.
+
+::::::
+
+In the following example the matrix does have a proper $LU$ decomposition, but we will row reduce it
+in another order than top-down to echelon form. It is a tiny example to illustrate that it is possible to deduce a $PLU$ decomposition from it when we keep record of both the multipliers and the positions of the pivots.
+
+::::::{prf:example}
+:label: Ex:LUdecomp:PLUexample-2
+
+We will row reduce the matrix $ A= \begin{bmatrix}2&4&3 \\ 1&2&3\\1&3&2 \end{bmatrix}$ in an alternative order than top-down and extricate a $PLU$ decomposition from it.
$$
-\begin{align*}
-\sum_{i=1}^{n-1}(n-i)(n-i+1) &= \sum_{i=1}^{n-1} i^{2} - 2 i n + n^{2} - i + n \\
-&=\sum_{i=1}^{n-1}(n^2-2in+i^2) + \sum_{i=1}^{n-1}(n-i) \\
-&=\sum_{i=1}^{n-1}(n-i)^2 + \sum_{i=1}^{n-1}(n-i)\\
-&=\sum_{i=1}^{n-1}i^2 + \sum_{i=1}^{n-1}i \\
-&=\frac{n(n-1)(2n-1)}{6}+\frac{n(n-1)}{2}\\
-&=\frac{n^3-n}{3}.
-\end{align*}
+\left[\begin{array}{rrr}2&4&3 \\ 1&2&3\\ \fbox{1}&3&2 \end{array} \right] \begin{array}{l}
+[R_1-\class{blue}{2}R_3] \\
+{[R_2-\class{blue}{1}R_3]} \\
+{[R_3]} \\
+\end{array}
+\sim
+\left[\begin{array}{rrr}0&\fbox{$-2$}&-1 \\ 0 & -1 &1 \\ 1&3&2\end{array} \right] \begin{array}{l}
+[R_1] \\
+{[R_2-\class{red}{\frac12}R_1]} \\
+{[R_3]} \\
+\end{array}
+\sim
+\left[\begin{array}{rrr}0&-2&-1 \\ 0 & 0 &\frac32 \\ 1&3&2\end{array} \right].
$$
-</li>
-</ul>
+If we put together the matrices that describe the row operations we get
+
+:::{math}
+:label: Eq:LUdecomp:PivotStructure
-So in total we have to perform
+ A = \begin{bmatrix}2&\fbox{4}&3 \\ 1&2&\fbox{3}\\ \fbox{1}&3&2\end{bmatrix} =
+ \begin{bmatrix} 1 & 0 & \class{blue}{2} \\ \class{red}{\frac12} &1 & \class{blue}{1} \\ 0 & 0 & 1\end{bmatrix}
+ \begin{bmatrix} 0 & -2 & -1 \\ 0 & 0 &\frac32 \\1 & 3 & 2\end{bmatrix} = \tilde{L}\tilde{U}.
+
+:::
+
+If we would have started from
$$
-\frac{4n^3-3n^2-7n}{6}
+ \begin{bmatrix} 1&3&2 \\2&4&3 \\ 1&2&3\end{bmatrix} = PA =
+ \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}
+ \begin{bmatrix} 2&4&3 \\ 1&2&3\\1&3&2 \end{bmatrix}
$$
-arithmetic operations to bring the matrix to echelon form.
+using the same pivots would have arrived at
+
+$$
+ \begin{bmatrix} 1&3&2 \\2&4&3 \\ 1&2&3\end{bmatrix} \sim
+ \begin{bmatrix} 1&3&2 \\0&-2&-1 \\ 0&-1&1\end{bmatrix} \sim
+ \begin{bmatrix} 1&3&2 \\0&-2&-1 \\ 0&0&\frac32\end{bmatrix},
+$$
-If we just count the number of arithmetic operations to compute the $LU$ decomposition, then we need
+so
$$
-\frac{4n^3-3n^2-n}{6}
+ PA = \begin{bmatrix} 1&3&2 \\2&4&3 \\ 1&2&3\end{bmatrix} =
+ \begin{bmatrix} 1 & 0 & 0 \\\class{blue}{2} & 1 & 0 \\
+ \class{blue}{1} & \class{red}{\frac12} & 1\end{bmatrix}
+ \begin{bmatrix} 1&3&2 \\0&-2&-1 \\ 0&0&\frac32\end{bmatrix} = LU.
$$
-arithmetic operations.
+The matrix $P$ is the inverse of (so also the transpose of) the matrix that describes the positions of the pivots as in Equation {eq}`Eq:LUdecomp:PivotStructure`.
+It asks for some careful analysis how to reconstruct $L$ from $\tilde{L}$.
-Using similar reasoning, we can calculate the number of arithmetic operations needed to solve an upper triangular linear system, which gives us $n^2$. And solving a lower triangular system with ones in the main diagonal requires $n^2-n$ arithmetic operations.
+::::::
-:::{prf:remark}
-The total number of arithmetic operations needed in order to solve a linear system with row reduction (without exchanging rows), and with $LU$ is the same. We leave the proof as an exercise for the reader.
-:::
-In many applications in engineering, it is required to solve $m$ linear systems, $[A|\mathbf{b}_1\,\mathbf{b}_2\,\dots \mathbf{b}_m]$, that have the same matrix of coefficients. In this situation is where the $LU$ Decomposition comes in handy. In {numref}`tbl:comparison_gausselim_LU` we can see the comparison in the number of operations need to solve several linear systems when using row reduction and $LU$ decomposition.
-:::{latextable} Comparison between solving linear systems with row reduction (RR) and with $LU$ decomposition ($LU$)
-:header-rows: 2
-:class: longtable table-bordered table-striped table-hover table
-:align: right
-:name: tbl:comparison_gausselim_LU
+## Efficiency Issues
-\begin{tabular}{crrrrrr}
-$n$ & \multicolumn{2}{c}{$m=5$} & \multicolumn{2}{c}{$m=10$} & \multicolumn{2}{c}{$m=50$} \\
-& RR & $LU$ & RR & $LU$ & RR & $LU$ \\
-$3$ & $140$ & $88$ & $280$ & $163$ & $1400$ & $763$ \\
-$5$ & $575$ & $295$ & $1,150$ & $520$ & $5,750$ & $2,320$ \\
-$10$ & $4,025$ & $1,565$ & $8,050$ & $2,515$ & $40,250$ & $10,115$ \\
-\end{tabular}
+To be filled in later.
-:::
-</li>
-</ul>
+## Grasple Exercises
-## Theoretical Exercises
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/cca386f8-a6ff-44b6-9b83-fe96482a4763?id=108857
+:label: grasple_exercise_3_6_2
+:dropdown:
+:description: To identify triangular matrices.
-::::::{exercise}
-:label: Exc:LUdecomp:Theory1
+::::::
-Prove {prf:ref}`Prop:LUDecomp:PropertiesTriangularMatricesInverse`.
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/4ef9b6fe-e204-44e7-9463-3e1c3537a10b?id=82913
+:label: grasple_exercise_3_6_3
+:dropdown:
+:description: To compute the $LU$-decomposition of a 2x2 matrix $A$.
-**Hint:** Write the matrix $[A\vert I]$ and apply row operations to compute $A^{-1}$. The idea is similar to the one used in the proof of {prf:ref}`thm:existence_and_uniqueness_LU`.
::::::
-::::::{exercise}
-:label: Exc:LUdecomp:Theory2
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/b167deea-922f-4a80-9b3e-7cbdf16f023f?id=106332
+:label: grasple_exercise_3_6_4
+:dropdown:
+:description: To compute the $LU$-decomposition of a 3x3 matrix $A$.
+
+::::::
-Check that the number of arithmetic operations needed to solve a linear system using row reduction (without exchanging rows) and with $LU$ decomposition is the same.
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/9708bce4-5c01-4486-8f44-7ea3a5157950?id=82914
+:label: grasple_exercise_3_6_5
+:dropdown:
+:description: To compute the $LU$-decomposition of a 3x3 matrix $A$ and use it to solve $A\vect{x} = \vect{b}$.
+
+::::::
+
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/d7ec03b4-32c4-4c3e-8c1e-2714878ef558?id=82917
+:label: grasple_exercise_3_6_6
+:dropdown:
+:description: To compute the $LU$-decomposition of a 3x3 matrix $A$ and use it to solve $A\vect{x} = \vect{b}$.
+
+::::::
+
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/9cbcf004-bfbe-428f-927f-5c64ca802946?id=82919
+:label: grasple_exercise_3_6_7
+:dropdown:
+:description: To decide solving $A\vect{x} = \vect{b}$ via (given) $A=LU$ or (given) $A^{-1}$.
+
+::::::
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/8fe1ca5c-3f24-4148-9725-a96c44e3f43a?id=82920
+:label: grasple_exercise_3_6_8
+:dropdown:
+:description: To compute $A^{-1}$ using $A = LU$.
::::::
-[^flopnote]: In some books they use the abbreviation _flop_ (floating point operations).
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/7d8c3553-18cd-4866-bfa1-9ff273ee18e8?id=82925
+:label: grasple_exercise_3_6_9
+:dropdown:
+:description: Explorative exercise about the $LDU$-decomposition
+
+::::::
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/5bb6dcac-4575-4953-bb3a-c0e8d4594798?id=82928
+:label: grasple_exercise_3_6_10
+:dropdown:
+:description: To compute the $LU$-decomposition of 3x4 matrix $A$ and use it to solve $A\vect{x} = \vect{b}$.
+
+::::::
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/9a2cb913-3462-4832-8e33-9f4b878f1da7?id=106804
+:label: grasple_exercise_3_6_11
+:dropdown:
+:description: To compute a $PLU$-decomposition of a 3x3 matrix.
+
+::::::
+
+::::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/48ef4fa5-cdcf-4449-8c5a-0d5f5d448025?id=106870
+:label: grasple_exercise_3_6_12
+:dropdown:
+:description: To solve a system $A\vect{x} = \vect{b}$ using $ PA = LU$.
+
+::::::
diff --git a/Chapter3/Linear_Transformations.md b/Chapter3/Linear_Transformations.md
index 94a3fda..824cf5c 100644
--- a/Chapter3/Linear_Transformations.md
+++ b/Chapter3/Linear_Transformations.md
@@ -204,6 +204,7 @@ This transformation "embeds" the plane $\mathbb{R}^2$ into the space $\mathbb{R}
:fig: Images/Fig-LinTrafo-EmbedR2R3.svg
:name: Fig:LinTrafo:EmbedR2R3
:status: reviewed
+:class: dark-light
$T$: embedding $\mathbb{R}^2$ into $\mathbb{R}^3$.
```
@@ -286,8 +287,9 @@ See {numref}`Figure %s <Fig:LinTrafo:SkewProjection>`.
::::{figure} Images/Fig-LinTrafo-SkewProjection.svg
:name: Fig:LinTrafo:SkewProjection
+:class: dark-light
-The transformation of {prf:ref}`Eq:LinTrafo:SkewProjection`
+The transformation of {prf:ref}`Eq:LinTrafo:SkewProjection`.
::::
::::::
@@ -425,7 +427,8 @@ Thus, if $ T:\mathbb{R}^n \to\mathbb{R}^m$ is a linear transformation, then $T(\
::::::
-::::::{dropdown} Solution to {numref}`Exc:LinTrafo:ImageofZeroVector` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:LinTrafo:ImageofZeroVector`
+:class: solution, dropdown
If $ T:\mathbb{R}^n \to\mathbb{R}^m$ is linear, and $\vect{v}$ is any vector in $\R^n$, then $\mathbf{0}_n = 0\vect{v}$. From the second property in {prf:ref}`Dfn:LinTrafo:LinTrafo` it follows that
@@ -620,7 +623,8 @@ a linear transformation?
::::::
-::::::{dropdown} Solution to {numref}`Exc:LinTrafo:T(x)=x+p` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:LinTrafo:T(x)=x+p`
+:class: solution, dropdown
The transformation defined by $T(\vect{x}) = \vect{x} + \vect{p}$, with $\vect{p}\neq \vect{0}$ does not have any of the two properties of a linear transformation.
@@ -660,7 +664,9 @@ Each matrix transformation is a linear transformation.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:LinTrafo:MatrixTrafoIsLinear`
+:class: myproof
+
This is a direct consequence of the two properties of the matrix-vector product ({prf:ref}`Prop:MatVecProduct:Linearity`) that say
$$
@@ -689,7 +695,9 @@ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^p$.
The transformation $S\circ T$ is called the **composition** of the two transformations $S$ and $T$. It is best read as _"$S$ after $T$"_.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:LinTrafo:CompositionLintrafos`
+:class: myproof
+
Suppose that
$$
@@ -739,7 +747,8 @@ Show that $S$ and $T_3$ are again linear transformations.
::::::
-::::::{dropdown} Solution to {numref}`Exc:LinTrafo:CombiningLinTrafos` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:LinTrafo:CombiningLinTrafos`
+:class: solution, dropdown
The properties of the linear transformatiuon $T_1$ and $T_2$ carry over to $S$ and $T_3$ in the following way.
We check the properties one by one.
@@ -812,7 +821,9 @@ c_1T(\mathbf{x}\_1)+c_2T(\mathbf{x}\_2)+\ldots +c_kT( \mathbf{x}\_k).
In words: for any linear transformation
_the image of a linear combination of vectors is equal to the linear combination of their images_.
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:LinTrafo:ExtendedLinearity`
+:class: myproof
+
Suppose $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear transformation.
So we have
@@ -1015,7 +1026,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:LinTrafo:MatrixForFirstExample` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:LinTrafo:MatrixForFirstExample`
+:class: solution, dropdown
Consider the linear transformation
$T:\mathbb{R}^2\rightarrow\mathbb{R}^3$ that sends each vector $ \begin{bmatrix}
@@ -1038,7 +1050,7 @@ $$
\begin{bmatrix} 1 & 0 \\ 0 & 1\\ 0 & 0 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}.
$$
-::::::
+::::
The reasoning of {prf:ref}`Ex:LinTrafo:StandardMatrixIntro` can be generalized. This is the content of the next theorem.
@@ -1062,7 +1074,9 @@ T(\mathbf{e}\_1) & T(\mathbf{e}\_2) & \ldots & T(\mathbf{e}\_n)
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:LinTrafo:LinTrafo=MatrixTrafo`
+:class: myproof
+
We can more or less copy the derivation in {prf:ref}`Ex:LinTrafo:StandardMatrixIntro`.
First of all, any vector $\mathbf{x}$ is a linear combination of the standard basis:
@@ -1115,7 +1129,7 @@ $$
For a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$, the matrix
:::{math}
-:label: Eq:LinTrafo:StandardMatrix
+:label: Eq:LinTrafo:StandardMatrix2
\begin{bmatrix}
T(\mathbf{e}\_1) & T(\mathbf{e}\_2) & \ldots & T(\mathbf{e}\_n)
@@ -1319,6 +1333,7 @@ $$
## Grasple Exercises
%::::::{grasple}
+:iframeclass: dark-light
%:url: https://embed.grasple.com/exercises/97a589a8-54f9-4688-bd4d-a17a9585813b?id=69465
%:label: grasple_exercise_3_1_1
%:dropdown:
@@ -1326,6 +1341,7 @@ $$
%::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3f14573a-1d4c-4a4b-ae48-ccb168005702?id=70373
:label: grasple_exercise_3_1_2
:dropdown:
@@ -1333,6 +1349,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b80d9889-bd46-45c6-a9cb-d056aa315232?id=70374
:label: grasple_exercise_3_1_3
:dropdown:
@@ -1340,6 +1357,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/be6a768d-c60d-4ed6-81a7-5dea71b4a1a5?id=70375
:label: grasple_exercise_3_1_4
:dropdown:
@@ -1347,6 +1365,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c8bb24f6-d357-4571-adb3-39ea0fa9e4ee?id=70395
:label: grasple_exercise_3_1_5
:dropdown:
@@ -1354,6 +1373,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/93048f7c-b755-4445-a532-949f34136096?id=70398
:label: grasple_exercise_3_1_6
:dropdown:
@@ -1361,6 +1381,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2af6559f-8871-494d-abce-d4263d530c69?id=70381
:label: grasple_exercise_3_1_7
:dropdown:
@@ -1368,6 +1389,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ce6e4a52-c985-43ee-92cb-2762a467ac5a?id=70383
:label: grasple_exercise_3_1_8
:dropdown:
@@ -1375,6 +1397,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/37b6bd46-8cfc-4c98-a5e8-53aa41c87dcf?id=70384
:label: grasple_exercise_3_1_10
:dropdown:
@@ -1382,13 +1405,15 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c5b2a642-fd50-43f6-9346-c37a0ffe1a40?id=70386
-:label: grasple_exercise_3_1_10
+:label: grasple_exercise_3_1_10b
:dropdown:
:description: Find vectors $\vect{w}$ for which $T(\vect{w}) = \vect{u}$.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c3d009c0-62d6-4ae3-8ca1-04a5d2730455?id=70406
:label: grasple_exercise_3_1_11
:dropdown:
@@ -1396,6 +1421,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b9a4b128-f2c2-4612-a7f5-271c4e69aa70?id=70418
:label: grasple_exercise_3_1_12
:dropdown:
@@ -1403,6 +1429,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4058e54a-74f2-414e-9693-420abbc62677?id=70391
:label: grasple_exercise_3_1_13
:dropdown:
@@ -1410,6 +1437,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/990bf561-629e-430f-b8d0-e757c63fe15c?id=70392
:label: grasple_exercise_3_1_14
:dropdown:
@@ -1417,6 +1445,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4e5d3f55-9257-4023-9739-5df0a1a9f277?id=70410
:label: grasple_exercise_3_1_15
:dropdown:
@@ -1424,6 +1453,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9efa96e2-483d-4b2c-a58a-ba197bc09a81?id=70411
:label: grasple_exercise_3_1_16
:dropdown:
@@ -1431,6 +1461,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/729cba57-72d1-4d54-8cf9-c9946952bf9d?id=70412
:label: grasple_exercise_3_1_17
:dropdown:
@@ -1438,6 +1469,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b4bb3730-f14c-4a60-a8b8-6b895cf93ac5?id=70413
:label: grasple_exercise_3_1_18
:dropdown:
@@ -1445,6 +1477,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/34bb6386-7e7c-411b-83a1-09bbaf1106c5?id=70415
:label: grasple_exercise_3_1_19
:dropdown:
@@ -1452,6 +1485,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ce8ba17c-0a17-4d5e-b4b7-5c277c7e8df8?id=70416
:label: grasple_exercise_3_1_20
:dropdown:
@@ -1459,6 +1493,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2de4f8d1-ab3d-4d3a-94e4-5e414e2da3d9?id=70372
:label: grasple_exercise_3_1_21
:dropdown:
@@ -1466,6 +1501,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3f992e7a-19e3-4b83-8d90-db86e323ea94?id=69296
:label: grasple_exercise_3_1_22
:dropdown:
@@ -1473,6 +1509,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/94d618e0-de21-491c-ad44-8e29974e0303?id=71098
:label: grasple_exercise_3_1_23
:dropdown:
@@ -1480,6 +1517,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f983b627-10c2-4dd6-a273-2a33e99d0ded?id=71101
:label: grasple_exercise_3_1_24
:dropdown:
diff --git a/Chapter3/MatrixInverse.md b/Chapter3/MatrixInverse.md
index 6a45a57..c5414a8 100644
--- a/Chapter3/MatrixInverse.md
+++ b/Chapter3/MatrixInverse.md
@@ -37,9 +37,16 @@ $$
The bad news:
-$$
- \frac{A}{B} \quad \text{cannot be defined in any useful way!}
-$$
+<p style="text-align:center;">
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+ <mstyle displaystyle="true" scriptlevel="0">
+ <mfrac>
+ <mi>A</mi>
+ <mi>B</mi>
+ </mfrac>
+ </mstyle>
+</math> cannot be defined in any useful way!
+</p>
First of all the corresponding matrix equation
@@ -167,7 +174,9 @@ If an inverse of a matrix $A$ exists, then it is unique.
The proof is very short, when we plug in the right idea at the right place.
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixInv:UniqueInverse`
+:class: myproof
+
Suppose $B$ and $C$ are two matrices that satisfy the properties of being an inverse of $A$, i.e.
$$
@@ -295,7 +304,9 @@ Also check that the first matrix in {prf:ref}`Ex:FirstInverse` illustrates the f
::::::
-::::::{dropdown} Solution to {numref}`Exc:MatrixInv:CheckBA=I` (_click to show_)
+
+::::::{admonition} Solution to {numref}`Exc:MatrixInv:CheckBA=I`
+:class: solution, dropdown
$$
\begin{array}{rcl} BA &=&
@@ -432,7 +443,9 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixInv:InvertibleIndepCols`
+:class: myproof
+
As in the proof in {prf:ref}`Rem:MatrixInvDetZeroDependentColumns` we have to prove two implications:
$$
@@ -546,7 +559,9 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:SolutionViaInverse`
+:class: myproof
+
We multiply both sides of the equation
$$
@@ -708,7 +723,9 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixInv:ElemProperties`
+:class: myproof
+
All statements can be proved by verifying that the relevant products are equal to $I$.
<ol type="i">
@@ -725,13 +742,12 @@ and likewise $\dfrac1c A^{-1}\cdot (cA) = I$,
which proves that indeed $\dfrac1c A^{-1} = (cA)^{-1}$.
</li>
+
<li>
Since it is given that $A^{-1}$ exists we can proceed as follows, where we make use of the characteristic property
$ B^TA^T = (AB)^T$.
-<BR>
-
$$
(A^{-1})^TA^T = ( AA^{-1})^T = I^T = I
$$
@@ -742,10 +758,13 @@ $$
A^T(A^{-1})^T =( A^{-1}A)^T = I^T = I,
$$
-which settles the second statement. To prove iii., see {numref}`Exc:MatrixInv:Ainvinv`.
+which settles the second statement.
</li>
-</ul>
+
+</ol>
+
+To prove iii., see {numref}`Exc:MatrixInv:Ainvinv`.
::::::
@@ -756,7 +775,8 @@ Prove the last statement of the previous proposition.
::::::
-::::::{dropdown} Solution to {numref}`Exc:MatrixInv:Ainvinv` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:MatrixInv:Ainvinv`
+:class: solution, dropdown
For the inverse $C = (A^{-1})^{-1}$ of $A^{-1}$, it should hold that
@@ -820,7 +840,9 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixInv:ProductRule`
+:class: myproof
+
Again we just check that the properties of the definition hold.
Suppose that $A$ and $B$ are invertible with inverses $A^{-1}$ and $B^{-1}$.
@@ -856,7 +878,8 @@ In case it is true, give an argument, when false, give a counterexample.
::::::
-::::::{dropdown} Solution to {numref}`Exc:MatrixInv:(AB)Tinv` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:MatrixInv:(AB)Tinv`
+:class: solution, dropdown
The statement is _true_. <BR>
From the two properties
@@ -1060,7 +1083,9 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixInv:Algorithm`
+:class: myproof
+
We have already seen ({prf:ref}`Prop:MatrixInv:InvertibleIndepCols`) that an invertible matrix linearly independent columns,
which implies that the reduced echelon form of $A$ is indeed the identity matrix. And then it is clear that via row operations we get
@@ -1279,7 +1304,8 @@ Make sure that you do not use $A^{-1}$ or $B^{-1}$ prematurely, i.e., before you
::::::
-::::::{dropdown} Solution to {numref}`Exc:MatrixInv:ConverseProdRule` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:MatrixInv:ConverseProdRule`
+:class: solution, dropdown
Suppose $A$ and $B$ are two $n \times n$ matrices for which $AB$ is invertible. Let $C=(AB)^{-1}$ be the inverse of $AB$. We claim that
$BC$ is the inverse of $A$.
@@ -1363,7 +1389,9 @@ $A$ can be written as a product of elementary matrices: $A = E_1E_2\cdots E_k$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:MatrixInv:InvertibilityCharacterizations`
+:class: myproof
+
It is a good exercise to find out where the evidence of each characterization is found,
and wherever necessary to fill in the missing details.
@@ -1412,6 +1440,7 @@ The first exercises are quite straightfordwardly computational.
The remaining exercises tend to be more theoretic.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6683a2f9-7b6b-4dd1-bec1-1e8b894fa3bb?id=71086
:label: grasple_exercise_3_4_1
:dropdown:
@@ -1420,6 +1449,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1bbca38b-a734-4049-b8a2-f79d4bf1b098?id=71087
:label: grasple_exercise_3_4_2
:dropdown:
@@ -1428,6 +1458,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/045cd1
:label: grasple_exercise_3_4_3
:dropdown:
@@ -1436,6 +1467,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/82c06a56-8ee8-4f36-8173-e5d56da1e8e3?id=71073
:label: grasple_exercise_3_4_4
:dropdown:
@@ -1444,6 +1476,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/551172d9-861c-4958-9b17-dfa828acdabe?id=71088
:label: grasple_exercise_3_4_5
:dropdown:
@@ -1451,6 +1484,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9174c68c-e2d5-4c23-af96-e3fe3dd36f42?id=71089
:label: grasple_exercise_3_4_6
:dropdown:
@@ -1459,6 +1493,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/800dc2f9-227e-401b-818b-093fc9647dd9?id=83083
:label: grasple_exercise_3_4_7
:dropdown:
@@ -1467,6 +1502,7 @@ The remaining exercises tend to be more theoretic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9146f49d-74a5-4fda-a641-181c4536fe01?id=83086
:label: grasple_exercise_3_4_8
:dropdown:
@@ -1477,15 +1513,16 @@ The remaining exercises tend to be more theoretic.
The remaining exercises have more theoretic flavour.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/677aa3ee-4594-4d77-ace6-583a1efcba59?id=71090
:label: grasple_exercise_3_4_9
:dropdown:
-:description: True/False question about invertibility versus
-consistent linear systems.
+:description: True/False question about invertibility versus consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f789ebd5-171b-4556-83a9-eefc5ef830ef?id=71092
:label: grasple_exercise_3_4_10
:dropdown:
@@ -1494,6 +1531,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/29dc7c2f-6636-493e-9c97-da1847a336b7?id=68908
:label: grasple_exercise_3_4_11
:dropdown:
@@ -1502,6 +1540,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8f3feb75-b41b-42e0-b574-f6442da253ce?id=70272
:label: grasple_exercise_3_4_12
:dropdown:
@@ -1510,6 +1549,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5185c5c0-4d92-4e0e-92a7-6dc5eed8f7cf?id=68896
:label: grasple_exercise_3_4_13
:dropdown:
@@ -1518,6 +1558,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ee4bb61e-6939-4074-a556-b82f3d0e8c28?id=71091
:label: grasple_exercise_3_4_14
:dropdown:
@@ -1525,6 +1566,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1732d75b-2027-4a92-b8bb-c98bda62475d?id=71093
:label: grasple_exercise_3_4_15
:dropdown:
@@ -1532,6 +1574,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f8602d4f-57b7-4752-9edc-69c83069fe36?id=71095
:label: grasple_exercise_3_4_16
:dropdown:
@@ -1539,6 +1582,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a8ea864d-1164-4afc-9a24-c0a126ee8e54?id=71097
:label: grasple_exercise_3_4_17
:dropdown:
@@ -1546,6 +1590,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/73a16f62-28d7-4a4c-baf5-7ce3be9272ce?id=71104
:label: grasple_exercise_3_4_18
:dropdown:
@@ -1553,6 +1598,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a8c2b8ed-9961-4779-8841-491a9529b71c?id=71466
:label: grasple_exercise_3_4_19
:dropdown:
@@ -1560,6 +1606,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dfe429bd-1ab9-47f7-8f6c-06150c468645?id=71468
:label: grasple_exercise_3_4_20
:dropdown:
@@ -1567,6 +1614,7 @@ consistent linear systems.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9af5928a-7ecb-478e-a896-7c66d16d9d09?id=71463
:label: grasple_exercise_3_4_21
:dropdown:
@@ -1576,6 +1624,7 @@ consistent linear systems.
In the last two exercises (non-)invertibility of non-square matrices is considered.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ca504661-cc62-454f-8035-04a9bef85f91?id=61170
:label: grasple_exercise_3_4_22
:dropdown:
@@ -1583,6 +1632,7 @@ In the last two exercises (non-)invertibility of non-square matrices is consider
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4e9b4ec1-f775-430f-b81f-c76c42fcbc76?id=60136
:label: grasple_exercise_3_4_23
:dropdown:
diff --git a/Chapter3/MatrixOperations.md b/Chapter3/MatrixOperations.md
index fa8aa73..49e1a7b 100644
--- a/Chapter3/MatrixOperations.md
+++ b/Chapter3/MatrixOperations.md
@@ -191,8 +191,11 @@ An operator of which the usefulness is not immediately clear, but which fits wel
The **transpose** of an $m \times n$ matrix $A$ with entries $a_{ij}$ is the
$n \times m$ matrix $B$ with entries $b_{ij}$ defined by
-<!-- prettier-ignore -->
-$ b_{ij} = a_{ji}$. It is denoted by $B = A^T$.
+$$
+ b_{ij} = a_{ji}, \quad i = 1,\ldots,n,\,\,j=1,\ldots, m
+$$
+
+It is denoted by $B = A^T$.
::::::
@@ -239,7 +242,9 @@ $(A^T)^T = A$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixOps:Transpose`
+:class: myproof
+
We will prove the second statement and leave the other two to the diligent reader. See
{numref}`Exc:MatrixOps:CheckTransposeRules`.
@@ -334,7 +339,8 @@ Prove statements (i) and (iii) of {prf:ref}`Prop:MatrixOps:Transpose`.
::::::
-::::::{dropdown} Solution to {numref}`Exc:MatrixOps:CheckTransposeRules` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:MatrixOps:CheckTransposeRules`
+:class: solution, dropdown
Suppose $A = \left[\begin{array}{cccc}
a_{11} & a_{12}& \ldots& a_{1n} \\
@@ -403,6 +409,7 @@ $$
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bc898154-3f5e-45bd-8993-28a74bf34b5f?id=70278
:label: grasple_exercise_3_2_1
:dropdown:
@@ -411,6 +418,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bf170c2b-127b-4ce7-bd75-c9c9bdfb12f9?id=70277
:label: grasple_exercise_3_2_2
:dropdown:
@@ -419,6 +427,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dd83bd83-0ce4-4dd7-84de-3472c24acbc0?id=70279
:label: grasple_exercise_3_2_3
:dropdown:
@@ -427,6 +436,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3e5f0674-1e9f-4349-867f-6b1d638e744b?id=82934
:label: grasple_exercise_3_2_4
:dropdown:
@@ -435,6 +445,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/52a8c0e8-09e8-4c46-aa43-0cee2a93e7c4?id=82931
:label: grasple_exercise_3_2_5
:dropdown:
@@ -452,7 +463,6 @@ $$
B = \mathbf{x} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},
$$
-
a vector in $\mathbb{R}^n$, which we can identify with an $n \times 1$ matrix. We want of course the definition of the general matrix product to be consistent with this.
::::::{prf:definition}
@@ -506,7 +516,9 @@ This is sometimes called the **row-column expansion** of the product.
::::::
-::::::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:MatrixOps:RowColExpansion`
+:class: myproof
+
We already saw this row-column expansion in {numref}`Sec:MatVecProduct`.
::::::
@@ -517,16 +529,16 @@ $$
\begin{array}{ccc}
&
\begin{bmatrix}
- b_{11} & b_{12}& \ldots& {\color{red}b_{1j}} & \ldots& b_{1p} \\
- b_{21} & b_{22}& \ldots& {\color{red}b_{2j}} & \ldots& b_{2p} \\
+ b_{11} & b_{12}& \ldots& \class{red}{b_{1j}} & \ldots& b_{1p} \\
+ b_{21} & b_{22}& \ldots& \class{red}{b_{2j}} & \ldots& b_{2p} \\
\vdots & \vdots& \ldots& & \ldots& \vdots \\
- b_{n1} & b_{n2}& \ldots& {\color{red}b_{nj}} & \ldots& b_{np}
+ b_{n1} & b_{n2}& \ldots& \class{red}{b_{nj}} & \ldots& b_{np}
\end{bmatrix} \\
\begin{bmatrix}
a_{11} & a_{12}& \ldots& \ldots& a_{1n} \\
a_{21} & a_{22}& \ldots& \ldots& a_{2n} \\
\cdots & \ddots& \ldots& \ldots& \vdots \\
- {\color{red}a_{i1}} & {\color{red}a_{i2}}& {\color{red}\cdots}& \ldots& {\color{red}a_{in}} \\
+ \class{red}{a_{i1}} & \class{red}{a_{i2}}& \class{red}{\cdots}& \ldots& \class{red}{a_{in}} \\
\vdots & \vdots& \ldots& \ldots& \vdots \\
a_{m1} & a_{m2}& \ldots& \ldots& a_{mn}
\end{bmatrix} \!\! & \!
@@ -534,7 +546,7 @@ $$
c_{11} & c_{12}& \ldots& c_{1j} &\ldots& c_{1p} \\
c_{21} & c_{22}& \ldots& c_{2j} &\ldots& c_{2p} \\
\vdots & \vdots& \ldots& & \ldots& \vdots \\
- c_{i1} & c_{i2}& \cdots&{\color{red}c_{ij}} &\ldots& c_{ip} \\
+ c_{i1} & c_{i2}& \cdots&\class{red}{c_{ij}} &\ldots& c_{ip} \\
\vdots & \vdots& \ldots& &\ldots& \vdots \\
c_{m1} & c_{m2}& \ldots& c_{mn} &\ldots& c_{mp}
\end{bmatrix}
@@ -709,7 +721,9 @@ The $i$-th row of the product $AB$ is the linear combination of the rows of the
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixOps:ProductRowCombinations`
+:class: myproof
+
The indicated linear combination yields:
$$
@@ -931,8 +945,9 @@ We need a good perspective to give a proof of the general case.
::::::
-::::::{prf:proof}
-(of {prf:ref}`Prop:MatrixOps:ProdProperties`)
+:::{admonition} Proof of {prf:ref}`Prop:MatrixOps:ProdProperties`
+:class: myproof
+
Rules i. and ii. are checked in a straightforward way. See {numref}`Exc:MatrixOps:RulesProduct`.
<ol type = "i" start = "3">
@@ -1133,8 +1148,9 @@ $$
::::{figure} Images/Fig-MatrixOps-NonCommutativity.svg
:name: Fig:MatrixOps:NonCommutativity
+:class: dark-light
-$ \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1&0 \end{bmatrix} \neq \begin{bmatrix} 0 & 1 \\ 1&0 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$
+$ \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1&0 \end{bmatrix} \neq \begin{bmatrix} 0 & 1 \\ 1&0 \end{bmatrix}\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$.
::::
Note that $T_A$ is a transformation that 'stretches' horizontally, and $T_B$ is a reflection. {numref}`Figure %s <Fig:MatrixOps:NonCommutativity>` visualizes the transformations corresponding to $AB$ and $BA$. When we apply the transformations one after another, the order in which we do this is important.
@@ -1257,13 +1273,14 @@ So $A^2\vect{x} = T_2(T_1(T_2(T_1(\vect{x})))) = \vect{0}$, for each vector $\ve
See {numref}`Figure %s <Fig:MatrixOps:NilPotent>`.
-:::::{figure} Images/Fig-MatrixOps-Nilpotent.svg
+```{applet}
+:url: matrix_operations/nilpotent
+:fig: Images/Fig-MatrixOps-Nilpotent.svg
:name: Fig:MatrixOps:Nilpotent
+:class: dark-light
Visualisation of $\vect{x} \mapsto A^2\vect{x}$.
-:::::
-
-::::::
+```
::::::{prf:remark}
The next list gives six situations where matrix multiplication acts differently than multiplication of numbers.
@@ -1365,7 +1382,7 @@ $$
AB = O,
$$
-and let $C$ be the zero matrix.
+and let $C$ be the zero matrix.
Then $B \neq C$, whereas
$$
@@ -1410,6 +1427,8 @@ Give a $2 \times 2$ matrix $B$ for which $B^2 = -I$.
The following property connects the two operations matrix transposition and matrix multiplication.
::::::{prf:proposition}
+:label: Prop:MatrixOperations:TransposeProduct
+
If $A$ is an $m\times n$ matrix and $B$ an $n\times p$ matrix, then
$$
@@ -1467,7 +1486,9 @@ $$
As {prf:ref}`Ex:TransposeProduct` illustrates the rule is not restricted to square matrices $A$ and $B$.
The proof for general matrices $A$ and $B$ for which the product $AB$ is well defined is as follows
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:MatrixOperations:TransposeProduct`
+:class: myproof
+
To show that
$$
@@ -1545,6 +1566,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/262bcea8-548b-45c2-8c37-b4cb3cb03ddc?id=70281
:label: grasple_exercise_3_2_6
:dropdown:
@@ -1553,6 +1575,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/718bda8a-9e75-495a-8aea-506788d46432?id=70282
:label: grasple_exercise_3_2_7
:dropdown:
@@ -1561,6 +1584,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e5799b3f-53f6-4095-bb96-bc2f4febde30?id=70284
:label: grasple_exercise_3_2_8
:dropdown:
@@ -1569,6 +1593,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9d1526f4-777b-4a41-8b8e-c0746f7503c9?id=70285
:label: grasple_exercise_3_2_9
:dropdown:
@@ -1577,6 +1602,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d03c79a5-4936-41ae-8129-96ea9dee875a?id=82963
:label: grasple_exercise_3_2_10
:dropdown:
@@ -1585,6 +1611,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9fd59a3b-bdc6-42c5-af90-da9b0541437b?id=70291
:label: grasple_exercise_3_2_11
:dropdown:
@@ -1593,6 +1620,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6e4d152b-1eae-480b-a40c-ca8846ed6612?id=70286
:label: grasple_exercise_3_2_12
:dropdown:
@@ -1601,6 +1629,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/65f960ef-01a1-4c81-b053-8c93c66504db?id=70287
:label: grasple_exercise_3_2_13
:dropdown:
@@ -1609,6 +1638,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d2ccfcf5-7aaf-4859-8219-392abad68e79?id=82853
:label: grasple_exercise_3_2_14
:dropdown:
@@ -1617,6 +1647,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2fc08e2c-b3ad-4a2b-8077-ce66abc466d7?id=82936
:label: grasple_exercise_3_2_15
:dropdown:
@@ -1625,6 +1656,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6bd96baf-1862-40c7-a21d-24c1dada9078?id=82937
:label: grasple_exercise_3_2_16
:dropdown:
@@ -1635,6 +1667,7 @@ We will dedicate {numref}`Sec:MatrixInv` to this topic.
The remaining exercises are less of a compuational character.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/786324ef-8706-4f4d-ac06-f6b4360a70d8?id=69285
:label: grasple_exercise_3_2_17
:dropdown:
@@ -1643,6 +1676,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/14b6af51-de5f-4e6c-bfb8-20a018fce053?id=69458
:label: grasple_exercise_3_2_18
:dropdown:
@@ -1651,6 +1685,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cdb5014d-eace-489e-9616-45e03bb6e95e?id=69295
:label: grasple_exercise_3_2_19
:dropdown:
@@ -1659,6 +1694,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7f91a5d2-e1c9-422e-b0f9-ba0b22936e2a?id=69456
:label: grasple_exercise_3_2_20
:dropdown:
@@ -1667,6 +1703,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/78c129ac-644d-4fbd-bf47-c2283d0e1f7a?id=69460
:label: grasple_exercise_3_2_21
:dropdown:
@@ -1675,6 +1712,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9fe5f92d-54f9-4794-a2e8-c21a24a5a8cf?id=70288
:label: grasple_exercise_3_2_22
:dropdown:
@@ -1683,14 +1721,16 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bbed8637-4110-4e90-a1dc-a5960a405caf?id=70289
:label: grasple_exercise_3_2_23
:dropdown:
-:description: Number of columns of $C$ is $AC=B$.
+:description: Number of columns of $C$ if $AC=B$.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/eb4b9e6e-0436-466c-bb1f-7e596b43ec34?id=70290
:label: grasple_exercise_3_2_24
:dropdown:
@@ -1699,6 +1739,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/deea79ca-ba41-46fc-b75b-4cd109fc0513?id=71118
:label: grasple_exercise_3_2_25
:dropdown:
@@ -1707,6 +1748,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/0b27fa70-e097-4090-b57e-7225019a4624?id=78589
:label: grasple_exercise_3_2_26
:dropdown:
@@ -1715,6 +1757,7 @@ The remaining exercises are less of a compuational character.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/958761d7-421f-40f1-b3be-3535bf71422b?id=82968
:label: grasple_exercise_3_2_27
:dropdown:
diff --git a/Chapter4/BasisAndDimension.md b/Chapter4/BasisAndDimension.md
index a36cec8..6bea066 100644
--- a/Chapter4/BasisAndDimension.md
+++ b/Chapter4/BasisAndDimension.md
@@ -163,8 +163,9 @@ with $\vect{e}_1$ pointing to the right and $\vect{e}_2$ pointing upwards. Likew
:url: basisdim/standardbasis
:fig: Images/Fig-BasisDim-StandardBasis.svg
:name: Fig:BasisDim:StandardBasis
+:class: dark-light
-The standard bases in $\R^2$ and $\R^3$
+The standard bases in $\R^2$ and $\R^3$.
```
::::::
@@ -199,7 +200,8 @@ is indeed a basis for $\R^4$.
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:CheckStandardBasis` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:CheckStandardBasis`
+:class: solution, dropdown
The set is linearly independent:
@@ -235,6 +237,7 @@ $\vect{x} = x_1\vect{e}_1 + x_2\vect{e}_2 + x_3\vect{e}_3 + x_4\vect{e}_4$.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7ecdf0d8-e529-4589-82f6-7207f229cd87?id=88187
:label: grasple_exercise_4_2_A
:dropdown:
@@ -298,7 +301,8 @@ So, there exists a set $\mathcal{B}$ containing $\mathcal{A}$ which is a basis f
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:Thinning`
+:class: myproof
The ideas are quite straightforward.
@@ -462,6 +466,7 @@ spans the null space. It is also linearly independent, so it is a basis.
::::{figure} Images/Fig-BasisDim-BasisColA.svg
:name: Fig:BasisDim:BasisColA
+:class: dark-light
The basis $\lbrace\vect{a}_1,\vect{a}_3\rbrace$ of $\Col{A}$.
::::
@@ -469,6 +474,7 @@ The basis $\lbrace\vect{a}_1,\vect{a}_3\rbrace$ of $\Col{A}$.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6c836722-f664-434d-bf7d-3a4f86ff0187?id=88188
:label: grasple_exercise_4_2_B
:dropdown:
@@ -580,7 +586,8 @@ The null space of $A$ is equal to the null space of $E$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:PivotColsBasis`
+:class: myproof
The main idea of the first statement: row operations do not alter the linear relations between the columns of a matrix. Now why is this so?
Let
@@ -755,7 +762,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:TF-ColA-ColE` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:TF-ColA-ColE`
+:class: solution, dropdown
The statement is _false_. <BR>
For instance, look at {prf:ref}`Ex:BasisDim:FourByFour`. In that example all vectors in $\Col{(E)}$ have a zero on position four, and there are (many) vectors in $\Col{(A)}$ that have a nonzero fourth entry. So definitely $\Col{(A)} \neq \Col{(E)}$.
@@ -822,7 +830,8 @@ Find a basis for the null space of the matrix in the previous example.
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:FinishExampleNulspace` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:FinishExampleNulspace`
+:class: solution, dropdown
We row reduce the matrix $E = \begin{bmatrix}
1 & 1 & 2 & 1 & -1 \\
@@ -867,14 +876,16 @@ Every basis of a fixed subspace $S$ in $\R^n$ has the same number of elements.
::::{margin}
-:::{admonition} {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep`.
+:::{admonition} {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep`
+:class: theorem
Let $\vect{u}_{1},...,\vect{u}_{k}$ and $\vect{v}_{1},...,\vect{v}_{\ell}$ be vectors in $\R^{n}$.
If $ k< \ell$ and $\Span{\vect{u}_{1},...,\vect{u}_{k}}$ contains $\Span{\vect{v}_{1},...,\vect{v}_{\ell}}$ then the set $\left\lbrace\vect{v}_{1},...,\vect{v}_{\ell}\right\rbrace$ is linearly dependent.
::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:BasisDim:EqualDim`
+:class: myproof
The proof is an immediate consequence of {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep`
from the section on linear independence. Because of its vital important we restate it here (check the side note).
@@ -1080,7 +1091,8 @@ The dimension of the vector $\vect{u} = \begin{bmatrix} 3\\4 \end{bmatrix} $ is
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:TF:DimOfVector` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:TF:DimOfVector`
+:class: solution, dropdown
The statement is _false_. The attribute dimension is only defined for _subspaces_.
@@ -1121,7 +1133,8 @@ Another way to put it: once it is known that the dimension of $S$ equals $k$, ea
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:TwoOfThreeSuffice`
+:class: myproof
We first show the "easy" part:
@@ -1257,7 +1270,9 @@ $$
::::::
-::::::{prf:proof}
+
+::::::{admonition} Proof of {prf:ref}`Thm:BasisDim:DimensionTheorem`
+:class: myproof
The proof consists of combining several properties of this section.
@@ -1430,7 +1445,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:EquivalentMatricesEqualRowspaces`
+:class: myproof
Recall that equivalence here means that by row operations we can transform $A$ into $B$ (and vice versa). So we have to show that row operations do not change the row space.
@@ -1457,7 +1473,8 @@ For each matrix $A$ the row space and the column space have the same dimension.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:EqualDimRowColSpace`
+:class: myproof
If a matrix $A$ is row reduced to an echelon matrix $E$ we know that
@@ -1576,7 +1593,8 @@ In the above {prf:ref}`Ex:BasisDim:Rowspace4x3` find out how the four rows of th
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:ExpressInRowsOfE` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:ExpressInRowsOfE`
+:class: solution, dropdown
It can be done 'by inspection'
@@ -1623,7 +1641,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:TF:RankAEqualsRankAT` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:TF:RankAEqualsRankAT`
+:class: solution, dropdown
The rank of $A^T$ is the dimension of the column space of $A^T$, which is the dimension of the row space of $A$. <BR>
The rank of $A$ is the dimension of the column space of $A$.
@@ -1680,7 +1699,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:BasisDim:RankABLeqRankA` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:BasisDim:RankABLeqRankA`
+:class: solution, dropdown
The statement in {numref}`Exc:Subspaces:ColABinColA` that says $\Col{AB} \subseteq \Col{A}$ immediately gives that
@@ -1714,7 +1734,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:BasisDim:RankAPEqualToRankPA`
+:class: myproof
Suppose $A$ and $P$ are as stated. Then by {numref}`Exc:BasisDim:RankABLeqRankA`
@@ -1765,6 +1786,7 @@ $B$ is an $n\times m$ matrix?
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9387d1c3-03b4-41ff-b7b6-bb0d0ee3c771?id=70632
:label: grasple_exercise_4_2_1
:dropdown:
@@ -1773,6 +1795,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7bf4ef59-9f95-4697-9309-f06078856988?id=70633
:label: grasple_exercise_4_2_2
:dropdown:
@@ -1781,6 +1804,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2ea2e118-a471-4688-9d89-87cd49cfddc2?id=70634
:label: grasple_exercise_4_2_3
:dropdown:
@@ -1789,6 +1813,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1c00677f-eadb-4a13-84aa-5c7b07774f21?id=88189
:label: grasple_exercise_4_2_4
:dropdown:
@@ -1797,6 +1822,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ae1a690e-1e4d-40be-94d7-cf91121134f0?id=70649
:label: grasple_exercise_4_2_5
:dropdown:
@@ -1805,6 +1831,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9440dca7-7d2d-4f74-8679-c966be28d73f?id=70638
:label: grasple_exercise_4_2_6
:dropdown:
@@ -1813,6 +1840,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3d5483ab-4a1f-4caf-b979-9a4518551416?id=70640
:label: grasple_exercise_4_2_7
:dropdown:
@@ -1821,6 +1849,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b85a9ef2-3b82-47d9-89f3-212a435b8be2?id=70642
:label: grasple_exercise_4_2_8
:dropdown:
@@ -1829,6 +1858,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/04d804ff-16ea-44ee-ac76-452a73a88859?id=70653
:label: grasple_exercise_4_2_9
:dropdown:
@@ -1837,6 +1867,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/789d912e-6ac2-4f48-b92b-11ab33d5c949?id=70655
:label: grasple_exercise_4_2_10
:dropdown:
@@ -1845,6 +1876,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/610e51bc-c62f-4da2-81f3-eeef106bba84?id=83409
:label: grasple_exercise_4_2_11
:dropdown:
@@ -1853,6 +1885,7 @@ $B$ is an $n\times m$ matrix?
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/98aae0f1-3cd9-43d8-aa9f-7c884f2c9527?id=83411
:label: grasple_exercise_4_2_12
:dropdown:
@@ -1863,6 +1896,7 @@ $B$ is an $n\times m$ matrix?
The exercises below are more theoretical.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9dc4b61f-1ded-496c-82e1-8413bc0fa5e7?id=70644
:label: grasple_exercise_4_2_13
:dropdown:
@@ -1871,6 +1905,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dc1ffd49-1c32-407a-9042-f386b85c771d?id=70645
:label: grasple_exercise_4_2_14
:dropdown:
@@ -1879,6 +1914,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3576e4c9-2084-4e3f-af61-08f09d63ad82?id=70646
:label: grasple_exercise_4_2_15
:dropdown:
@@ -1887,6 +1923,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/37a433f6-b15f-4b8a-8232-4098fe82e6c9?id=70647
:label: grasple_exercise_4_2_16
:dropdown:
@@ -1895,6 +1932,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2940572e-f3bf-40e9-bf01-c2244c6f9aa5?id=70648
:label: grasple_exercise_4_2_17
:dropdown:
@@ -1903,6 +1941,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2f4a3540-9454-4b49-ac4c-0aae22fd5b50?id=70657
:label: grasple_exercise_4_2_18
:dropdown:
@@ -1911,6 +1950,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b07a06d8-ea95-4798-9ae7-7d4bdea040de?id=70658
:label: grasple_exercise_4_2_19
:dropdown:
@@ -1919,6 +1959,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/547dd05d-c9da-4e7a-8deb-e4e8715e0f02?id=70660
:label: grasple_exercise_4_2_20
:dropdown:
@@ -1927,6 +1968,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e8051821-b278-4895-ad3e-bc40ba99e1dc?id=70659
:label: grasple_exercise_4_2_21
:dropdown:
@@ -1935,6 +1977,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/edb07654-fb52-4049-af08-e6333fd2d96e?id=83377
:label: grasple_exercise_4_2_22
:dropdown:
@@ -1943,6 +1986,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/af70726a-1610-4515-8f43-c11e109ca5cd?id=83396
:label: grasple_exercise_4_2_23
:dropdown:
@@ -1951,6 +1995,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3a957728-2365-4e80-820b-2d0f9dcf6ec3?id=83399
:label: grasple_exercise_4_2_24
:dropdown:
@@ -1959,6 +2004,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/429fda86-ad72-402d-ab47-58ae103b36fc?id=83402
:label: grasple_exercise_4_2_25
:dropdown:
@@ -1967,6 +2013,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/608b1140-590f-468c-b704-eb922aa7fca4?id=83407
:label: grasple_exercise_4_2_26
:dropdown:
@@ -1975,6 +2022,7 @@ The exercises below are more theoretical.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d88ce866-54b2-48aa-bfe2-26116c2c6c34?id=83404
:label: grasple_exercise_4_2_27
:dropdown:
diff --git a/Chapter4/ChangeOfBasis.md b/Chapter4/ChangeOfBasis.md
index cdcf616..1cfc940 100644
--- a/Chapter4/ChangeOfBasis.md
+++ b/Chapter4/ChangeOfBasis.md
@@ -29,8 +29,9 @@ See {numref}`Figure %s <Fig:ChangeOfBasis:Reflection>`.
:::{figure} Images/Fig-ChangeOfBasis-Reflection.svg
:name: Fig:ChangeOfBasis:Reflection
+:class: dark-light
-Reflection along the line $\mc{L}$
+Reflection along the line $\mc{L}$.
:::
From the geometry involved it follows that
@@ -73,7 +74,8 @@ for _unique_ constants $c_1,c_2,\ldots,c_m$ in $\R$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:ChangeOfBasis:UniqueCoords`
+:class: myproof
From the definition of a basis it follows that
@@ -185,11 +187,14 @@ $$
{numref}`Figure %s <Fig:ChangeOfBasis:AlternativeBasis>` may help to see what is going on geometrically.
-:::{figure} Images/Fig-ChangeOfBasis-AlternativeBasis.svg
+```{applet}
+:url: change_of_basis/alternative_basis
+:fig: Images/Fig-ChangeOfBasis-AlternativeBasis.svg
:name: Fig:ChangeOfBasis:AlternativeBasis
+:class: dark-light
The basis $\{\vect{b}_1,\vect{b}_2\}$ of {prf:ref}`Ex:ChangeOfBasis:BasicExampleR2`.
-::::
+```
::::::{prf:remark}
:label: Rem:ChangeOfBasis:ConventionBasis
@@ -318,7 +323,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:ChangeOfBasis:ToStandardBasis`
+:class: myproof
Suppose
@@ -346,7 +352,8 @@ $$
Show that every change-of-coordinates matrix $P_{\mc{B}}$ is invertible.
::::
-::::{dropdown} Solution to {numref}`Exc:ChangeOfBasis:InvertiblePB` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ChangeOfBasis:InvertiblePB`
+:class: solution, dropdown
Let $\mc{B} = \{\vect{b}_1,\vect{b}_2, \ldots, \vect{b}_n\}$ be any basis of $\R^n$.
<BR>
@@ -461,7 +468,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:ChangeOfBasis:MatrixOfTrafo`
+:class: myproof
We use the definition of the coordinate vector and the linearity of the transformation. In fact we can play copycat with the proof of
{prf:ref}`Thm:LinTrafo:LinTrafo=MatrixTrafo` in the section of linear transformations.
@@ -622,6 +630,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d6aa929c-7614-4af2-b015-31e2e6fd7b80?id=104847
:label: grasple_exercise_4_3_T1
:dropdown:
@@ -676,8 +685,9 @@ The vector $\vect{b}_1+\vect{b}_2 = \begin{bmatrix} 1 \\ \sqrt{3} \end{bmatrix}
:::{figure} Images/Fig-ChangeOfBasis-TriangularGrid.svg
:name: Fig:ChangeOfBasis-TriangularGrid
+:class: dark-light
-Rotation in disguise
+Rotation in disguise.
:::
@@ -877,7 +887,8 @@ $$
::::
-::::{dropdown} Solution to {numref}`Exc:ChangeOfBasis:PinvAPversusPBinvP` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ChangeOfBasis:PinvAPversusPBinvP`
+:class: solution, dropdown
Starting from $B = P^{-1}A\,P$ multiply both sides from the left by $P$ and from the right by $P^{-1}$,
and use that $P^{-1}P = PP^{-1} = I$:
@@ -904,6 +915,7 @@ See {numref}`Figure %s <Fig:ChangeOfBasis:Projection>`.
:url: change_of_basis/projection
:fig: Images/Fig-ChangeOfBasis-Projection.svg
:name: Fig:ChangeOfBasis:Projection
+:class: dark-light
Projection with respect to a suitable basis.
```
@@ -959,7 +971,8 @@ $A^2=A$. Show that the matrix $[T]_{\mc{E}}$ of {prf:ref}`Ex:ChangeOfBasis:Matri
::::
-::::{dropdown} Solution to {numref}`Exc:ChangeOfBasis:CheckIdempotent` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ChangeOfBasis:CheckIdempotent`
+:class: solution, dropdown
Obviously the matrix $B = [T]_{\mc{B}} = \begin{bmatrix}1 & 0 & 0\\0& 1& 0\\0&0&0 \end{bmatrix}$ has the property $B^2 = B$.
@@ -1013,7 +1026,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:ChangeOfBasis:CoBmatrix`
+:class: myproof
Again we will make use of the identity
@@ -1067,7 +1081,8 @@ P_{\mc{B}\leftarrow\mc{D}} \quad \text{and} \quad P_{\mc{C}\leftarrow\mc{D}}$?
::::
-::::{dropdown} Solution to {numref}`Exc:ChangeOfBasis:RelationMatricesCoB` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:ChangeOfBasis:RelationMatricesCoB`
+:class: solution, dropdown
The defining relation of a change-of-coordinates matrix like $P_{\mc{B}\leftarrow\mc{D}}$ is that for every vector $\vect{x}$ in $\R^n$ we have
@@ -1127,7 +1142,8 @@ $$
\end{array}
$$
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Thm:ChangeOfBasis:MatrixChangeGeneralBasis`
+:class: myproof
<!-- prettier-ignore -->
To find $[T(\vect{x})]_{\mc{C}'}$ when $ [\vect{x}]_{\mc{B}'}$ is given one can either
@@ -1194,6 +1210,7 @@ must be equal.
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b19d38ef-60cf-4581-a4b2-9aa2efcf6f30?id=90868
:label: grasple_exercise_4_3_1
:dropdown:
@@ -1202,6 +1219,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3008a6aa-58ee-4182-8fd1-9d452ac3e9f0?id=90867
:label: grasple_exercise_4_3_2
:dropdown:
@@ -1210,6 +1228,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/18d96cbf-6158-4800-9a10-3ec7f6e933f8?id=90885
:label: grasple_exercise_4_3_3
:dropdown:
@@ -1218,6 +1237,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dfad4903-b2a8-4f67-979e-5651cf4072ec?id=90872
:label: grasple_exercise_4_3_4
:dropdown:
@@ -1226,6 +1246,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d6e53d94-578e-48fe-96b6-aca26f4eca1c?id=90870
:label: grasple_exercise_4_3_5
:dropdown:
@@ -1234,6 +1255,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/75c6d18b-8a54-4592-97f4-edd77169cc10?id=90876
:label: grasple_exercise_4_3_6
:dropdown:
@@ -1242,6 +1264,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c45ea239-1e90-4d1f-ae08-323e595bd53a?id=104827
:label: grasple_exercise_4_3_7
:dropdown:
@@ -1253,6 +1276,7 @@ must be equal.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/39e1af19-0a32-456b-a414-20056e6b7f16?id=85157
:label: grasple_exercise_4_3_8
:dropdown:
@@ -1261,6 +1285,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b03d9983-3ef2-4d45-82b3-6c1762510561?id=90875
:label: grasple_exercise_4_3_9
:dropdown:
@@ -1269,6 +1294,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/784bba91-a0be-4076-9918-63b8ab2fbc49?id=90881
:label: grasple_exercise_4_3_10
:dropdown:
@@ -1276,6 +1302,7 @@ must be equal.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d8894445-4426-4694-8500-229cd47a5288?id=85165
:label: grasple_exercise_4_3_11
:dropdown:
@@ -1286,6 +1313,7 @@ must be equal.
The remaining exercises are about matrix representations of linear transformations.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2de2a3b5-1d3f-4e79-9421-393d59b9dc87?id=93047
:label: grasple_exercise_4_3_12
:dropdown:
@@ -1294,6 +1322,7 @@ The remaining exercises are about matrix representations of linear transformatio
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/01e5c371-0139-458c-8f0a-25f35bc03fcb?id=93053
:label: grasple_exercise_4_3_13
:dropdown:
@@ -1302,6 +1331,7 @@ The remaining exercises are about matrix representations of linear transformatio
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dd1af96d-34d7-407e-9877-c1e8b6495e6f?id=85167
:label: grasple_exercise_4_3_14
:dropdown:
@@ -1310,6 +1340,7 @@ The remaining exercises are about matrix representations of linear transformatio
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/933f3e07-36b6-4db5-a47d-91e276185269?id=85159
:label: grasple_exercise_4_3_15
:dropdown:
@@ -1318,6 +1349,7 @@ The remaining exercises are about matrix representations of linear transformatio
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7985ac3f-c432-4b48-9ef1-8ce0914b0f97?id=85162
:label: grasple_exercise_4_3_16
:dropdown:
diff --git a/Chapter4/Images/Fig-TheGame.svg b/Chapter4/Images/Fig-TheGame.svg
new file mode 100644
index 0000000..02444d2
--- /dev/null
+++ b/Chapter4/Images/Fig-TheGame.svg
@@ -0,0 +1,16 @@
+<svg width="666" height="329" viewBox="0 0 666 329" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<mask id="path-1-outside-1_1_2" maskUnits="userSpaceOnUse" x="412" y="125" width="190" height="84" fill="black">
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uiN4KuUUXQNX4Ius4PQdcZOhMQdJ0mgq4zJEEjgKAj6Ai6toYatDeCruFG0DV+RfRG0DWqCLrGD0HX+SHoOkNnAoKu00TQdYYkaAQQdAQdQdfWUIP2RtA13Ai6xq+I3gi6RhVB1/gh6Do/BF1n6ExA0HWaCLrOkASNAIKOoCPo2hpq0N4IuoYbQdf4FdEbQdeoIugaPwRd54eg6wydCQi6ThNB1xmSoBFA0BF0BF1bQw3aG0HXcCPoGr8ieiPoGlUEXeOHoOv8EHSdoTMBQddpIug6QxI0Agg6go6ga2uoQXsj6BpuBF3jV0RvBF2jiqBr/BB0nR+CrjN0JiDoOk0EXWdIgkYAQUfQEXRtDTVobwRdw42ga/yK6I2ga1QRdI0fgq7zQ9B1hs4EBF2niaDrDEnQCCDoCDqCrq2hBu2NoGu4EXSNXxG9EXSNKoKu8UPQdX4Ius7QmYCg6zQRdJ0hCRoBBB1BR9C1NdSgvRF0DTeCrvErojeCrlFF0DV+CLrOD0HXGToTEHSdJoKuMyRBI4CgI+gIuraGGrQ3gq7hRtA1fkX0RtA1qgi6xg9B1/kh6DpDZwKCrtNE0HWGJGgEEHQEHUHX1lCD9kbQNdwIusaviN4IukYVQdf4Ieg6PwRdZ+hMQNB1mgi6zpAEjQCCjqAj6Noaes/eW7duja5du9q+oeyCvmnTpjj99NNtxztv3rwYN26cLa9Lly6xbds2W17btm1j7969trw8IdfU1NjyMmjt2rVx9tln2zI3b94cPXr0sOVdfPHF8cwzz9jyBg4cGC+++KIt74477oibbrrJlvf1r389pk2bZss7//zz46WXXrLl9e3bt3bOuFrnzp2ta+7QoUPRokUL1/Bqc1q2bBmZ62qrV6+Ofv36ueLiwgsvjBdeeMGW554zOb7nn3/eNr5cb7nuXM19znKN6+0c9/iqQdBXrVoV5557rg1lz549I+8/XO173/tefPazn3XF2XPKfi/ovve1A6yCQAT9nUVqVuO+O27ASfDEE0/ERz/6Ues3unG0adPGOr4yhx07dizuv//+uPrqq23DLPtJ+d57740bb7wxTjrpJMsxX3DBBfGTn/zEkpUhGzZsqJXL5s2bWzJ37NgR7dq1s+WlJHTr1s2Wd/jw4Tj11FPjyJEjluPNvBSuN954w5KXayQfYG3fvt2Wd+utt8YXvvAFS16GTJ48OR5++GFb3jnnnBMpcK410qdPn3jllVdsee3bt49du3bZ8lq3bh27d++2zem8acl14lrDOac3btwYzmvTWWedZXtwl2vkBz/4QXzkIx+xzMGjR4/GVVddFUuXLrXk5fhyTq9bt86SlyF5H5M7mK42e/bs+OM//mPbnD7jjDPiF7/4hWt4sWLFivjwhz9sm9P5ACvntKvlGsnr0imnnGKJzLw85g9+8IO2vHyw6LyO/OM//mP89//+3y3jKyKk7PeCc+fOjWuvvda25opg6M7ct2+fNRJBR9Dfc0IVIej79++3TuIyhz344INNTtCvv/56W0nyRtJ5o2YbWEFBud5c4vb2EFOA82m2q/Xv3z9y98PVhg4dapOFHNOsWbNi4sSJruHFzTffHHfeeact77LLLouFCxfa8sr+ivuZZ55Z+wDB1VKk3TdCe/bsiXwbxtVyZ3DNmjWuuFi0aFGMHDnSlnfNNdfEAw88YMsr+yvuKeiTJk2yHe+YMWNi/vz5tryU1cGDB9vyunfvHrkr72x5XXLeD+YDhNz1drW8jixbtswVV3vfkfcfZW3VIOjjx48vKz77uPJBtPu6hKC/s0zsoP/GtHWekDM6b64Q9Pd/bij7STl30BH0919fBP39s3u7J4KuMcy3VpyvUyPoWj2yN4KuMUTQNX7ZG0HXGToTyn4vmDvoCLpWcQQdQX/PGYSgawuMHXSNHzvoGr/szQ66xpAddI0fgq7xQ9B1fgi6zhBB1xk6ExB0J009ix10nWFdCeyg/wYhBL2uKfPeP0fQNX4IusYPQdf5IegaQwRd44eg6/wQdJ0hgq4zdCYg6E6aehaCrjOsKwFBR9DrmiMn9HME/YRwvevDCLrGD0HX+SHoGkMEXeOHoOv8EHSdIYKuM3QmIOhOmnoWgq4zrCsBQUfQ65ojJ/RzBP2EcCHo/JE4bcLwR+JkfvwOuoyw9n8fxR+Je/8c3f+bNQT9/dfi7Z4Ius7QmYCgO2nqWQi6zrCuBAQdQa9rjpzQzxH0E8KFoCPo2oRB0GV+CLqMEEEXESLoGkD+irvGL3vzV9w1hvyROI1f9uaPxL2TIYKOoOur6rgEBF3DySvuGr/szR+J0xjyirvGj1fcNX7Zm7/irjFkB13jl73ZQdcZOhPYQXfS1LPYQdcZ1pWAoCPodc2RE/o5gn5CuN71YQRd44eg6/wQdI0hgq7xQ9B1fgi6zhBB1xk6ExB0J009C0HXGdaVgKAj6HXNkRP6OYJ+QrgQdF5x1yYMr7jL/HjFXUbIK+4iQl5x1wDyirvGL3vzirvGkFfcNX7Zm1fc38kQQUfQ9VV1XAKCruFkB13jl715xV1jyA66xo8ddI1f9uYVd40hO+gav+zNDrrO0JnADrqTpp7FDrrOsK4EBB1Br2uOnNDPEfQTwvWuDyPoGj8EXeeHoGsMEXSNH4Ku80PQdYYIus7QmYCgO2nqWQi6zrCuBAQdQa9rjpzQzxH0E8KFoPOKuzZheMVd5scr7jJCXnEXEfKKuwaQV9w1ftmbV9w1hrzirvHL3rzi/k6GCDqCrq+q4xIQdA0nO+gav+zNK+4aQ3bQNX7soGv8sjevuGsM2UHX+GVvdtB1hs4EdtCdNPUsdtB1hnUlIOgIel1z5IR+jqCfEK53fRhB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0HWGzgQE3UlTz0LQdYZ1JSDoCHpdc+SEfo6gnxAuBJ1X3LUJwyvuMj9ecZcR8oq7iJBX3DWAvOKu8cvevOKuMeQVd41f9uYV93cyRNARdH1VHZeAoGs42UHX+GVvXnHXGLKDrvFjB13jl715xV1jyA66xi97s4OuM3QmsIPupKlnsYOuM6wrAUFH0OuaIyf0cwT9hHC968MIusYPQdf5IegaQwRd44eg6/wQdJ0hgq4zdCYg6E6aehaCrjOsKwFBR9DrmiMn9HME/YRwvevDl1xySSxZskQLOa733r17o23btrY8d9CxY8fi5JNPtsa2adMm9u3bZ8vs3bt3bNiwwZZ37rnnxurVq215s2bNiokTJ9ryyi7o559/frz00ku24+3bt2+sXbvWltejR4/YvHmzLa9FixZx6NAhW14Gbd++PU477TRbZq6RV1991Zb36KOPxtixY215l19+eTz++OO2vFGjRtXu8rta2V9x//CHPxw///nPXYcbP/vZz2L48OG2vLzG7dmzx5aXQW5BX7VqVe2vgrha//79rdeR733ve/HZz37WNTx7DoJuRyoFIugSvnp1RtB/A1NNTU29wNX3QykL+/fvr+/Hq/5zTU3Qr7/++njsscdsdcub5qeffjpatWplyVy2bFn81m/9li3vyJEj8etf/9oytrdFIQWpffv2lsw33ngj7r///vjgBz9oyTt48GCteOQYHS3z8kYtf+/Z1fKmKoXL1T7ykY9E3gy52tSpU+P55593xdXOle9///vRsmVLS+Yvf/nLOPvss215KR9//ud/bsvLh01Hjx61rpHly5fHWWedZeF34MCBGDBgQOTDNkfLNfLNb37T9tApxzdy5MjYsmWLY3i1GRdeeGFMnz7dljdt2rRaaXW1zp07x//5P//HFVf7MOLLX/6y7TqSD5zyn2sN53UpH9p16tTJcsw7d+6sXR/O69KPf/zjuOiiiyzjyzk9cODAyLXiaJmTwpUPJVzt2muvtV5HEHRXZTw5CLqH43ulIOi/QQdB1yZdUxP0e++9N1LSXc39insK+tChQ13Di549e8bGjRttebnenDcFObDcvcw/GtRU2qRJkyJfaXW1O+64I3JHz9VmzJgRKemuNmHChJgzZ44rzp6zZs0a606Z+42QPODcbXS+WZM7g3ncrlb230FPOZ8yZYrrcOPOO++MfHPF1fKNmnyzxtUWLFhgfaMhd3/zQaWr8UfidJLDhg2LZ599Vg96K2HmzJkxefJkWx6CbkNpCULQLRjfMwRB/w08CLo26RB0jR+CrvHL3gi6xhBB1/gh6Bq/7I2gawwRdI1f9na/4p4PtvMBt6vlg/d8AO9qCLpGkr/irvHL3vwV93cyRNARdH1VHZeAoGs4EXSNH4Ku80PQNYYIusYPQdf5Ieg6QwRdY8gOusav7L3ZQS++Qgg6gm6dZQi6hhNB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGr+y9EfTiK4SgI+jWWYagazgRdI0fgq7zQ9A1hgi6xg9B1/kh6DpDBF1jiKBr/MreG0EvvkIIOoJunWUIuoYTQdf4Ieg6PwRdY4iga/wQdJ0fgq4zRNA1hgi6xq/svRH04iuEoCPo1lmGoGs4EXSNH4Ku80PQNYYIusYPQdf5Ieg6QwRdY4iga/zK3htBL75CCDqCbp1lCLqGE0HX+CHoOj8EXWOIoGv8EHSdH4KuM0TQNYYIusav7L0R9OIrhKAj6NZZhqBrOBF0jR+CrvND0DWGCLrGD0HX+SHoOkMEXWOIoGv8yt4bQS++Qgg6gm6dZQi6hhNB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGr+y9EfTiK4SgI+jWWYagazgRdI0fgq7zQ9A1hgi6xg9B1/kh6DpDBF1jiKBr/MreG0EvvkIIOoJunWUIuoYTQdf4Ieg6PwRdY4iga/wQdJ0fgq4zRNA1hgi6xq/svRH04iuEoCPo1lmGoGs4EXSNH4Ku80PQNYYIusYPQdf5Ieg6QwRdY4iga/zK3htBL75CCDqCbp1lCLqGE0HX+CHoOj8EXWOIoGv8EHSdH4KuM0TQNYYIusav7L0R9OIrhKAj6NZZhqBrOBF0jR+CrvND0DWGCLrGD0HX+SHoOkMEXWOIoGv8yt4bQS++Qgg6gm6dZQi6hhNB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGr+y9EfTiK4SgI+jWWYagazgRdI0fgq7zQ9A1hgi6xg9B1/kh6DpDBF1jiKBr/MreG0EvvkIIOoJunWUIuoYTQdf4Ieg6PwRdY4iga/wQdJ0fgq4zRNA1hgi6xq/svRH04iuEoCPo1lmGoGs43YK+ePHiGDlypDao43p36dIltm7dasurqamJvBFyts2bN0f37t2dkaXOmjRpUsyePds2xqYm6Bs3boxevXrZ+D333HMxbNgwW16rVq3iwIEDtrwM2r59e5x22mm2zN69e8err75qy/vXf/3X+PSnP23Lu+aaa+KBBx6w5U2fPj2mTJliy7vzzjvj5ptvtuWVXdBzvmzYsMF2vJ06dYodO3bY8jKo7ILev3//WL16te2Y85z17LPP2vIQdBvKUgYh6MWXBUFH0K2zrKkJ+kMPPRQpra42Z86cWLZsmSsuUlZTgtu0aWPJ3LVrV3To0KH2n6PlTdr//t//2xH1/zNSPM477zxb5sUXXxzNmze35V133XXxB3/wB7a8HN+bb75pyzvzzDPj4YcftuX97u/+brz88su2vJ07d8YZZ5xhy8s1kv9OPvlkS2bKdN7cux48HT58uHZsrrxjx45F/kvxd7SjR49GjvGUU05xxNWO7Y/+6I/ib/7mbyx5GTJkyJA4cuSILW/AgAFW4b/yyiutDzjyfLV06VLb8f7Zn/1Z/PjHP7bl7d27N/7hH/7Bdl3KgX3oQx+yXZfWr19vfYCQ48uHJt26dbMw3L17d+zfv9/Kb8uWLdYH24MGDYr77rvPcrwZ8rWvfS2++tWv2vIya9q0aba8uXPnxvjx4215ZQ9C0IuvEIL+G4xTZpwtxShPpE2lNTVBd9d13rx5MW7cOFts3kiuXLnSlpc703khd7VmzZrV3pCXuQ0ePDhWrFhhG+L8+fNjzJgxtrz151Q8AAAgAElEQVSmtoN+6aWXxpIlS2z8+vbtG2vXrrXlpag6ZTDXiPu6ZDvYt4Lcx3z33XfHDTfcYBsmO+gaygULFsTYsWO1kON6p7wtX77cllcNQfmWzmuvvWYbqvvazg66VhoEXeOXvfNa52yPP/54jB492hnZoFkIOoJunXAIuoYTQdf4FdEbQdeozpgxI6ZOnaqFHNcbQbehtAUh6BrKpvaKO4KuzZfsjaBrDNlB1/ixg67xq09vBB1Br888qfdnEPR6o/pPP4iga/yK6I2ga1QRdI0fO+gav+zNDrrGkB10jV/2ZgddY8gr7ho/d28E3U303XkIOoJunWUIuoYTQdf4FdEbQdeoIugaPwRd44eg6/wQdJ0hgq4xRNA1fu7eCLqbKIJeJ1H37/rxO+h1In/PD5T9pKwd3bt7I+huonoegq4xRNA1fgi6xg9B1/kh6DpDBF1jWPZ7QX4HXatv9uZ30N/JkB3035hTCLq2yNhB1/gh6Bq/Inoj6BpVBF3jh6Br/BB0nR+CrjNE0DWGCLrGz92bHXQ30XfnIegIunWWIegaTgRd41dEbwRdo4qga/wQdI0fgq7zQ9B1hgi6xhBB1/i5eyPobqIIep1E2UGvE9F7fgBB1/gh6Bq/Inoj6BpVBF3jh6Br/BB0nR+CrjNE0DWGCLrGz90bQXcTRdDrJIqg14kIQdcQvWdvBL1AuO8zGkF/n+De6oaga/wQdI0fgq7zQ9B1hgi6xhBB1/i5eyPobqIIep1EEfQ6ESHoGiIE/TgCKR/Hjh0rkKgejaBrDBF0jR+CrvFD0HV+CLrOEEHXGCLoGj93bwTdTRRBr5Mogl4nIgRdQ4SgI+gxZswY2yyaNGlSzJ4925Z3xx13xE033WTLQ9A1lAi6xg9B1/kh6DpDBF1jiKBr/Ny9EXQ3UQS9TqIIep2IEHQNEYKOoCPowhq69NJLY8mSJULCO7v27ds31q5da8s75ZRT4siRI7Y8BF1Hec0118QDDzygB72VMH369JgyZYot784774ybb77Zljdx4sSYNWuWLQ9B11Ei6BpDBF3j5+6NoLuJIuh1EkXQ60SEoGuIEHQEHUEX1hCCLsArqKv7ocTdd98dN9xwg220CLqGEkHX+GVvBF1jiKBr/Ny9EXQ3UQS9TqIIep2IEHQNEYKOoCPowhpC0AV4BXVF0DWw7KBr/KqhN4KuVQlB1/i5eyPobqIIep1EEfQ6ESHoGiIEHUFH0IU1hKAL8ArqiqBrYBF0jV819EbQtSoh6Bo/d28E3U0UQa+TKIJeJyIEXUOEoCPoCLqwhhB0AV5BXRF0DSyCrvGrht4IulYlBF3j5+6NoLuJIuh1EkXQ60SEoGuIEHQEHUEX1hCCLsArqCuCroFF0DV+1dAbQdeqhKBr/Ny9EXQ3UQS9TqIIep2IEHQNEYKOoCPowhpC0AV4BXVF0DWwCLrGrxp6I+halRB0jZ+7N4LuJoqg10nULeinnnpqHDhwoM7vbSwfePDBB+Pqq6+2HU7ZT8q2A30raN68eTFu3Dhb7IABA2LlypW2vE6dOsXOnTttefm/kDp27Jgtr4igwYMHx4oVK2zR8+fPR9AFmmUX9JNPPjmOHj0qHOE7u1bD/2bNfcz8FXdt+jS1/83aoUOHokWLFhq0gnt37do1tm3bZvsW97V92LBh8eyzz9rGN3PmzJg8ebItr+z3gnPnzo3x48fbjrfsQa1atYo333zTOsy81jnb448/HqNHj3ZGNmhWsxq3kTbg8J944on46Ec/av1GJ46tW7fGokWLrOOrhjAE/f1X6aGHHrL+P5TzBOVcI2vWrInu3btH+/bt3/9B/kbPUaNGRd68lLXl/ye7W7dutuFt2bIlhgwZYstr3ry59cZg+/bt8fu///u28Y0ZMyZ2795ty0t+3/nOd2x5t956a2zYsMGWd/Dgwfj85z8fHTt2tGTm8bZs2dKWt2vXrtqHxj169LCML/P+5//8n7VjdLXckf/Yxz7miovnnnsuTj/9dFtehw4dIv/XY652+eWXW292N23aFBdddJFreLXr40tf+pItb+HChfHaa6/Z8tatWxft2rWzXUfy3i3vL13CUMS94I9+9KP4xCc+YWP42GOPxRVXXGHLy3PM5z73OVteNQi67WCrJMh97+Zab2/jQ9ArOJHKLugVRNNovrrsJ+VGA5oD+S8JuHfQhw4dGkuXLrURnzVrVuSOWVnbjBkzYurUqbbhTZgwIebMmWPLu+eee+LGG2+05eUDk9zRa0otdy8PHz5sO+R8KLZ27Vpbnvuti+nTp8eUKVNs43O/4j5ixIhYvHixbXz5kC3f/HG1fCMpz6uulg94U4KdLd/scguDc3xNLYt7wcZfcfd6Q9ArOGcQ9ArCb6Cv5qTcQKD5GgS9oDmAoBcEtkSxCLpWDARd44ega/yqoTf3gtVQJW2MCPo7+fGK+2/MJ+cr7tpUpXcS4KTMPKg0AXbQtQog6Bq/auiNoGtVQtA1fgi6xq8aenMvWA1V0saIoCPo7zmDEHRtgbl7c1J2EyXvRAkg6CdK7J2fR9A1ftXQG0HXqoSga/wQdI1fNfTmXrAaqqSNEUFH0BF0bQ01aG9Oyg2Kmy/7Twgg6Nq0QNA1ftXQG0HXqoSga/wQdI1fNfTmXrAaqqSNEUFH0BF0bQ01aG9Oyg2Kmy9D0O1zAEG3Iy1dIIKulQRB1/gh6Bq/aujNvWA1VEkbI4KOoCPo2hpq0N6clBsUN1+GoNvnAIJuR1q6QARdKwmCrvFD0DV+1dCbe8FqqJI2RgQdQUfQtTXUoL05KTcobr4MQbfPAQTdjrR0gQi6VhIEXeOHoGv8qqE394LVUCVtjAg6go6ga2uoQXtzUm5Q3HwZgm6fAwi6HWnpAhF0rSQIusYPQdf4VUNv7gWroUraGBF0BB1B19ZQg/bmpNyguPkyBN0+BxB0O9LSBSLoWkkQdI0fgq7xq4be3AtWQ5W0MSLoCDqCrq2hBu3NSblBcfNlCLp9DiDodqSlC0TQtZIg6Bo/BF3jVw29uReshippY0TQEXQEXVtDDdqbk3KD4ubLEHT7HEDQ7UhLF4igayVB0DV+CLrGrxp6cy9YDVXSxoigI+gIuraGGrQ3J+UGxc2XIej2OYCg25GWLhBB10qCoGv8EHSNXzX05l6wGqqkjRFBR9ARdG0NNWhvTsoNipsvQ9DtcwBBtyMtXSCCrpUEQdf4Iegav2rozb1gNVRJGyOCjqAj6NoaatDenJQbFDdfhqDb5wCCbkdaukAEXSsJgq7xQ9A1ftXQm3vBaqiSNkYEHUFH0LU11KC9OSk3KG6+DEG3zwEE3Y60dIEIulYSBF3jh6Br/KqhN/eC1VAlbYwIOoKOoGtrqEF7c1JuUNx8GYJunwMIuh1p6QIRdK0kCLrGD0HX+FVDb+4Fq6FK2hgRdAQdQdfWUIP25qTcoLj5MgTdPgcQdDvS0gUi6FpJEHSNH4Ku8auG3twLVkOVtDEi6Ag6gq6toQbtzUm5QXHzZQi6fQ4g6BrSQ4cORQqws7kzEXStOgi6xg9B1/hVQ2/uBauhStoYEXQE/T1n0Fe/+lVthtHbSuCJJ56I/OdqWd9p06a54uIb3/hGbN682Za3a9eu+F//63/Z8h599NGYP3++LW/btm3xve99z5ZXU1MTV199dfTq1cuWed1118WHP/xhW94555xjHd/Bgwfj2muvtY3v+9//flxwwQW2vHbt2sUdd9xhy7vyyitj3759trwdO3bEiy++aMsbMWJE/Pu//7stL+v75ptv2vK2bNkSw4cPjzZt2lgysxZPPfVUdO/e3ZKXIaeeemq0bNnSlte+ffv40pe+ZMv71re+FWeffbYtL2uR51ZX+/jHPx6HDx92xcWvfvUrK7/FixdHSrCrLVu2LH7v937PFVd7j7B161Zb3u7du+MHP/hB9O7d25aZx9ulSxdb3tixYyPPra6W182OHTu64qJHjx7x5S9/2ZbnFvTRo0dH/qOVh4Dbvx5//PGqrnGzmrxDrtKWJ+WPfvSjVTp6hl0JAm5Bv/fee+P666+3HcpVV10VjzzyiC0vb4SGDh1qy+vZs2ds3LjRlpenn5NOOsmWl0H5wMQpH4MHD44VK1bYxpgPTMaMGWPLmzRpUsyePduWl3J+00032fLYQddQpqDnza6zudeIewf97rvvjhtuuMF2yNdcc0088MADtrzp06fHlClTbHnuHfSJEyfGrFmzbONbsGBBpBC62qBBg2L58uWuuChijRw7diycO3r5EPq1116zHfPSpUtjyJAhtrxx48bFvHnzbHkzZ86MyZMn2/Lcgm4bGEGlJYCgV7A0CHoF4VfpVyPoWuEQdI1f9kbQNYYTJkyIOXPmaCHH9b7nnnvixhtvtOU1b9488hVyVytCPhB0rToIusYPQdf4ZW8EXWdIQuMmgKBXsL4IegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYIegXhV+lXI+ha4RB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXedHQuMngKBXsMYp6AsXLqzgCPjqaiNw2WWXxejRo23D/uu//uvYtWuXLe/JJ5+Mj3/847a8F154Ifr372/LW7NmTXzoQx+y5e3cuTN+8pOfRMeOHW2Z119/fVx77bW2vD/+4z+Ozp072/KeeuqpuPzyy215yW/EiBG2vObNm8dXvvIVW96VV14Z+/fvt+W9/PLL8Yd/+Ie2vHvuuScOHDhgyzt48KD1eH/605/GtGnTbOPLoK997WvWOdi6deto2bKlbYznnXde/OxnP7Pludfw4sWL42Mf+5htfLNmzYo+ffrY8tauXRt/8Ad/YMvLOTh8+HBb3vbt2+Pv/u7vbHn3339/zJw505aX1/S8DjuvS+4aHzt2LK644grbMf/93/999OvXz5aX7P7t3/7Nlsf9vg1lkwlyXzcbGlyzmpqamob+Ur4PAhD4zwnMmzcvxo0bZ8MzZMiQWLp0qS3vtddei169etnymjVrFu5TkHt30HawbwWNHTs2FixYYIvNG7+JEyfa8txBM2bMiKlTp9piL7300liyZIktr2/fvpFC42r5gOPQoUOuuKrIadGiRRw+fNg21rvvvjtuuOEGW547KOdzzmtXy4fGKSCulg/s8iGCq40ZMybmz5/virPnbNmyJXr06GHNdV+bunfvHjlOVxswYECsXLnSFRfDhg2LZ5991paXD0wmT55syyMIAk2NAILe1CrO8ZaaAIKulwdB1xk6ExB0J81yZiHoWl0QdI0fgq7xy94Ius6QBAg4CSDoTppkQUAkgKCLACMCQdcZOhMQdCfNcmYh6FpdEHSNH4Ku8UPQdX4kQMBNAEF3EyUPAgIBBF2A91ZXBF1n6ExA0J00y5mFoGt1QdA1fgi6xg9B1/mRAAE3AQTdTZQ8CAgEEHQBHoKuwysgAUEvAGrJIhF0rSAIusYPQdf4Ieg6PxIg4CaAoLuJkgcBgQCCLsBD0HV4BSQg6AVALVkkgq4VBEHX+CHoGj8EXedHAgTcBBB0N1HyICAQQNAFeAi6Dq+ABAS9AKgli0TQtYIg6Bo/BF3jh6Dr/EiAgJsAgu4mSh4EBAIIugAPQdfhFZCAoBcAtWSRCLpWEARd44ega/wQdJ0fCRBwE0DQ3UTJg4BAAEEX4CHoOrwCEhD0AqCWLBJB1wqCoGv8EHSNH4Ku8yMBAm4CCLqbKHkQEAgg6AI8BF2HV0ACgl4A1JJFIuhaQRB0jR+CrvFD0HV+JEDATQBBdxMlDwICAQRdgIeg6/AKSEDQC4BaskgEXSsIgq7xQ9A1fgi6zo8ECLgJIOhuouRBQCCAoAvwEHQdXgEJCHoBUEsWiaBrBUHQNX4IusYPQdf5kQABNwEE3U2UPAgIBBB0AR6CrsMrIAFBLwBqySIRdK0gCLrGD0HX+CHoOj8SIOAmgKC7iZIHAYEAgi7AQ9B1eAUkIOgFQC1ZJIKuFQRB1/gh6Bo/BF3nRwIE3AQQdDdR8iAgEEDQBXgIug6vgAQEvQCoJYtE0LWCIOgaPwRd44eg6/xIgICbAILuJkoeBAQCCLoAD0HX4RWQgKAXALVkkQi6VhAEXeOHoGv8EHSdHwkQcBNA0N1EyYOAQABBF+Ah6Dq8AhIQ9AKgliwSQdcKgqBr/BB0jR+CrvMjAQJuAgi6myh5EBAIIOgCPARdh1dAAoJeANSSRSLoWkEQdI0fgq7xQ9B1fiRAwE0AQXcTJQ8CAgEEXYCHoOvwCkhA0AuAWrJIBF0rCIKu8UPQNX4Ius6PBAi4CSDobqIF502bNi369Olj+5YuXbrEb//2b9vy3EHf+MY3okePHrbYXbt2xZ//+Z/b8txBf/RHfxQvvfSSLXbt2rXx9a9/3Za3cuXKeOyxx6JDhw6WzN27d9fmOPPGjh0b/fr1s4wvQ5588skYOXKkLe/222+P3r172/K2bdtmndM/+MEP4rd+67ds4/vOd74T3bp1s+X98pe/jPPOO8+Wt3Tp0mjevLktb+/evXHdddfZ8l5//fVo2bKldY0cPHjQWpPvfve70bp1a9sxd+3aNf7qr/7KlveTn/wkPvaxj9ny7rvvPus5Zvny5TF48GDb+H784x9b7xN27twZL774om187qAU9EmTJsXpp59uid60aVNtjjOvpqYmevbsaRlfhrhrvHnz5hg1apRtfHmNy/tV2vsj8NBDD0Ve211tyZIlcemll7rianOuvPJK2xqxDqyRhCHoVVbIvn37xrp162yjXrx4cQwfPtyW5w76kz/5k7j77rttsbfcckvcdttttjx3kHsHfcCAAZFS7WqdOnWKvFlztWbNmkXeuDhbx44dIx/EuFr//v1j1apVrrgYOnRopBS62ogRIyLXsatddtllsXDhQldcuPPyJiNvNlwtz6n5IMvVTjnllDhy5IgrLopYI7bBvRXkPmZ3TdxzZvr06TFlyhQ3Rlve7Nmza4XV1caMGRPz5893xdlzDh06FPkWR5mbe4x5HVm2bJntkB955JG46qqrbHkEaQSeeuqpyGu7q+UDkw0bNrjiolWrVvHmm2/a8gh6NwEEvcpmBYKuFQxB1/h17949crfC1YqQj9x927p1q2uIgaBrKBF0jV8Ra0Qb0bt7I+huolpeUxN0jVZ19kbQq7Nu9R112QU935jat29ffQ+Hz70PAgj6+4BWyS4IukYfQdf4Iegav+zNDrrG0L0b6t6tdcsqgq7Nl+ztnjPsoOs1IUEjgKBr/MreG0Eve4WKHx+CXjxj6zcg6BpOBF3jh6Br/BB0nZ9bthB0vSbuhxLumrjnDIKuzxkSNAIIusav7L0R9LJXqPjxIejFM7Z+A4Ku4UTQNX4IusYPQdf5uWXLLYNuWWUHvXxzBkHXa0KCRgBB1/iVvTeCXvYKFT8+BL14xtZvQNA1nAi6xg9B1/gh6Do/BF1n6E5wP5RwPzRxzxkE3T2DyDtRAgj6iRKrrs8j6NVVryJGi6AXQbXATARdg4uga/wQdI0fgq7zc8uWWwbdssoOevnmDIKu14QEjQCCrvEre28EvewVKn58CHrxjK3fgKBrOBF0jR+CrvFD0HV+CLrO0J3gfijhfmjinjMIunsGkXeiBBD0EyVWXZ9H0KurXkWMFkEvgmqBmQi6BhdB1/gh6Bo/BF3n55Yttwy6ZZUd9PLNGQRdrwkJGgEEXeNX9t4IetkrVPz4EPTiGVu/AUHXcCLoGj8EXeOHoOv8EHSdoTvB/VDC/dDEPWcQdPcMIu9ECSDoJ0qsuj6PoFdXvYoYLYJeBNUCMxF0DS6CrvFD0DV+CLrOzy1bbhl0yyo76OWbMwi6XhMSNAIIusav7L0R9LJXqPjxIejFM7Z+A4Ku4UTQNX4IusYPQdf5Ieg6Q3eC+6GE+6GJe84g6O4ZRN6JEkDQT5RYdX0eQa+uehUxWgS9CKoFZiLoGlwEXeOHoGv8EHSdn1u23DLollV20Ms3ZxB0vSYkaAQQdI1f2Xsj6GWvUPHjQ9CLZ2z9BgRdw4mga/wQdI0fgq7zQ9B1hu4E90MJ90MT95xB0N0ziLwTJYCgnyix6vo8gl5d9SpitAh6EVQLzETQNbgIusYPQdf4Ieg6P7dsuWXQLavsoJdvziDoek1I0Agg6Bq/svdG0MteoeLHh6AXz9j6DQi6hhNB1/gh6Bo/BF3nh6DrDN0J7ocS7ocm7jmDoLtnEHknSgBBP1Fi1fV5BL266lXEaBH0IqgWmImga3ARdI0fgq7xQ9B1fm7ZcsugW1bZQS/fnEHQ9ZqQoBFA0DV+Ze+NoJe9QsWPD0EvnrH1GxB0DSeCrvFD0DV+CLrOD0HXGboT3A8l3A9N3HMGQXfPIPJOlACCfqLEquvzCHp11auI0SLoRVAtMBNB1+Ai6Bo/BF3jh6Dr/Nyy5ZZBt6yyg16+OYOg6zUhQSOAoGv8yt4bQS97hYofX1UL+pIlS2LHjh02Sq+88kp07tw52rVrZ8scPHhw9OzZ05K3ffv2GD58eLRv396SlyEf+MAHYvLkyba8+fPnx9ixY2153/zmN6NDhw62vL1798YXv/hFW962bdvi85//vC3vn//5n+Ohhx6y5W3atClatGhRO68dbc2aNXHSSSfZ1siePXuiY8eOtvHlGtm8eXN07drVcbiR48s5c/rpp1vyMmTr1q1x/vnn2/JefPHFyAcnrrZx48bI85arvfTSS7Z65JjyPH3mmWe6hherV6+OlGBXe/PNN6N169bRvHlzS+S+fftq17Ar7/Dhw7F///449dRTLeM7cuRIHD161HZOyEElP+ca2bBhQ/Tu3dtyvG+Pz3nenzNnTlxzzTW28X3rW9+KAwcO2PJef/31uOeee2x5GXTJJZfYzvvWgRUQltelp59+2pac16Uvf/nLVn55j3DjjTfaxvjII4/EuHHjbHnuNZLucN1119nGl/dt+XDW1fJe8LXXXnPFRc7Bbt262ebMrl27YvHixbbxEfRuAlUt6KtWrYoBAwbY6tqmTZvImyFny7y82XC1Xr16WRftwIEDI2/wXS0vus4L0V133WW9aNx+++1x6623ug639uHGzJkzbXnuoJTLtm3b2mJTjs466yxbXorRsWPHbHkZdNppp8XOnTttmXljnzf4rpYPscaMGeOKi0mTJsXs2bNteXfccUfcdNNNtrwZM2bE1KlTbXnuHfQLLrggXnjhBdv48qGnc74cOnSoVtCdLa9J+SDB1Z599tm46KKLXHGlz8n5nPPa1UaPHh1PPPGEKy5GjBhhvXnO2j733HO28Z1zzjmRD3ubUsuHxvlA39X69OkT69evd8XFsGHDItexq7nn4GWXXRYLFy50Da/2vjLvL11t7ty5MX78eFdcuO/Nc6MwH77TqocAgn5crXJn+o033rBWzy3o7lfc3ScBBN06fUoXlk908yGRqxUh6D169IgtW7a4hhj9+/ePfBjoagi6RrLsgt6vX7/aXfkyt3xLLB/euVoebx53U2kIulbpQYMGxfLly7WQKuvt3lzJzamVK1faKCDoGsqyC/rZZ58da9eu1Q6S3g1KAEFH0NlBF5Zc2XfQhUP7T7si6DpRBF1jiKBr/LI3gq4xRNA1fgi6xi97I+gaw6a2g46ga/OlEr0RdAQdQRdWHoIuwIuo/d1f9yvu7KBrNeEVd40fO+gav2rojaBrVULQNX4Ius4PQdcZklAsAQQdQUfQhTWGoAvwEHQN3lu9+R10DaP7d9ARdK0e1dAbQdeqhKBr/BB0nR+CrjMkoVgCCDqCjqALawxBF+Ah6Bo8BN3CD0HXMfI76BrDpvZH4hB0bb4g6Do/BF1nSEKxBBB0BB1BF9YYgi7AQ9A1eAi6hR+CrmNE0DWGCLrGrxp680fitCrxV9w1fvwOusavEr0RdAQdQRdWHoIuwEPQNXgIuoUfgq5jRNA1hgi6xq8aeiPoWpUQdI0fgq7xq0RvBB1BR9CFlYegC/AQdA0egm7hh6DrGBF0jSGCrvGrht4IulYlBF3jh6Br/CrRG0FH0BF0YcAC2GsAACAASURBVOUh6AI8BF2Dh6Bb+CHoOkYEXWOIoGv8qqE3gq5VCUHX+CHoGr9K9EbQEXQEXVh5CLoAD0HX4CHoFn4Iuo4RQdcYIugav2rojaBrVULQNX4IusavEr0RdAQdQRdWHoIuwEPQNXgIuoUfgq5jRNA1hgi6xq8aeiPoWpUQdI0fgq7xq0RvBB1BR9CFlYegC/AQdA0egm7hh6DrGBF0jSGCrvGrht4IulYlBF3jh6Br/CrRG0FH0BF0YeUh6AI8BF2Dh6Bb+CHoOkYEXWOIoGv8qqE3gq5VCUHX+CHoGr9K9EbQEXQEXVh5CLoAD0HX4CHoFn4Iuo4RQdcYIugav2rojaBrVULQNX4IusavEr0RdAQdQRdWHoIuwEPQNXgIuoUfgq5jRNA1hgi6xq8aeiPoWpUQdI0fgq7xq0RvBB1BR9CFlYegC/AQdA0egm7hh6DrGBF0jSGCrvGrht4IulYlBF3jh6Br/CrRG0FH0BF0YeUh6AI8BF2Dh6Bb+CHoOkYEXWOIoGv8qqE3gq5VCUHX+CHoGr9K9EbQEXQEXVh5CLoAD0HX4CHoFn4Iuo4RQdcYIugav2rojaBrVULQNX4IusavEr2rXtA/9alPRdu2bS3sXn/99TjppJOidevWlrxt27bF4sWLo3///pa8vXv3xoUXXhgtWrSw5O3fvz+OHDkS7du3t+RlyBtvvBE9evSw5X3oQx+KqVOn2vK++c1vxtq1a215p59+ejzyyCO2PHfQsmXLrJFbt26NKVOm2Nbc5s2bY+PGjdYxnnvuudGuXTtLZq65Q4cORatWrSx5ueb+5E/+JP7iL/7CkpchQ4YMiTfffNOW98EPfjDmzp1ry/vsZz8bzz//vC0vj7Vz5862vJ07d8bRo0dt5/3Dhw/HihUrbGvEdqDHBZ155pk2hrlGvvWtb0WfPn1sQ02GF198sS3PHfTpT3861q9fb4vdtWuX7bqeg3KvkTwH5j/XvVaeB7/73e/a+GVQnvdd49u0aVPkP1fL483zoOu6lHl5r3raaadZhphr+NixY3HKKadY8jJk9+7dkfdHrpbzz3Xvm2PKa/pdd93lGl584xvfiF/84he2vD179sTJJ59suy4lv7wOu9ZInv/ScVwt53SeB3v27OmKrJ0v6RDV2qpa0LOgLpnOAm7fvj26dOlirWVOuA4dOtgy3U9h86naunXrbON78MEH4+qrr7blpczcfffdtrxbbrklbrvtNlte2YNS0IcOHWobZq6PlPSm1Pr16xcvv/yy7ZAfeOCB+MxnPmPLmzRpUsyePduWd8cdd8RNN91ky5sxY4b1IduECRNizpw5tvEtWLAgxo4da8v7wAc+EBs2bLDlVUNQ796949VXX7UNddGiRTFy5EhbXtmD7rzzzrj55pttw5w4cWLMmjXLlrdjxw6bDOag8gHW4MGDbePr1KlT5BidLQW4pqbGFpljzIeBrvbkk0/GiBEjXHFx6aWXxtNPP23LmzlzZuQbhmVteV+Z95eudskll1j5ue/NU3ydmyHpX23atHHhq83JvH379tkyV65cadsgtQ3qBIKqWtBP4Djr9dF84texY8d6fba+H8rJ5nyI0LdvX6tQDxw40PqKO4Je35nRMJ9zC7r7JN8wFLRvyRvJvKF0tfnz58eYMWNccYGgayjzRnfUqFFayHG984FOvvLdlFruXq5Zs8Z2yAi6htIt6Npo3t3bLejdu3ePfBvL2dyCnmPcsmWLbYhLly6tfXvK1caNGxfz5s1zxQWCrqF035u7X3EvQtDzbd58C9fVEHQXyRLkIOh6ERB0naEzAUHXaSLoGsOy76Aj6Fp9szeCrjEs+w66dnQIehJA0N2zSMsr+w46gq7VN3sj6DrD0iQg6HopEHSdoTMBQddpIugaQwRd41cNvRF0rUoIusaPHXSNX/ZmB11j6H7FHUHX6oGg6/xKlYCg6+VA0HWGzgQEXaeJoGsMEXSNXzX0RtC1KiHoGj8EXeOHoOv8EHSdIa+4v5Mhv4N+HA8EXV9gCLrO0JmAoOs0EXSNIYKu8auG3gi6ViUEXeOHoGv8EHSdH4KuM0TQEfT/chYh6PoCQ9B1hs4EBF2niaBrDBF0jV819EbQtSoh6Bo/BF3jh6Dr/BB0nSGCjqAj6McRcP+eC4Kun6ScCQi6ThNB1xgi6Bq/auiNoGtVQtA1fgi6xg9B1/kh6DpDBB1BR9ARdP1MUiUJCLpeKARdY4iga/yqoTeCrlUJQdf4IegaPwRd54eg6wwRdAQdQUfQ9TNJlSQg6HqhEHSNIYKu8auG3gi6ViUEXeOHoGv8EHSdH4KuM0TQEXQEHUHXzyRVkoCg64VC0DWGCLrGrxp6I+halRB0jR+CrvFD0HV+CLrOEEFH0BF0BF0/k1RJAoKuFwpB1xgi6Bq/auiNoGtVQtA1fgi6xg9B1/kh6DpDBB1BR9ARdP1MUiUJCLpeKARdY4iga/yqoTeCrlUJQdf4IegaPwRd54eg6wwRdAQdQUfQ9TNJlSQg6HqhEHSNIYKu8auG3gi6ViUEXeOHoGv8EHSdH4KuM0TQEXQEHUHXzyRVkoCg64VC0DWGCLrGrxp6I+halRB0jR+CrvFD0HV+CLrOEEFH0BF0BF0/k1RJAoKuFwpB1xgi6Bq/auiNoGtVQtA1fgi6xg9B1/kh6DpDBB1BR9ARdP1MUiUJCLpeKARdY4iga/yqoTeCrlUJQdf4IegaPwRd54eg6wwRdAQdQUfQ9TNJlSQg6HqhEHSNIYKu8auG3gi6ViUEXeOHoGv8EHSdH4KuM0TQEXQEHUHXzyRVkoCg64VC0DWGCLrGrxp6I+halRB0jR+CrvFD0HV+CLrOEEFvRIL+q1/9Klq2bKnPircSfv3rX8cVV1wRbdq0sWTu27cvFi5cGD169LDk7d27N0aPHh3Hjh2z5B06dChatWplzfvSl74Uf/VXf2UZX4b83u/9XixatMiWd+WVV8Y///M/2/LKHvRv//ZvMWnSpGjRooVtqGvWrIl27drZ8pxBe/bsifznarlGRo0aFflfR8uc2267LW644QZHXO24PvWpT8WKFSsseRkyZMiQmDlzpi1v2rRp8eijj9ry+vTpE0uWLLHl3XffffGFL3zBtkZOOumkeOKJJ6Jt27a2MeZ6c6059xrJg7z44out15F/+qd/qs10tfXr10feQJe1feYzn4knn3zSNrxBgwbFww8/bJvTtoG9FZTnq0984hO28R09ejTyYbSr5b3W0KFDrfeCea96yimnWIaY5/1Zs2bF7/zO79jyLrroonj99ddteXlO/cpXvmLJKyIk71W/+93v2qLTG3bu3Gmb03kdSYdw3bs1a9as1kdc16VNmzbV+ojTl/JY00kcLdfI4sWLo3///o64imQ0q6mpqanINxu+dNWqVTFgwABD0n9E5An04MGDtrwMat68eRw+fNiW2aFDh9i9e7ct7+yzz45169bZ8vKiMXHiRFseQRqBPEGNHDlSCzmud0rCG2+8YcsrIijHmDdYrta7d+/YsGGDKy4eeOCByBtyV8uHTvPnz3fFxWWXXVZ7IXe1KVOmxPTp011x9pwFCxbE2LFjbbkdO3aMXbt22fLypsV9XcprneuhUx7o6tWro1+/frZjvvDCC+OFF16w5c2ZMycmTJhgy3MHuXfQcz47H4q5jzfPzy5RyLFt2bLFthHy9rGm0Dhvj/N4ndelfKAzYsQIW2kuvfTSePrpp215f/3Xfx1/9md/Zssre9DWrVuja9eutmE+9thj8clPftKW53YHt9u8vY67detmO+b9+/dH69atbXkNHYSgH0fc/XpFRufkyEniam5ZGDhwYLz44ouu4cWDDz4YV199tS2PII1AU3zFPd9YyRs2V8snsPkw0NVSpseMGeOKq31DYvbs2ba8piboeaObb0m42plnnhmvvPKKK65WZJxvheTA3A+x3IKe9XDuKDc1Qc+H5PmwvKm0ahD0fA3feV1aunRp7dtOrjZu3LiYN2+eK672LazJkyfb8ppa0FNPPWV9AON2B7fbZH3zwXY+SKD9BwEEHUFH0Bvx2QBB14uLoGsMy76DjqBr9c3eCLrG0L2DjqBr9ai9OTbvoCPoek2aUgKC3pSq/Z8fK4KOoCPojfg8gKDrxUXQNYYIusaPHXSNX/ZmB11nWOYEdtD16rCDrjN0JiDoTprVmYWgI+gIenWu3XqNGkGvF6b3/BCCrjFE0DV+CLrGD0HX+ZU9AUHXK4Sg6wydCQi6k2Z1ZiHoCDqCXp1rt16jRtDrhQlBP44Av4OuzRl+B13jl735HXSNIa+4a/yyN6+4awz5HXSNH4Ku8WsMvRF0BB1Bbwwr+b84BgRdLy476BpDdtA1fuyga/zYQdf5lT2BHXS9Quyg6wydCQi6k2Z1ZiHoCDqCXp1rt16jRtDrhYkddHbQ9YnyVgI76DpKdtA1huyga/zYQdf5sYOuMUTQNX6NoTeCjqAj6I1hJbOD/v8J8L9Z0yY0r7hr/BB0jV/2RtA1hgi6xg9B1/kh6BpDBF3j1xh6I+gIOoLeGFYygo6gm+Yxgq6BRNA1fgi6zg9B1xnyO+gaQwRd44ega/waQ28EHUFH0BvDSkbQEXTTPEbQNZAIusYPQdf5Ieg6QwRdY4iga/wQdI1fY+iNoCPoCHpjWMkIOoJumscIugYSQdf4Ieg6PwRdZ4igawwRdI0fgq7xawy9EXQEHUFvDCsZQUfQTfMYQddAIugaPwRd54eg6wwRdI0hgq7xQ9A1fo2hN4KOoCPojWElI+gIumkeI+gaSARd44eg6/wQdJ0hgq4xRNA1fgi6xq8x9EbQEXQEvTGsZAQdQTfNYwRdA4mga/wQdJ0fgq4zRNA1hgi6xg9B1/g1ht4IOoKOoDeGlYygI+imeYygayARdI0fgq7zQ9B1hgi6xhBB1/gh6Bq/xtAbQUfQEfTGsJIRdATdNI8RdA0kgq7xQ9B1fgi6zhBB1xgi6Bo/BF3j1xh6I+gIOoLeGFYygo6gm+Yxgq6BRNA1fgi6zg9B1xki6BpDBF3jh6Br/BpDbwQdQUfQG8NKRtARdNM8RtA1kAi6xg9B1/kh6DpDBF1jiKBr/BB0jV9j6F31gn7eeefZ6tCyZcs4ePCgLS+DmjdvHocPH7Zltm/fPt544w1bXrdu3eL111+35T3wwANx9dVX2/JeffXV+MAHPmDLy6C88DaVtmzZsrjoootsh9uuXbvYtWuXLa+mpqY2y1mTXCP79u2zjbFz586xfft2W968efNizJgxtryrrroqFixYYMu78MIL4/nnn7fl/e7v/m48+OCDtrw8X3Xt2tWWl2O75pprbHkdO3a0rpEWLVrE/v37rWvk1FNPjUOHDtmOeeXKldGvXz9b3llnnRXr16+35X3ve9+LCRMm2PI2btxovS795V/+ZXznO9+xjS/PL48++qgtzx20efPm6N69uy123bp11vlnG9hxQW3atLFel/Kcf8UVV9iG+pGPfCSefvppW94//dM/xeTJk215r7zySuTDT2dz3nds3bo1unTpYhvev/7rv8ZnPvMZW57bHdxukwe6Y8eO6NChg+2Y877NWRPbwOoZVNWCvm3bNiv8vPFzXjSyBu5JnJN39+7d9Sxv3R+bO3du5A20q+Xu26JFi1xx4ZaF6667Lu677z7b+MoetGnTpjj99NNtw3zttdeiV69etjz3LkUOzH0j9KMf/Sg+/vGP24754osvjmeeecaWN2vWrMgdM1dLMerTp48rzp6TDwGdQp3n/C1bttjGWYSgO2U6D3TPnj3Rtm1b2zHnTZDzIVZeQ0aOHGkb3+WXXx6PP/64LW/69OkxZcoUW14K+re//W1b3tixY0st6CtWrIjBgwfbjtd9zs+Bua9Nud727t1rO+Yf/vCHVkG3DeytoHxw7HxI5L4X/PSnPx0pwa6W99Ljx493xdW6iPO61KlTp1oBdrXceEy/cbZ8cHzgwAFbZD447t+/vy2voYOqWtDdsFJ88+bK2Vq3bl27++FqvXv3jg0bNrjiYvHixTF8+HBbXp6g8kTlaqNGjbIK/y233BK33Xaba3hNLqcaBD13V/NptqstX748Bg0a5IqLvHl27ni7Bd12oAUFuQX9ggsuiBdeeME2Wvcr7kXIh1vQzz333FizZo2NoVvQ84FOzhtXcwv6nXfeGTfffLNreLUP7PK8UNbmFnT3OT+5uQXdLVxLly6NIUOGlLXEcf3118e9995rG5/7V69uvPHGuOuuu2zjcwv6wIEDrb9+evbZZ8fatWttx1tEUPqXcwMSQS+iShXKRNB18Ai6zrDMCQi6Xh0EXWOIoGv8sjeCrjFE0DV+CLrGL3sj6BpDBF3jV0RvBP2dVNlBP44Hgq4vOQRdZ1jmBARdrw6CrjFE0DV+CLrOD0HXGCLoGj8EXeeHoOsM3QkIOoL+X84pBF1fbgi6zrDMCQi6Xh0EXWOIoGv8EHSdH4KuMUTQNX4Ius4PQdcZuhMQdAQdQT+OAL+D7j7FNO48BF2vL4KuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoCDqC7j6vNJk8BF0vNYKuMUTQNX4Ius4PQdcYIugaPwRd54eg6wzdCQg6go6gI+ju80qTyUPQ9VIj6BpDBF3jh6Dr/BB0jSGCrvFD0HV+CLrO0J2AoCPoDSrozZs3j8OHD9vmcYcOHWL37t22vB/+8IdxxRVX2PIuu+yyWLRokS1v1KhR1rxbbrklbrvtNtv4Nm3aFKeffrotb/v27dG5c2db3t69e6Nt27a2vDVr1sS5555ry2vWrFnU1NTY8jKoTZs2sW/fPlvm8uXLY9CgQba8iy++OJ555hlb3t/+7d/GF77wBVte2YPcgt63b99Yu3at7bDzJmPXrl22vBYtWsShQ4dseUUIepcuXSLPXa726KOPRj7IcrXLL788Hn/8cVdclF3Qk10ydLUdO3bEaaed5oqLn/3sZzF8+HBbnvucnwNzX5vyOpzXY1dbunRpDBkyxBVXu36d9x5XXXWVdQ7mveXChQttx1t2QT/77LNj3bp1tuPt1KlT5Dp2tfSa9BtnO/XUU+PAgQO2yJUrV0b//v1teQ0d1KzGfXfc0Edg/L48QR05csSWmJIwePDgaNmypSVz//79kRM4LxyOlgvsnnvuif/23/6bI672JjLlY+PGjZa8DElBzzG62l/8xV/Ej370I1dc9OzZM1555ZXIm2hHy4t4nkRdeTn39uzZY8s7duxY5Dxs3bq143Brs1588UVbXq65zOzatatlfBmS/M455xxLXq6RvKl6/fXXLXm5hnNOf/nLX7bkFRGS47v//vtt0T169Ih8sOhqq1evjrPOOst2s7FkyZK49tprbXM650w+JHKtueR23nnn2c4JuebyvHX06FFLSXJOZ57rwXby+9a3vhWf+tSnLOPLkN///d+Pn//857a8rK1T3nKNvPrqq7Yau69LJ598cu152nWDn/dtWWfXGsmxZXPmPfzww3H++efb5swll1wSb775piUv2eUDGNcczLxbb73Vdm+ZB/m5z30uXnjhBcvxZsjAgQPjJz/5iS3v2WefjT59+tjycjPkzDPPtK2RHFg+yHJt2OQa6d69e22mo+V1ZNmyZdG+fXtHXG1Gyv4ZZ5xhy2voIAS9YOI5ed8+2Tu+qnfv3rFhwwZHVG3G4sWLrU+yx48fH3PnzrWN76677op80ulqt99+e+2Fw9VGjhwZTz75pCsuhg0bFnmid7UBAwZEPkV0tTwhb9myxRVX+7Appb8ptab2ivuMGTNi6tSpthJPmDAh5syZY8tzB7nfMskbqnxI5Gzt2rWz3YznuPJGMh9UutoFF1xgvRnP+ZLzxtVyPue8drXRo0fHE0884YqLESNG1F7bXe2iiy6K5557zhVXu6u1atUqW14+kN26dastL4PcO+i5cZEP9F1t6NChtULjau57j5kzZ8bkyZNdw6u9b8v7N1dz76C7xlUtOek1Ljl/+5jzzbN8S5j2HwQQ9IJnAoKuAUbQNX4IusaviN4IukYVQdf4ZW8EXWOIoGv8EHSNX/ZG0DWGCLrGD0HX+NWnN4JeH0rCZxB0AV5EIOgaPwRd41dEbwRdo4qga/wQdJ0fgq4xRNA1fgi6zg9B1xgi6Bq/+vRG0OtDSfgMgi7AQ9A1eBGBoMsI7QEIuoYUQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaDXh5LwGQRdgIega/AQdJlfEQEIukYVQdf4Ieg6PwRdY4iga/wQdJ0fgq4xRNA1fvXpjaAfR+nAgQPRqlWr+nCr92datGgRhw8frvfn6/pgt27d4vXXX6/rY/X++Q9/+MO44oor6v35uj44fvz4mDt3bl0fq/fP77rrrsgTqavdfvvtceutt7riYuTIkfHkk0/a8gYMGBArV6605fXp0yfWr19vy+vUqVPs3LnTltesWbM4duyYLS+DDh06FLnuXG3v3r3Rtm1bV1xcfPHF8cwzz9jyZs2aFRMnTrTluYNmzJgRKTSu5hb03bt3R4cOHVzDizVr1sS5555ry8u5fPDgQVteBrVs2bJ2nbhajx49YvPmza646Nu3b6xdu9aWN2fOnMh542pNTdDdQt27d+/YsGGDqxzRsWPH2LVrly0vg/LaVFNTY8vcuHFj9OzZ05aXNVm9erUtz33vMXPmzJg8ebJtfHnflvdvrtbUBN3tN3ndzHXnbFu2bIl0HNp/EEDQj5sJOeHyJJU3G46WNyz/9//+X5v05xOra665JnJX3tEy7ytf+Upcf/31jrjajEGDBkWeCFwts1IKXe2ss86qPWZX+9KXvmQV4JTBk046KVq3bm0ZYt7Yf/e737XlrVixopafa3zbt2+PU045xbbmkt/Ro0dtwpV53//+92vntaNl3qhRoyLXnqt95CMfiZT0sraHHnoozjjjDNvw8oHTpz71KVtePmCbNGmS7SHM23PGNcCcK1dffbVtjeR1KcXDdXOV48vz4IgRIyyHnHnr1q2LD33oQ5a8DMlrXJ4XXC1lMB9yuNqePXuiXbt2rrjYtm1bdOnSxZaX82X27Nm28/6qVasiJd11HUnZ79y5sy0v+WVzMsz7tn79+llqkueYl19+2ZKVIbnm8iFvPpRwtUsuuST+5V/+xRUX48aNsx5zrjmnDL7xxhvRvn172/Hmg9mf//zntrw8p37yk5+0XedyozAf8ua6c7S8F/zpT38aAwcOdMQ1igwE/bgyFvFEaN++fbaLRg41dxZyobna4sWLY/jw4a64cO+gp8wsWrTINr5bbrklbrvtNlvevffea33AMWzYsHj22Wdt4xsyZEgsXbrUlvfaa69Fr169bHnuXYocWNeuXWPr1q22MS5fvtwm6DmopvaKu60QBQWloOd5q1qijgAAIABJREFUxtXyJty5s5U34055y+NMWchrk6vl8brkwzWm43PywfYDDzxgi3Zfl0aPHh1PPPGEbXz5sCSv7a42ZsyYmD9/viuOnBISSAGeN2+ebWRl30F3r+F8IPH000/b+OV9uXMNpzekP7haPlxzbjTkuPLBp/NtNtexVioHQT+OPIKuT0MEXWOIoGv8EHSdX1NLQND1iiPoGkMEXeNHb50Agq4xRNA1fgj6u/kh6Ag6O+jCeYUddAFeAb/nh6Br9WiKvRF0veoIusYQQdf40VsngKBrDBF0jR+CjqC/5wxiB11fYOygawzZQdf4Ieg6v6aWgKDrFUfQNYYIusaP3joBBF1jiKBr/BB0BB1B/w0C/A66dlJhB13jx++ga/yyd9n/irt+hMUmIOg6XwRdY4iga/zorRNA0DWGCLrGD0FH0BF0BJ0/EiecR/kjcQK8t7ryR+J0hs4EBF2niaBrDBF0jR+9dQIIusYQQdf4IegIOoKOoCPownkUQRfgIeg6vAISEHQdKoKuMUTQNX701gkg6BpDBF3jh6Aj6Ag6go6gC+dRBF2Ah6Dr8ApIQNB1qAi6xhBB1/jRWyeAoGsMEXSNH4KOoCPoCDqCLpxHEXQBHoKuwysgAUHXoSLoGkMEXeNHb50Agq4xRNA1fgg6go6gI+gIunAeRdAFeAi6Dq+ABARdh4qgawwRdI0fvXUCCLrGEEHX+CHoCDqCjqAj6MJ5FEEX4CHoOrwCEhB0HSqCrjFE0DV+9NYJIOgaQwRd44egI+gIOoKOoAvnUQRdgIeg6/AKSEDQdagIusYQQdf40VsngKBrDBF0jR+CjqAj6Ag6gi6cRxF0AR6CrsMrIAFB16Ei6BpDBF3jR2+dAIKuMUTQNX4IOoKOoCPoCLpwHkXQBXgIug6vgAQEXYeKoGsMEXSNH711Agi6xhBB1/gh6Ag6go6gI+jCeRRBF+Ah6Dq8AhIQdB0qgq4xRNA1fvTWCSDoGkMEXeOHoCPoCDqCjqAL51EEXYCHoOvwCkhA0HWoCLrGEEHX+NFbJ4CgawwRdI0fgo6gI+gIOoIunEcRdAEegq7DKyABQdehIugaQwRd40dvnQCCrjFE0DV+CDqCjqAj6Ai6cB5F0AV4CLoOr4AEBF2HiqBrDBF0jR+9dQIIusYQQdf4IegI+nvOoO3bt0eXLl30WXZcwhtvvBHt2rWzZfbq1StSklxt8eLFMXz4cFdcjB8/PubOnWvLGzVqVCxatMiWd8sttzQpQT/33HNj1apVNn6vvPJKnHXWWba8Zs2aRU1NjS0vgzp27Bi7du2yZS5fvjwGDRpkyxs7dmwsWLDAljdr1qyYOHGiLa+pBbkF/QMf+EBs2LDBhnH37t21c9rZWrVqFQcOHLBFIugaSgRd49cUe+/duzfatm1rO3QEXUNZdkH/5S9/GR/84Ae1gzyut/saUoSg79+/P1q3bm075oYOalbjvjtu6CMwft/TTz8dn//8521CvWfPnlr56Nu3r2WUO3bsiK1bt8Zpp51mycuQzHMu2l/84hfRtWtX2/hatmwZt956qy3v29/+dmSmq7mPd/369dGnTx/X8GpFtXPnzrY5s27dutqxuR465U1GrhFXXq65SZMmxYUXXmhj+Jd/+ZfRs2dPW95LL70U3bp1s+Xlg8UBAwbY8n71q19ZH8Lk/Lv//vtt43MH/fSnP41Dhw5ZYz/84Q/b1txzzz0Xn/vc56xr5MYbb4x+/frZjvmLX/xi5IMJV3OfB3P9fuYzn3ENL/70T/802rRpY8tbu3at7T7h7Rvdu+++2za+hx56KPLhrKvleT8fjDWlNmTIkOjevbvlkPNe8O///u8jM13tggsuiBYtWrjiYtu2bXHeeefZ8p5//nnrdTg3z2bMmGEb3/z58yMfvrta5uWDT1dz37tt2bIl/uZv/sZ2XcrjzPPqmWeeaTnkXCP33Xdf9O/f35JXiRAE/TjqRexU5NObfIrjar1797buzgwcODBefPFF1/DC/RTxrrvuiryZdLXbb7/dKvwjR4603mgMGzYsnn32Wdfh1orbypUrbXl5g5EnZlcrYgd98+bNthuhPM7BgwfHihUrXIccQ4cOjaVLl9ryRowYEfkmjKtddtllsXDhQldcTJkyJaZPn27La2pBKTOuB1hvs8sHWc7dt3xTZ82aNbbSpCy88MILtrw5c+bEhAkTbHlTp0613ty7d9DzjZp8s8bV8o0fp3zkG0n5ZlJTau63H/Ma4hR09w562a9LeV+Z95dlbU899VQkQ1dzu0O6zb59+1zDq83JN8XSw1wt730RdBfNCucg6HoBEHSNIYKu8cveCLrGEEHX+Ll7I+g6UQRdY4iga/yyN4KuMXRflxB0rR4IusavPr3ZQT+OEoJenynz3p9B0DWGCLrGD0HX+blvhNhB12qCoGv8sjeCrjFE0DV+CLrOz31dQtC1miDoGr/69EbQEXReca/PSvkvPsMr7gK8iOAVd41f9i77q4QIulZjBF3jh6Dr/BB0nSE76BpDBF3jxyvuGr9K9EbQEXQEXVh5CLoAD0HX4L3VG0G3YCxtCIKul4YddI0hgq7xYwdd54egawwRdI1fJXoj6Ag6gi6sPARdgIega/AQdAu/socg6HqFEHSNIYKu8UPQdX4IusYQQdf4VaI3go6gI+jCykPQBXgIugYPQbfwK3sIgq5XCEHXGCLoGj8EXeeHoGsMEXSNXyV6I+gIOoIurDwEXYCHoGvwEHQLv7KHIOh6hRB0jSGCrvFD0HV+CLrGEEHX+FWiN4KOoCPowspD0AV4CLoGD0G38Ct7CIKuVwhB1xgi6Bo/BF3nh6BrDBF0jV8leiPoCDqCLqw8BF2Ah6Br8BB0C7+yhyDoeoUQdI0hgq7xQ9B1fgi6xhBB1/hVojeCjqAj6MLKQ9AFeAi6Bg9Bt/ArewiCrlcIQdcYIugaPwRd54egawwRdI1fJXoj6Ag6gi6sPARdgIega/AQdAu/socg6HqFEHSNIYKu8UPQdX4IusYQQdf4VaI3go6gI+jCykPQBXgIugYPQbfwK3sIgq5XCEHXGCLoGj8EXeeHoGsMEXSNXyV6I+gIOoIurDwEXYCHoGvwEHQLv7KHIOh6hRB0jSGCrvFD0HV+CLrGEEHX+FWiN4KOoCPowspD0AV4CLoGD0G38Ct7CIKuVwhB1xgi6Bo/BF3nh6BrDBF0jV8leiPoCDqCLqw8BF2Ah6Br8BB0C7+yhyDoeoUQdI0hgq7xQ9B1fgi6xhBB1/hVojeCjqAj6MLKQ9AFeAi6Bg9Bt/ArewiCrlcIQdcYIugaPwRd54egawwRdI1fJXoj6Ag6gi6sPARdgIega/AQdAu/socg6HqFEHSNIYKu8UPQdX4IusYQQdf4VaI3gl6woLdq1SoOHDhgq617kQ0cOBBBF6rT1AS9U6dOsXPnToHYO7s2a9YsampqbHkZtHnz5ujevbstc/DgwbFixQpb3tChQ2Pp0qW2vBEjRsTixYttee4boSlTpsT06dNt42tqQbt3746OHTtaD3vXrl3RocP/Y+/Mw7Qqzzz9gLKvoiCyCyIktrigiSiCnekk0tiJu2mjURGvmIlgerJ0y6Udt4Z04rStkpgZBdLRdKLBpR0VE+0kLA4qgopDNzsERNlll0WouZ5qnSGdGIr6/U7V+Yr7vS7/8nt+9X73c857zn3eU0U7W+bxxx8fixYtsuUNGDAg5s6da8tD0DWUZRf03bt3R9OmTbUvWXB1x44dY/369bafkteQU0891ZZ33nnnxTPPPGPLK/t16YYbboj77rvP9n3dQS+++GIkQ9dwu0O6zXvvveeaXnVO69atY/v27bbM+fPnR79+/Wx5dR1U0YL+/PPPx09/+lMbs7fffjsuu+wyW17eWE2cONF6I/Rv//Zv8fGPf9w2xyZNmsSVV15py3vhhRfiz/7sz2x5P/rRj6JPnz62vGXLlsWXvvQlW973vve9OPLII215CxcujLzZdY0VK1bErbfe6oqLXPCmTJliO6ZTFK699lpbXn7RT37yk9Zz5I477ohu3brZGGY/evToYcvbuHFjfOMb37Dl3XnnndGlSxdbXt48v/zyy7a8Qy1o3rx58Zd/+Ze2cySvS3ndPOGEE2wob7vtNusxfcstt8Sxxx5rm1/fvn2rr8WukWuMUwjz3uPmm292TS9y3f/2t79ty7vxxhtjzpw5tryVK1fGpz71KVte8mvcuHF07tzZkpkPefPB8THHHGPL+/SnP2190JbfddiwYZb5Zch3v/vdyIcIrpFrQkqha6xZsyZuuukmV1w8/PDD0bNnT1verl274ic/+Ykt76mnnooNGzbY8vIhvlP487r+L//yL9ZzLtcF58PoPD/GjBljY1jXQRUt6AsWLIj+/fvbmOWOQgqDc7Rq1Sp27Nhhi3Q/BcuT9qyzzrLNzx00atSoGD9+vC02b4JSuFxjwoQJMXLkSFdcnH766TFr1ixbXj5hd+7W5o1Q165dbfPLHfR9+/bZ8iohKC8auSPlGike11xzjSuu+oI2btw4Wx476BrKSnjFXfuGv189ZMiQmD59ui227DvoKR5jx461fV93kHsHPXe18v7NNVIs161b54qrznG/3bVq1Srrg0/rly0gLO+L8v7INfK+zfkQK+8r8/7SNS666KKYPHmyK670Oek16TfO0bZt29iyZYstkh10G8qDD0LQD57Zf65A0DWGCLrGD0HX+GU1gq4zLHMCgq53B0HXGCLoGr+sRtA1hgi6xs9djaC7if5+Hjvo+zFhB734A+5gfwI76AdL7Hc/zw66xq+IanbQi6DacDMRdL23CLrGEEHX+CHoOj8EXWfoTEDQnTT/cBaCjqDzirtwnrGDLsD74DVCXnHXGLKDrvErezWCrncIQdcYIugaPwRd54eg6wydCQi6kyaCfkCa7KAfEFGdf4AddA05O+gavyKq2UEvgmrDzUTQ9d4i6BpDBF3jh6Dr/BB0naEzAUF30kTQD0gTQT8gojr/AIKuIUfQNX5FVCPoRVBtuJkIut5bBF1jiKBr/BB0nR+CrjN0JiDoTpoI+gFpIugHRFTnH0DQNeQIusaviGoEvQiqDTcTQdd7i6BrDBF0jR+CrvND0HWGzgQE3UkTQT8gTQT9gIjq/AMIuoYcQdf4FVGNoBdBteFmIuh6bxF0jSGCrvFD0HV+CLrO0JmAoDtpIugHpImgHxBRnX8AQdeQI+gavyKqEfQiqDbcTARd7y2CrjFE0DV+CLrOD0HXGToTEHQnTQT9gDQR9AMiqvMPIOgacgRd41dENYJeBNWGm4mg671F0DWGCLrGD0HX+SHoOkNnAoLupImgH5Amgn5ARHX+AQRdQ46ga/yKqEbQi6DacDMRdL23CLrGEEHX+CHoOj8EXWfoTEDQnTQR9APSRNAPiKjOP4Cga8gRdI1fEdUIehFUG24mgq73FkHXGCLoGj8EXeeHoOsMnQkIupMmgn5Amgj6ARHV+QcQdA05gq7xK6IaQS+CasPNRND13iLoGkMEXeOHoOv8EHSdoTMBQXfSRNAPSBNBPyCiOv8Agq4hR9A1fkVUI+hFUG24mQi63lsEXWOIoGv8EHSdH4KuM3QmIOhOmgj6AWki6AdEVOcfQNA15Ai6xq+IagS9CKoNNxNB13uLoGsMEXSNH4Ku80PQdYbOBATdSRNBPyBNBP2AiOr8Awi6hhxB1/gVUY2gF0G14WYi6HpvEXSNIYKu8UPQdX4Ius7QmYCgO2ki6AekiaAfEFGdfwBB15Aj6Bq/IqoR9CKoNtxMBF3vLYKuMUTQNX4Ius4PQdcZOhMQdCdNBP2ANBH0AyKq8w8g6BpyBF3jV0Q1gl4E1YabiaDrvUXQNYYIusYPQdf5Ieg6Q2cCgu6kiaAfkCaCfkBEdf4BBF1DjqBr/IqoRtCLoNpwMxF0vbcIusYQQdf4Ieg6PwRdZ+hMQNCdNBugoN97773x+OOP2yi99dZbsXjxYlteBvXp0ye6d+9uy2zWrFmcc845tryePXvG5Zdfbsv78pe/HEceeaQt76WXXopPf/rTtrw9e/bE3/7t39ryzj///Ni0aZMt74033oiTTjrJlrdhw4Z48803bXn/9E//FJMmTbLlbd68OU444YTo0aOHLfOiiy6KgQMH2vLcQddff33keecarVu3jnyQ5Rpnn312HHbYYa642LlzZ+R57Bq57q9evdoVVz23M844w5bnPoeXLVsWAwYMsM0vgy688MLo16+fLfO6666Ljh072vIeeuih6munaxx99NGRku4aebw0b97cFRd79+6N6dOn2/Luuuuu2Lhxoy1v8uTJ0aVLF1veokWLom/fvra8FStWRB6DrpHX9F/84hfRvn17V2Q0atQoBg0aZMsbOnRofPazn7XlXXvttZHniWu4z+G8Jv3rv/6ra3rxD//wD7Fr1y5b3vPPP2+9jhyK16UHHnjAei+Y17nRo0fbelzXQY2qqqqq6vqHun7eggULon///q64qIQd9CVLlkTv3r1t39kddPHFF8djjz1mi73vvvvihhtusOW5gyZMmBAjR460xZ5++ukxa9YsW14l7KC7l6CUN+eNhq0ZFRI0ZsyYGDdunG22eYG85557bHmPPvpoXHbZZba8lN+5c+fa8nr16hXLly+35eUDmK1bt9ryigg6/vjjI6XLNdw9OdR20PMh6ogRI1ztiNNOOy1effVVW14+HMr7N9fI9d750C7n1bhx43Bem3KOa9ascX3lmD17duT13TXOO++8eOaZZ1xxMXjw4JgxY4Ytz72DbpvYB0F535v3v65x4oknWjdXcqNw5cqVrulFy5YtY/v27ba8DMoHYrlp4xrz58+3Poh2zaumOQj6fqQQ9JoeNh/9OQRdY4iga/yyGkHXGCLoGj8EXeOX1Qi6xhBB1/gh6Do/BF1jiKBr/LIaQdcZ1jqBHfRaoyusEEHX0CLoGj8EXeeHoGsMEXSNH4Ku80PQdYbsoGsMEXSNH4Ku8UPQdX5SAoIu4SukGEHXsCLoGj8EXeeHoGsMEXSNH4Ku80PQdYYIusYQQdf4IegaPwRd5yclIOgSvkKKEXQNK4Ku8UPQdX4IusYQQdf4Ieg6PwRdZ4igawwRdI0fgq7xQ9B1flICgi7hK6QYQdewIugaPwRd54egawwRdI0fgq7zQ9B1hgi6xhBB1/gh6Bo/BF3nJyUg6BK+QooRdA0rgq7xQ9B1fgi6xhBB1/gh6Do/BF1niKBrDBF0jR+CrvFD0HV+UgKCLuErpBhB17Ai6Bo/BF3nh6BrDBF0jR+CrvND0HWGCLrGEEHX+CHoGj8EXecnJSDoEr5CihF0DSuCrvFD0HV+CLrGEEHX+CHoOj8EXWeIoGsMEXSNH4Ku8UPQdX5SAoIu4SukGEHXsCLoGj8EXeeHoGsMEXSNH4Ku80PQdYYIusYQQdf4IegaPwRd5yclIOgSvkKKEXQNK4Ku8UPQdX4IusYQQdf4Ieg6PwRdZ4igawwRdI0fgq7xQ9B1flICgi7hK6QYQdewIugaPwRd54egawwRdI0fgq7zQ9B1hgi6xhBB1/gh6Bo/BF3nJyUg6BK+QooRdA0rgq7xQ9B1fgi6xhBB1/gh6Do/BF1niKBrDBF0jR+CrvFD0HV+UgKCLuErpBhB17Ai6Bo/BF3nh6BrDBF0jR+CrvND0HWGCLrGEEHX+CHoGj8EXecnJSDoEr5CihF0DSuCrvFD0HV+CLrGEEHX+CHoOj8EXWeIoGsMEXSNH4Ku8UPQdX5SAoIu4SukGEHXsCLoGj8EXeeHoGsMEXSNH4Ku80PQdYYIusYQQdf4IegaPwRd5yclIOgSvkKKEXQNK4Ku8UPQdX4IusYQQdf4Ieg6PwRdZ4igawwRdI0fgq7xQ9B1flICgi7hK6QYQdewIugaPwRd54egawwRdI0fgq7zQ9B1hgi6xhBB1/gh6Bo/BF3nJyUg6BK+QooRdA0rgq7xQ9B1fgi6xhBB1/gh6Do/BF1niKBrDBF0jR+CrvFD0HV+UsKvf/3rePLJJ6WM/YtXr14djzzyiC0vgy699NI45phjbJkXXXRRDBkyxJZ3yy23RMeOHW15kydPjlNOOcWW9+abb8b5559vy3vuuefi3HPPteX94Ac/iM6dO9vyli1bZv2+s2fPrj4GXSPPuXfffdcVF5s3b46qqqpo3769LbNDhw5xzjnn2PJ69uwZn//85215t956a+QcXaNp06Zx/fXXu+Li7LPPjsMOO8yWt3PnznjppZdseXfddVesXLnSlpdrgnONXrx4cRx33HG2+W3ZsiWeeuqp6Natmy3THXTllVdaj+n8vnneucbatWut58jdd99tnd/evXtj+vTprq8b48aNi7yfcQ33ObJo0aLIh/musXz58vgv/+W/uOJi06ZN8cQTT0S7du1smXv27InTTjvNlpfXYWee+97tscces66DeU3613/9Vxs/d9A//MM/xG9/+1tb7CuvvBKf+MQnrHnNmjWz5bmvS2+99Vb1fVabNm1sc7zwwgtj9OjRtry6DmpUlXfHjEOWQD7lfPHFF23fPxf5fIjgGqNGjYrx48e74mLo0KExdepUW17KjPPGavjw4fH000/b5jdnzpwYOHCgLe/oo4+ONWvW2PIaNWpULejOkQ+c1q1bZ4t87bXX4uSTT7blDRs2LPKG1zUmTpwY11xzjSsuyr6DbvuiHwTl+et86OneQW/RokXs2LHD/bVLnZf9cK6rgwYNipkzZ9q+c85v2rRptrybbropxo4da8tzB+V6leuWa+R6muuqa+Q1yfmgPOflvjatWrUqunTp4vrK1df1vL67Rt535P2Ha4wcOTImTJjgios77rgjbr75ZlveoRa0dOnS6NOnj+1rN2/ePN577z1bXga1bt06tm/fbsucP39+9OvXz5ZX10EIel0TL9nPQ9C1hiDoGj/3TVDOBkHXeoKga/zcgp43LVu3btUmVWHVCHq5Goag6/1A0DWGCLrGzy3oLVu2tMp0frt8kzLfqnQNBN1Fkpx6IYCga9gRdI0fgq7xy2p20DWGZd9BR9C1/mY1O+gaQwRd45fVCLrGEEHX+CHoGr/6qGYHvT6ol+hnIuhaMxB0jR+CrvFD0HV+CLrO0J3ADrqbqJaHoGv8EHSdH4KuMUTQNX71UY2g1wf1Ev1MBF1rBoKu8UPQNX4Ius4PQdcZuhMQdDdRLQ9B1/gh6Do/BF1jiKBr/OqjGkGvD+ol+pkIutYMBF3jh6Br/BB0nR+CrjN0JyDobqJaHoKu8UPQdX4IusYQQdf41Uc1gl4f1Ev0MxF0rRkIusYPQdf4Ieg6PwRdZ+hOQNDdRLU8BF3jh6Dr/BB0jSGCrvGrj2oEvT6ol+hnIuhaMxB0jR+CrvFD0HV+CLrO0J2AoLuJankIusYPQdf5IegaQwRd41cf1Qh6fVAv0c9E0LVmIOgaPwRd44eg6/wQdJ2hOwFBdxPV8hB0jR+CrvND0DWGCLrGrz6qEfT6oF6in4mga81A0DV+CLrGD0HX+SHoOkN3AoLuJqrlIegaPwRd54egawwRdI1ffVQj6PVBvUQ/E0HXmoGga/wQdI0fgq7zQ9B1hu4EBN1NVMtD0DV+CLrOD0HXGCLoGr/6qEbQ64N6iX4mgq41A0HX+CHoGj8EXeeHoOsM3QkIupuoloega/wQdJ0fgq4xRNA1fvVRjaDXB/US/UwEXWsGgq7xQ9A1fgi6zg9B1xm6ExB0N1EtD0HX+CHoOj8EXWOIoGv86qMaQa8P6iX6mQi61gwEXeOHoGv8EHSdH4KuM3QnIOhuoloegq7xQ9B1fgi6xhBB1/jVRzWCXh/US/QzEXStGQi6xg9B1/gh6Do/BF1n6E5A0N1EtTwEXeOHoOv8EHSNIYKu8auPagS9PqiX6Gci6FozEHSNH4Ku8UPQdX4Ius7QnYCgu4lqeQi6xg9B1/kh6BpDBF3jVx/VCHp9UC/Rz0TQtWYg6Bo/BF3jh6Dr/BB0naE7AUF3E9XyEHSNH4Ku80PQNYYIusavPqoR9PqgXqKfiaBrzUDQNX4IusYPQdf5Ieg6Q3cCgu4mquUh6Bo/BF3nh6BrDBF0jV99VCPo9UG9RD8TQdeagaBr/BB0jR+CrvND0HWG7gQE3U1Uy0PQNX4Ius4PQdcYIugav/qoRtDrg3qJfuaf//mfR5s2bWwzOuecc+IrX/mKLW/EiBGxfft2W96LL74YPXr0sOUtXrw4jjvuOFteftc33njDlvfkk0/GT3/6U1teft927dpFx44dLZnr1q2rznHmbd682dqTnFsKg2v89V//dRxzzDGuuDj22GPjJz/5iS0vH9rt27fPlrd37954+eWXbXmPPvqoLevDBxxbtmyxZa5cubL6HGnbtq0lc9u2bfHCCy9Ep06dLHlr166N3/zmN5asD0PyeM6Hla4xfPjwaN26tSuu+vjr0qWLLS97fOaZZ9ryOnToEPfff78tL9eDJk2a2PJ+8YtfRB6HrrFkyZL41re+5YqLXPPzOue8jlRVVdnOubzOPfzww9ZjsH///pHHjWsMHDgw7rvvPldcfOlLX4pdu3bZ8o4++ui49957bXmTJk2KVq1a2fJ++ctfxmc+8xlb3ltvvRX/7b/9N1teXjdvv/1223Upr5lf+9rXrHl333139bXTMXJ+f/M3fxNXXHGFI65eMhD0esHOD60vAnfeeWfccsstth9f9h102xf9IGj37t3RtGlTd6w1zz3HU045JV5//XXbHPNGaPbs2ba8iRMnxjXXXGPLGzNmTIwbN86WN3r06LjnnntseXmjcdlll9nyBgwYEHPnzrXlde7cOVavXm3La9GiRezYscOWl0EtW7aM9957z5a5cOHC6Nu3ry3PHZTHi/PBTh7PeVyXdaR85MNt1zj33HNjypQprrjq9TTXVdc44ogjYuP1iRJOAAAgAElEQVTGja64QnLc16W8jsyZM8c216effjrywdihMsaPHx+jRo2yfd0zzjgjXnrpJVveWWedFTNmzLDl/fu//3t8/OMft+U1b948du7cacvLoHxg4tyQy3XmpJNOss6xLsMQ9Lqkzc+qdwIIer23oOImgKBrLTvUBL1Xr16xfPlyDdp+1bmTvHXrVlteBuVbU84dUQTd2h457FAT9NxddT4UkxtQBwEIugb5UBP0Il5xdz84zrfOnG+zzZ8/P/r166cdKPVYjaDXI3x+dN0TQNDrnnml/0QEXesggq7xQ9A1flnNDrrGsOw76Ai61t+sZgddY1j2HXQEXetvfVQj6PVBnZ9ZbwQQ9HpDX7E/GEHXWoega/wQdI0fgq7zQ9B1hu4EdtA1ouyga/zy16TYQdcYHqgaQT8QIf5/gyKAoDeodtbJl0HQNcwIusYPQdf4Ieg6PwRdZ+hOQNA1ogi6xg9B1/jVpBpBrwklPtNgCCDoDaaVdfZFEHQNNYKu8UPQNX4Ius4PQdcZuhMQdI0ogq7xQ9A1fjWpRtBrQonPNBgCCHqDaWWdfREEXUONoGv8EHSNH4Ku80PQdYbuBARdI4qga/wQdI1fTaoR9JpQ4jMNhgCC3mBaWWdfBEHXUCPoGj8EXeOHoOv8EHSdoTsBQdeIIugaPwRd41eTagS9JpT4TIMhgKA3mFbW2RdB0DXUCLrGD0HX+CHoOj8EXWfoTkDQNaIIusYPQdf41aQaQa8JJT7TYAgg6A2mlXX2RRB0DTWCrvFD0DV+CLrOD0HXGboTEHSNKIKu8UPQNX41qUbQa0KJzzQYAgh6g2llnX0RBF1DjaBr/BB0jR+CrvND0HWG7gQEXSOKoGv8EHSNX02qEfSaUOIzDYYAgt5gWllnXwRB11Aj6Bo/BF3jh6Dr/BB0naE7AUHXiCLoGj8EXeNXk2oEvSaU+EyDIYCgN5hW1tkXQdA11Ai6xg9B1/gh6Do/BF1n6E5A0DWiCLrGD0HX+NWkGkGvCSU+02AIIOgNppV19kUQdA01gq7xQ9A1fgi6zg9B1xm6ExB0jSiCrvFD0DV+NalG0GtCic80GAIIeoNpZZ19EQRdQ42ga/wQdI0fgq7zQ9B1hu4EBF0jiqBr/BB0jV9NqhH0mlDiMw2GAILeYFpZZ18EQddQI+gaPwRd44eg6/wQdJ2hOwFB14gi6Bo/BF3jV5NqBL0mlPhMgyGAoDeYVtbZF0HQNdQIusYPQdf4Ieg6PwRdZ+hOQNA1ogi6xg9B1/jVpBpBrwklPtNgCCDoDaaVdfZFEHQNNYKu8UPQNX4Ius4PQdcZuhMQdI0ogq7xQ9A1fjWpRtBrQonPNBgCCHqDaWWdfREEXUONoGv8EHSNH4Ku80PQdYbuBARdI4qga/wQdI1fTaoR9P0oLVy4MNasWVMTbjX+zDHHHBPHHXdcjT//xz64efPmmDt3riUrQ9avXx/Tp0+PPn362DJ/9rOfxRe+8AVb3j/90z/FVVddZcv7zne+E82aNbPlrV27Njp16mTL27t3byxbtsyW99JLL8WePXtseRk0YMCAaNeunTWzzGFnnHFGNG3a1DZF9zEzbNiwuOmmm2zzu+KKK2LFihW2vFwDH3nkEVve9ddfHzNnzrTlbdiwIZo3bx6tWrWyZOa62qJFC2vehAkTbHnbt2+Pa6+9No466ijL9/0wb/Dgwba8JUuWVK8zrnH11VdHPuhwjczKa4lrbNy4MT7/+c+74mLcuHExZcoUW16ufy+88IIt7/XXX6++rruuI3kOz5s3zza/SggaNGhQNGnSxDbVDh06xKc//Wlb3kMPPRRXXnmlLW/ixIkxYsQIW95//+//PRo3bmzLc1/Xc2JTp06Nrl27Wua4dOnSuOCCC2zn3Ntvv119TXKdw+k3eR4feeSRlu+b16Xvfe971d+5UgeCvl/n8gBp3769tZd5kOSTJtfIkzVPDNfIG9OdO3e64qpvgrZt22bLy5M/++Ia2d9Nmza54qoXJ+f8UvadD4nmzJkT+aTdNfKmft26da44ciBw0AQeffTRuOyyyw667qMK8gFlCqFrdO7cOVavXu2Kq3544Fyjc2LuTPd3dvckZcb5UGfIkCExbdo0W4/zAdvYsWNteWUPynsE5wOTsn/fSpjf0UcfHSmZrtG2bdvYsmWLK6763rzM927ue9+UX+e9dF5Dct13jX379lkfcOS8kmE6k2vkg8CTTjrJFVfnOQj6fsgrQdDzxiWfhLmG+zUV9yLlFuCyL/JlF/QuXbrEqlWrXIcfORA4aAJuQc+dWuebSb169Yrly5cf9Pf6qIK8UXPetOTPcWe6v7O7Jwi67XAkqIESQNC1xrrvffMBh3PzR/t2dVOd9+fO7zx//vzo169f3Uy+gJ+CoCPosWPHDtuh5V6kEHStNe4ddARd6wfVOgEEXWeIoGsM2UHX+FFdPgIIutYT970vgq71I6sRdJ1haRLYQddb4V6kEHStJwi6xo/q8hFA0PWeIOgaQwRd40d1+Qgg6FpP3Pe+CLrWDwRd51eqBARdb4d7kULQtZ4g6Bo/qstHAEHXe4KgawwRdI0f1eUjgKBrPXHf+yLoWj8QdJ1fqRIQdL0d7kUKQdd6gqBr/KguHwEEXe8Jgq4xRNA1flSXjwCCrvXEfe+LoGv9QNB1fqVKQND1drgXKQRd6wmCrvGjunwEEHS9Jwi6xhBB1/hRXT4CCLrWE/e9L4Ku9QNB1/mVKgFB19vhXqQQdK0nCLrGj+ryEUDQ9Z4g6BpDBF3jR3X5CCDoWk/c974IutYPBF3nV6oEBF1vh3uRQtC1niDoGj+qy0cAQdd7gqBrDBF0jR/V5SOAoGs9cd/7IuhaPxB0nV+pEhB0vR3uRQpB13qCoGv8qC4fAQRd7wmCrjFE0DV+VJePAIKu9cR974uga/1A0HV+pUpA0PV2uBcpBF3rCYKu8aO6fAQQdL0nCLrGEEHX+FFdPgIIutYT970vgq71A0HX+ZUqAUHX2+FepBB0rScIusaP6vIRQND1niDoGkMEXeNHdfkIIOhaT9z3vgi61g8EXedXqgQEXW+He5FC0LWeIOgaP6rLRwBB13uCoGsMEXSNH9XlI4Cgaz1x3/si6Fo/EHSdX6kSEHS9He5FCkHXeoKga/yoLh8BBF3vCYKuMUTQNX5Ul48Agq71xH3vi6Br/UDQdX6lSkDQ9Xa4FykEXesJgq7xo7p8BBB0vScIusYQQdf4UV0+Agi61hP3vS+CrvUDQdf5lSoBQdfb4V6kEHStJwi6xo/q8hFA0PWeIOgaQwRd40d1+Qgg6FpP3Pe+CLrWDwRd51eqBARdb4d7kULQtZ4g6Bo/qstHAEHXe4KgawwRdI0f1eUjgKBrPXHf+yLoWj8QdJ1fqRIQdL0d7kUKQdd6gqBr/KguHwEEXe8Jgq4xRNA1flSXjwCCrvXEfe+LoGv9QNB1fqVKQND1drgXKQRd6wmCrvGjunwEEHS9Jwi6xhBB1/hRXT4CCLrWE/e9L4Ku9QNB1/lJCQsXLoxGjRpJGfsXb926NT7zmc9Ehw4dLJkbN26MFStWRMuWLS157733XnzsYx+zfeedO3fGhg0boqqqyjK/zGncuLEl68OQww47LPbu3WvLPPzww+P999+35WWQi19mNWnSJKZNmxb5YMIx5s2bF6NGjYoWLVo44mLPnj2xbNkyS1aG5PGX54lrbNq0KXbs2BFdunSxROY5snLlyujbt68tb8GCBXHSSSdZ8jLksccei4suusiWt27dujjzzDNtee6gFPSvf/3rtmN6165dsXv37mjevLllqnmO5HXJdc5t2bKles1yXpdyHcwbQMfI61Kugbl2OUaec3kdadq0qSMuMi+/b/7nGtlfZ96AAQNi3LhxrunF008/Heedd54tL4O6detmO6atE4uovs/K89g18jqSx/QRRxxhicy87du3R9euXS157utSbk7lQ6dcu1zDfT/ovndz31smt3379tnuz3MNfPHFF233gmU/h3N+eX7kw2PHyOvSCy+8EP369XPE1UtGoyqnXdTxV8gb3f79+9t+at6gZVOdIxc8143Lhwfw22+/bZtifudc7F1j5MiR8cADD7jiSp9zzTXXxI9+9CPbPN1PYY8//vjI86Ssw30Rz+/p3h3s3r17taS7RvYkHy66xuDBg2PGjBmuuBg7dmzcdNNNtjx30Nq1a6NTp0622Oeeey6GDRtmy0uRcR4v+fDAJasffkl3Zp4jb731lo1hnz59YsmSJba8E044IfJhpWvcc889MXr0aFdctZyPGTPGludeEz7xiU/Eyy+/bJufO2jNmjXRuXNna2w+hHHeHruv7T169Kh+MOEa7vldffXVMWnSJNf0Sp9z3XXXxYMPPmibp/vePDctVq1aZZtfJQTlZo1rg7Q+vi+Cvh/13LXMJ51lHnnjsnTpUtsU8+DNg9g1vvGNb8T3vvc9V1zpc9yC3r59e+sxeOqpp8bs2bNLy7EIQe/YsWPkLrBr5BNY50OOgQMHWnvivhkvu6C7+vphzvTp06t3j1wj37ZwPoBxzavInHzotGjRItuPyB3luXPn2vIGDRoUM2fOtOUdaoJ+7rnnxpQpU2z83EGVIOj5CnnO0zVyc2r+/PmuuHDfexxqgv7Nb34z7rrrLls/3PfmvXv3tj70tH1Rgj6SAIKOoCPowgKBoAvwPvj1APevRSDoWk8QdI0fgq7xy2oEXWPofmiHoGv9yGoEXWdY5gQEvczdqcy5IegIOoIunLsIugAPQdfgfVDtvhlH0LW2IOgaPwRd5+deExB0vScIus6wzAkIepm7U5lzQ9ARdARdOHcRdAEegq7BQ9At/HjFXcfIK+4aw7L/DjqCrvWXHXSdX9kTEPSyd6jy5oegI+gIunDeIugCPARdg4egW/gh6DpGBF1jiKBr/PgddI1fVvM76BpDBF3jR/XvE0DQEXQEXVgZEHQBHoKuwUPQLfwQdB0jgq4xRNA1fgi6xg9B1/kh6DpDEn6XAIKOoCPowqqAoAvwEHQNHoJu4Yeg6xgRdI0hgq7xQ9A1fgi6zg9B1xmSgKB/5DHAP7Omnx78M2saQ/drZvwza1o/spp/Zk1nWOYEBF3vDoKuMUTQNX4IusYPQdf5Ieg6QxIQdAR9PwLuf2sRQdeWGARd45fV/DNrGkP+irvGj7/irvHLav6ZNY0hf8Vd45fVjRo1iqqqKj3ogwT+irsNZSmDEPRStqWiJ8Ur7vu1jx10/VhG0DWGCLrGD0HX+SHoGkMEXeOHoOv8EHSdIYKuMbz66qtj0qRJWkgFVSPoFdSsCpkqgo6g8zvowsnK76AL8PgddA3eB9Xum3EEXWsLgq7xQ9B1fu41gX9mTe8JO+g6wzInIOhl7k5lzg1BR9ARdOHcRdAFeAi6Bg9Bt/Djd9B1jPwOusaQ30HX+PE76Bq/rHa/vccOutYT96+f9u7dO5YsWaJNiuo6JYCgI+gIunDKIegCPARdg4egW/gh6DpGBF1jiKBr/BB0jR+CrvNjB11nSMLvEkDQEXQEXVgVEHQBHoKuwUPQLfwQdB0jgq4xRNA1fgi6xg9B1/kh6DpDEhD0jzwG+CNx+unBH4nTGLpfM+OfWdP6kdX8M2s6wzInIOh6dxB0jSGCrvFD0DV+CLrOD0HXGZKAoCPo+xFw/54Lgq4tMQi6xi+r+WfWNIb8kTiNH38kTuOX1fwzaxpD/kicxi+r+SvuGkN+B13j574353fQtX7URzWvuO9HnR10/RBE0DWGCLrGD0HX+SHoGkMEXeOHoOv8EHSdIYKuMUTQNX4IusavIVQj6Ag6v4MunMn8DroAj99B1+B9UO2+GUfQtbYg6Bo/BF3n514T+GfW9J7wz6zpDMucwCvuZe5OZc6t4gU9X4Vr2rSphX4+MX377bctWRmyZ8+e6v/ySZhjZFb+PuyOHTsccbF3797q/95//31LXoZcccUV8eMf/9iSt3v37nj33XctWR+GzJs3Lz71qU/ZMocPHx5Tpkyx5eWx3Lhx4zjssMMsmUcccUT827/9myUrQzZu3BitW7e2nXNVVVVx1FFH2fLymMnzzXVMZ16HDh1i69atFoZ5vuX33bRpky0vf//X+c+n5Dl8//33W+aXIYsWLYpjjjnGlrdy5cro0qWL7RyZPHlyfOUrX7Hl5fmxYsUK2zG9bds2G7v9g3KejpHnSI8ePcI1zzxH8pzbsmWLY3rV17jjjjsuli1bZsv7zne+E6NHj7bkZcill14azz77rC2vZ8+esXTpUtsxnefbrFmzbHn5RfNa57p3W7BgQfWvRTRp0sTCMO+18n7QlZfnSKtWrarvBx2jiOtSXosz1zWGDRsWzzzzjCsufvWrX8UJJ5xgy8ugvD9yHYNf+tKX4uGHH7bN7/DDD68+31z3gnn85bXT9X23b98eecy4Rp4bu3btqr6/dI28huTaVamjogV9/fr11Te7rrF58+bqfwvSOXKBdy3KOa98DT/n6RrNmzePnTt3uuJi5MiR8cADD9jy8sbKKR+5s5B/FMo1LrzwwnjiiSdccdUXjJRg15gzZ04MHDjQFVe9eLpuxHNSeRO0b98+2/wyKOfnXORTLlevXm2b46OPPhqXXHKJLS/n1rlzZ1ueO+jee++NG2+80RabN2n5oM01+vTpY11jjjzyyMhrk2vk8dymTRtXXHVOPnByniN5Hd6wYYNtjvnQM3dtXWPt2rXRqVMnV1y89dZb0a1bN1ue+4/EnXjiifHmm2/a5uf+Q5m5Xr3zzju2+eXDiDyPnSOvS3l9co0855zncYqHk6H72n7BBRfE448/7sIXZ599dsyYMcOWl8fL4sWLbXnXXXddPPjgg7Y89725+94yNwpT+p2jWbNm1ZLuGvPnz6/e1KzUUdGC7oZehKC7f4+ke/fu1U/BXMM9P/fvoJ900kkxd+5c19cNt6C7X3HPm8j8i7Su4RZ092t6RQi6i92HOaecckq8/vrrtli3fNgmVlCQW9AHDRoUM2fOtM3W/QfJ3K+4V4Kgu/+K+7Rp06pvyA+V4RZ09yvup512Wrz66qu2dpx88snx2muv2fKK+CvubkG3fdkPgvLBe17fXcP992/cv4PuFvRc99944w0XvjjUXnEvQtDbtm1re3MqG4ug2w7v+g9C0PUeIOgaQwRd41dENYKuUUXQNX4IusavEqoRdK1LCLrGL6sRdI0hgq7xy2oE/XcZsoO+Hw8EXT/BEHSNIYKu8SuiGkHXqCLoGj8EXeNXCdUIutYlBF3jh6Dr/BB0nSGCjqB/5FGEoOsnGIKuMUTQNX5FVCPoGlUEXeOHoGv8KqEaQde6hKBr/BB0nR+CrjNE0BF0BH0/AvwOurao8DvoGj9+B13jVwnVCLrWJQRd41cJ1Qi61iUEXeOHoOv8EHSdIYKOoCPoCLq+knyQgKBrKBF0jV8lVCPoWpcQdI1fJVQj6FqXEHSNH4Ku80PQdYYIOoKOoCPo+kqCoFsYIugWjKUOQdC19iDoGr9KqEbQtS4h6Bo/BF3nh6DrDBF0BB1BR9D1lQRBtzBE0C0YSx2CoGvtQdA1fpVQjaBrXULQNX4Ius4PQdcZIugIOoKOoOsrCYJuYYigWzCWOgRB19qDoGv8KqEaQde6hKBr/BB0nR+CrjNE0BF0BB1B11cSBN3CEEG3YCx1CIKutQdB1/hVQjWCrnUJQdf4Ieg6PwRdZ4igI+gIOoKuryQIuoUhgm7BWOoQBF1rD4Ku8auEagRd6xKCrvFD0HV+CLrOEEFH0BF0BF1fSRB0C0ME3YKx1CEIutYeBF3jVwnVCLrWJQRd44eg6/wQdJ0hgo6gI+gIur6SIOgWhgi6BWOpQxB0rT0IusavEqoRdK1LCLrGD0HX+SHoOkMEHUFH0BF0fSVB0C0MEXQLxlKHIOhaexB0jV8lVCPoWpcQdI0fgq7zQ9B1hgg6go6gI+j6SoKgWxgi6BaMpQ5B0LX2IOgav0qoRtC1LiHoGj8EXeeHoOsMEXQEHUFH0PWVBEG3METQLRhLHYKga+1B0DV+lVCNoGtdQtA1fgi6zg9B1xki6Ag6go6g6ysJgm5hiKBbMJY6BEHX2oOga/wqoRpB17qEoGv8EHSdH4KuM0TQEXQEHUHXVxIE3cIQQbdgLHUIgq61B0HX+FVCNYKudQlB1/gh6Do/BF1niKAj6HUq6E2aNIk9e/boR+4HCe3atYvNmzfb8po3bx47d+605Y0cOTIeeOABW95xxx0XS5YsseUNHjw4pk+fbsu78MIL44knnrDlHXHEEbFx40Zb3i9/+cv47Gc/a8tr3bp1pDC4RhGCnvPLebrGMcccE6tXr3bFxaOPPhqXXHKJLS/n1rlzZ1ueO+j222+Pb3/727bYE044IebNm2fL69Onj3WNOfLII2P9+vW2+VWCoB911FGxYcMG23d+9tlnY9iwYbY8d9Bbb70V3bp1s8W6Bf3EE0+MN9980za/fv36xYIFC2x5uV698847trxDUdC7dOliZei+tl9wwQXx+OOP23p89tlnx4wZM2x5ue4vXrzYlnfdddfFgw8+aMtz35u77y3TQ9q3b2/7vhnUrFmz2LVrly1z/vz5kWtXpY5GVVVVVZU6efe8t2zZEo0bN7bF7t69O7p37x4pIY6Rop+LqOsA3rt3b+R/zgcIV1xxRTz00EOOrxvJ791337VkfRiSN/af+tSnbJnDhw+PvJl0jZYtW1Z/56ZNm1oin3nmmciHCPmgyDEyZ8eOHba8D4891/fNY2bFihVx9NFHO75u9TGYD4lcD03yfEuZ2bRpk2V+mff9738/rrnmGktehlx11VXWG6tjjz02Fi1aFIcddphljrmmLl++3JbXpk2b6n645pdrdB6DrmPa+UBs/wa4HmLlOdKjRw/bg7s8pidNmhRf+MIXLMdLhnz605+Ol156yZKX88ubPufN/XnnnWd9sP2P//iP8Xd/93e2YzplcNasWba8bESeH65zJNdnV9aHB0le61IYXCPXfde9W55zHTp0iK1bt1qml8d0qoBrfjmpP//zP4+8/3CNX/3qV5EPZ50jpdV13Fx55ZXx8MMP26aX88qH7657t7xmDhw40PZ99+3bF++//75tfnkvmA8Vc61xjTw/coOlUgeCXnDnWrVqVS00rpE3pytXrnTFRQqhc37f+MY34nvf+55tfmUPSjH60Y9+ZJtmp06dIncDXGPOnDnVi7JrpPg655cPr9zPCPOi5hL05HbKKafE66+/7kJY3Y/Zs2fb8iZOnGgV9DFjxkTu6LnG0KFDY+rUqa64GDRoUMycOdOWN2DAgJg7d64tr2/fvrFw4UJbXiUEHX/88dUPYVxj2rRpkTtmrnHZZZdVv7niGkOGDImco2vcdNNNMXbsWFdc9QOOESNG2PLOPffcmDJlii3vUAzq2rVrvP3227av3r9//8gdQtfI3VDXg+Oc09VXX119HB4qw/2Ke77u7XxbdunSpZFvDbiG2x1yXnn85VvCjP8ggKAXfCQg6AUDrud4BF1rAIKu8ctqBF1jiKBr/LIaQdcYIugav0qoRtAroUu1nyOCXnt2H1Yi6L/LEEHXj6k/moCgFwy4nuMRdK0BCLrGD0HX+SHoOkMEXWOIoGv8KqEaQa+ELtV+jgh67dkh6H+YHYKuH1MI+n4EeMVdO6B4xV3jl9W84q4x5BV3jR+vuGv8sppX3DWGvOKu8SuiGkEvgmp5MhF0vRfsoP8uQwRdP6YQdATddhQh6DpKBF1jiKBr/BB0jR+CrvND0HWG7gQE3U20XHkIut4PBB1B14+ig0jgFfeDgFWBH+UVd61pvOKu8ctqfgddY8gr7hq/rOYVd40hr7hr/CqhGkGvhC7Vfo4Ieu3ZfViJoCPo+lF0EAkI+kHAqsCPIuha0xB0jR+CrvND0HWGCLrGEEHX+FVCNYJeCV2q/RwR9NqzQ9D/MDtecdePqT+agKAXDLie4xF0rQEIusYPQdf5Ieg6QwRdY4iga/wqoRpBr4Qu1X6OCHrt2SHoCLp+9NQiAUGvBbQKKkHQtWYh6Bo/BF3nh6DrDBF0jSGCrvGrhGoEvRK6VPs5Iui1Z4egI+j60VOLBAS9FtAqqARB15qFoGv8EHSdH4KuM0TQNYYIusavEqoR9EroUu3niKDXnh2CjqDrR08tEhD0WkCroBIEXWsWgq7xQ9B1fgi6zhBB1xgi6Bq/SqhG0CuhS7WfI4Jee3YIOoKuHz21SEDQawGtgkoQdK1ZCLrGD0HX+SHoOkMEXWOIoGv8KqEaQa+ELtV+jgh67dkh6Ai6fvTUIgFBrwW0CipB0LVmIegaPwRd54eg6wwRdI0hgq7xq4RqBL0SulT7OSLotWeHoCPo+tFTiwQEvRbQKqgEQdeahaBr/BB0nR+CrjNE0DWGCLrGrxKqEfRK6FLt54ig154dgo6g60dPLRIQ9FpAq6ASBF1rFoKu8UPQdX4Ius4QQdcYIugav0qoRtAroUu1nyOCXnt2CDqCrh89tUhA0GsBrYJKEHStWQi6xg9B1/kh6DpDBF1jiKBr/CqhGkGvhC7Vfo4Ieu3ZIegIun701CIBQa8FtAoqQdC1ZiHoGj8EXeeHoOsMEXSNIYKu8auEagS9ErpU+zki6LVnh6Aj6PrRU4sEBL0W0CqoBEHXmoWga/wQdJ0fgq4zRNA1hgi6xq8SqhH0SuhS7eeIoNeeHYKOoOtHTy0SmjZtGnv27KlF5R8u6dSpU6xdu9aW17x589i5c6ctb+TIkfHAAw/Y8soe5Bb0I444IjZu3Gj72jNmzIizzz7blte6devYtm2bLa8SBL1v376xePFi23dOmVm4cKEtb+LEiZHHoWuMGTMmxo0b54qLoUOHxtSpU215J5xwQsybN8+W16dPn1iyZIktr1u3brFy5Upb3u7duyOvI87hzuzevXu89atL0UMAACAASURBVNZbtik+++yzMWzYMFvepz71qfj1r39tyxsyZEhMmzbNlld2Qf/EJz4RL7/8su375jUkryWu4c5zzWv/nLZt28bWrVtt0T169IgVK1bY8tzX9quvvjomTZpkm1/Zg6677rp48MEHbdPMzT3nvdbSpUsjr3Wu0aRJE6vb5Lw2bdoU7dq1c02x4nMaVVVVVVX8tyjpF0jR6t+/f+SJ5hgp0tmuFi1aOOKqxXz9+vWWrAzJuTVu3NiWl0GHHXZY7N2715Z5+OGHx/vvv2/L27dvX6RkukbeiOeNboq6Y/z85z+P0aNHRz6IcYw8Zp566qlo3769I656Qc7hysusP/3TP42WLVta5vfee+/F97///fjYxz5myUt+Kee5a+saX/ziF/8fR0dmnh/Oy0I+oMyLuWtkb3/2s5/ZjunJkyfHvffea8vL75trgmud3rJlS/Wa1aFDBwvCvC7lOpjC4Bh5juTx4upxniPNmjWzPZTI+eXullP4f/GLX8RnP/tZB77qjNtuuy1eeeUVW14eM67jLyeVN80zZ860Zb7++uvxF3/xF7a87HH+5zxH1q1bZ7u25zmXa77zHMk1Js8Tx8hzLueY9zOu4bwvyjm5793c95Y5R+f9YN5L54ac6zzOh7LPP/+8q73V7pA9cd67pSt16dLFNsdKD0LQC+7gofaKu/spbN4YbN682dalXEw+lEJHaN7k5s2Qa+SCvGbNGldczJkzJwYOHGjLy8Vz1apVtrwigjp37mxl+Nprr8XJJ59cxFQtmSNGjLDuVLh3vN15l156aTzyyCMWdhkyffr0yB1R1+jVq1csX77cFVf9gHf79u22vAxyZ7q/s/vXDvJ4yeOmrCPfWMk3V1xj8ODBkW9Puca5554bU6ZMccVFCvopp5xiy+vYsWOkUDuHU7ZyXofaK+7ueyP3vZv73tJ975sPonfs2GE7pHv37m19U8w2MYI+kgCCXvDBgaBrgN2LqHuRd1+EEHTteMlqBF1j6BZqdx6CrvUXQdf5uRMQdI0ogq7xy+qy3xu55+e+t0TQ9WOQhN8lgKAXfEQg6Bpg9yLqXuQRdK2/RVQj6BpVt1C78xB0rb8Ius7PnYCga0QRdI0fgq7zQ9B1hiQg6HV6DCDoGm4EXePHK+4av6zmFXeNIYKu8XO/jo6ga/0oohpB16gi6Bo/BF3nh6DrDElA0Ov0GEDQNdwIusYPQdf4Ieg6PwRdY4iga/yymt9B1xjyO+gav6zmd9A1hu63H933lgi61l+qf58Ar7gXfFQg6Bpg9yLqXuR5xV3rbxHVvOKuUXULtTuPV9y1/rKDrvNzJ7CDrhFlB13jxw66zg9B1xmSwA56nR4DCLqGG0HX+LGDrvFjB13nh6BrDNlB1/ixg67zYwddZ8gOusbQvbnivrdE0LX+Us0Oep0fAwi6hty9iLoXeXbQtf4WUc0OukbVLdTuPHbQtf6yg67zcyewg64RZQdd48cOus4PQdcZksAOep0eAwi6hhtB1/ixg67xYwdd54egawzZQdf4sYOu82MHXWfIDrrG0L254r63RNC1/lLNDnqdHwMIuobcvYi6F3l20LX+FlHNDrpG1S3U7jx20LX+soOu83MnsIOuEWUHXePHDrrOD0HXGZLADnqdHgMIuoYbQdf4sYOu8WMHXeeHoGsM2UHX+LGDrvNjB11nyA66xtC9ueK+t0TQtf5SzQ56nR8DCLqG3L2Iuhd5dtC1/hZRzQ66RtUt1O48dtC1/rKDrvNzJ7CDrhFlB13jxw66zg9B1xmSwA56nR4DCLqGG0HX+LGDrvFjB13nh6BrDNlB1/ixg67zYwddZ8gOusbQvbnivrdE0LX+Us0Oep0fAwi6hty9iLoXeXbQtf4WUc0OukbVLdTuPHbQtf6yg67zcyewg64RZQdd48cOus4PQdcZksAOep0eAwi6hhtB1/ixg67xYwdd54egawzZQdf4sYOu82MHXWfIDrrG0L254r63RNC1/lLNDnqdHwMIuobcvYi6F3l20LX+FlHNDrpG1S3U7jx20LX+soOu83MnsIOuEWUHXePHDrrOD0HXGZLADnqdHgMIuoYbQdf4sYOu8WMHXeeHoGsM2UHX+LGDrvNjB11nyA66xtC9ueK+t0TQtf5SzQ56nR8DCLqG3L2Iuhd5dtC1/hZRzQ66RtUt1O48dtC1/rKDrvNzJ7CDrhFlB13jxw66zg9B1xmSwA76Rx4D77//fhx++OHWY6RFixaxc+dOW2anTp1i7dq1trzmzZtb5+depBB0rdXuHfSjjjoq1q1bp02q4OoOHTrEu+++a/spr732Wpx88sm2PHfQiBEjYtKkSbZYt1C789yC/txzz8WwYcNs/PIB0erVq2157jU6J+bOdH/nPn36xJIlS2wMH3nkkcjjpqwDQdc6437wnrPZu3dvNG7cWJvYftX5EGH9+vW2vB49esSKFStseW6G7s0L9/zc95bue1/3Gt2lS5dYtWqV7XjZt2+f9fzIiRXhYLYvXA9Bjaqqqqrq4eeW8kdu3rw5hgwZEnniOkbm3XXXXdU3Q46xbdu2uP766yMXKsfYvn17nH766TF48GBHXHXGt7/9bdv8Mi8Xga9//eu2+d15552RbzW4xjvvvBPHHHOMKy727NkTv/3tb215L7/8cuzevduWlzcYt956q+0cyRuM5cuX2+aX59zcuXNteRn0la98JVL6XeOLX/xifPnLX3bFxSc/+UnrQ7aWLVvGd7/7Xdv8nnzyyTj//PNteXmT8YUvfMGW9/3vfz/uvvtu27qQ50g+mHWtM5k3YcIEW15eRzZt2hTdunWzMMy8XAf79u1ryxs1apTt++akBg0aFD/84Q8t8ysi5Cc/+UmkcLnGU089FZ/73OdccdUimOuWa7zyyiuxa9cuV1z1mu88h9esWVN93+bqSV6X8rqZD7gdI8+5r371q9GmTRtHXHXGypUrrde5Ro0axde+9jXb/PIccR6Df//3fx9Nmza1zW/Dhg2R95eukQ8ozz77bNsx8/bbb8fYsWNt926Zl9c4py9NmzbNlufqQ33mIOj70c9F1CW/H8amBOcNr2vkzsLSpUtdcTFjxow466yzbHkXX3xxPPbYY7a8++67L2644QZbXi6gt9xyiy0vF9Dp06fb8oYPHx5PP/20Lc8dlIty/i6da+RFPB/ClHmccsop8frrr9umOGXKlMjf6XQN9w56XsRvuukm1/RKn5Pnbz6YdY1evXpZHzrlzszWrVtd06uInOyHc10t+w56RTSlxJNMoc63OJwjr03O/at8sJi7mK4xcODAyDfkXCM3a2bNmuWKiwcffDCuvfZaW547aPz48ZEPAl3joosuismTJ7vi7DnpDekPrpFes2PHDldcdU4+OHYJv3Vi9RSGoO8HHkHXj0IEXWOIoGv8iqhG0IugWp5MBL08vfhwJgh6+XpS5hkh6Hp3EHSNIYKu8UPQf58fgo6gs4MurCvsoAvwIoIddI1fVrODrjFE0DV+RVQj6EVQbbiZCLreWwRdY4iga/wQdAT9jx5B7KDrJxg76BpDdtA1fkVUs4NeBNXyZCLo5ekFO+jl60UlzAhB17uEoGsMEXSNH4KOoCPo/4kAv4OuLSrsoGv82EHX+LGDrvND0HWG7gR20N1EG3Yegq73F0HXGCLoGj8EHUFH0BF0/kicsI7yR+IEeB+U8kfidIbOBATdSdOThaB7OB4qKQi63mkEXWOIoGv8EHQEHUFH0BF0YR1F0AV4CLoOr4AEBL0AqGIkgi4CPMTKEXS94Qi6xhBB1/gh6Ag6go6gI+jCOoqgC/AQdB1eAQkIegFQxUgEXQR4iJUj6HrDEXSNIYKu8UPQEXQEHUFH0IV1FEEX4CHoOrwCEhD0AqCKkQi6CPAQK0fQ9YYj6BpDBF3jh6Aj6Ag6go6gC+sogi7AQ9B1eAUkIOgFQBUjEXQR4CFWjqDrDUfQNYYIusYPQUfQEXQEHUEX1lEEXYCHoOvwCkhA0AuAKkYi6CLAQ6wcQdcbjqBrDBF0jR+CjqAj6Ag6gi6sowi6AA9B1+EVkICgFwBVjETQRYCHWDmCrjccQdcYIugaPwQdQUfQEXQEXVhHEXQBHoKuwysgAUEvAKoYiaCLAA+xcgRdbziCrjFE0DV+CDqCjqAj6Ai6sI4i6AI8BF2HV0ACgl4AVDESQRcBHmLlCLrecARdY4iga/wQdAQdQUfQEXRhHUXQBXgIug6vgAQEvQCoYiSCLgI8xMoRdL3hCLrGEEHX+CHoCDqCjqAj6MI6iqAL8BB0HV4BCQh6AVDFSARdBHiIlSPoesMRdI0hgq7xQ9ARdAQdQUfQhXUUQRfgIeg6vAISEPQCoIqRCLoI8BArR9D1hiPoGkMEXeOHoCPoCDqCjqAL6yiCLsBD0HV4BSQg6AVAFSMRdBHgIVaOoOsNR9A1hgi6xg9BR9ArXtC7du0aKUmuMWPGjDjrrLNccXHxxRfHY489Zsu777774oYbbrDl3XnnnQi6QBNBF+Ah6Dq8AhLcgt65c+dYvXq1baYtWrSIHTt22PIqIQhBr4QulWeOlSDoy5Yti169etmgDRw4MObMmWPLQ9A1lIeaoDdv3jx27typQftP1Zs2bYp27dpZMys5rFFVVVVVJX8B59xzAZ01a5YzMn7wgx/E0Ucfbclct25dXHjhhdGpUydLXobcddddceyxx9rymjVrFuedd54tLwW9S5cutrwXX3wxevToYctbu3ZtjB071paXN/ajR4+25T355JPx05/+1Ja3YMGCGDNmjC0vg8455xzrMX3GGWdEz549bXPMh1jOvOxx3gy5RqNGjarXBdfYuHFjXH/99a64+OY3vxkrVqyw5WXQI488YsvL4/k3v/mNLS/7m5lt27a1ZbrPEdvEPggaOnRo5IMJ1/jf//t/R/fu3V1x8bGPfSwmTJhgy7v22mtj27ZttrwOHTrE/fffb8s71IJeeeWVWL58ue1rb9myJf7xH//Rdg5n3ne+8x3rvdFxxx1nvW4uWrQo+vbta2OYDyP++Z//2ZbnDpo0aVK0atXKFvv000/Hrl27bHlbt26NZ5991paX9zHOzb3169dH06ZNbedIftG8L3L6iA1ePQUh6AWDzwXAufuxZMmS6N27t23WgwcPjpRW15g8eXLkk0TXGDVqVIwfP94VF3kjOXXqVFte3qg9+OCDtjx3UD5hzyftrpEPS1atWuWKKyQnRSF3VFyjX79+kQ8mXCP7MXv2bFdcTJw4Ma655hpbnjvo3nvvjRtvvNEWe+mll1oF3b2Dnje5CxcutH3fSgg6/vjjI2/wXWPAgAExd+5cV1z18ZLHjWvk8ZzHtWvcdNNN1ge9rnkdyjmNGzcO5/5VXjedmw3soJfr6Mw3R/MNUtfIN1tTqhmHLgEEveDeI+gaYARd44ega/yyGkHXGCLoGr9KqEbQtS4h6Bq/IqoRdI1qblzkBsahMhD0Q6XTdfc9EfSCWSPoGmAEXeOHoGv8EHSdH4KuMyx7AoKudQhB1/gVUY2ga1QRdI0fO+gav4ZQjaAX3EUEXQOMoGv8EHSNH4Ku80PQdYZlT0DQtQ4h6Bq/IqoRdI0qgq7xQ9A1fg2hGkEvuIsIugYYQdf4IegaPwRd54eg6wzLnoCgax1C0DV+RVQj6BpVBF3jh6Br/BpCNYJecBcRdA0wgq7xQ9A1fgi6zg9B1xmWPQFB1zqEoGv8iqhG0DWqCLrGD0HX+DWEagS94C4i6BpgBF3jh6Br/BB0nR+CrjMsewKCrnUIQdf4FVGNoGtUEXSNH4Ku8WsI1Qh6wV1E0DXACLrGD0HX+CHoOj8EXWdY9gQEXesQgq7xK6IaQdeoIugaPwRd49cQqhH0gruIoGuAEXSNH4Ku8UPQdX4Ius6w7AkIutYhBF3jV0Q1gq5RRdA1fgi6xq8hVCPoBXcRQdcAI+gaPwRd44eg6/wQdJ1h2RMQdK1DCLrGr4hqBF2jiqBr/BB0jV9DqEbQC+4igq4BRtA1fgi6xg9B1/kh6DrDsicg6FqHEHSNXxHVCLpGFUHX+CHoGr+GUI2gF9xFBF0DjKBr/BB0jR+CrvND0HWGZU9A0LUOIegavyKqEXSNKoKu8UPQNX4NoRpBL7iLCLoGGEHX+CHoGj8EXeeHoOsMy56AoGsdQtA1fkVUI+gaVQRd44ega/waQjWCXnAXEXQNMIKu8UPQNX4Ius4PQdcZlj0BQdc6hKBr/IqoRtA1qgi6xg9B1/g1hGoEveAuIugaYARd44ega/wQdJ0fgq4zLHsCgq51CEHX+BVRjaBrVBF0jR+CrvFrCNUIesFdRNA1wAi6xg9B1/gh6Do/BF1nWPYEBF3rEIKu8SuiGkHXqCLoGj8EXePXEKoR9IK7iKBrgBF0jR+CrvFD0HV+CLrOsOwJCLrWIQRd41dENYKuUUXQNX4IusavIVQj6AV3EUHXACPoGj8EXeOHoOv8EHSdYdkTEHStQwi6xq+IagRdo4qga/wQdI1fQ6hG0AvuYosWLWLnzp22n7JkyZLo3bu3LW/w4MHx4osv2vImT54cF110kS0PQddQIugaPwRd54eg6wzLnoCgax1C0DV+RVQj6BpVBF3jh6Br/BpCdUUL+q9+9av4l3/5F1sfVq9eHY888ogt76233oq/+Iu/iLZt29oy//Iv/zKuv/56W94tt9wSHTt2tOXld7366qttebfeemt06NDBlpey0LVrV1tePjBxPpBIoT711FNt83v11Vfjsssus+WtWLEi7rrrLlteVVVVXHLJJdae/K//9b+ie/futjnOmzcvTjjhhNLm5fnxxBNP2Ob3zW9+M3bv3m3LyzW6Z8+etrzFixfHxRdfbMubPXt2XHrppba8VatWxd///d/b8ioh6K//+q+t5/B3v/vd6NOnj+2rr1y5svpa7BqPP/649UH5vn37Yvr06a7pkSMSyHu38847L9q1aycm/f/yESNGxFVXXWXLO+6446znXG4kffGLX7TNr2nTptZ7VdvEPgi67bbbYuPGjbbYp59+Orp162bLy3utz33uc7Y8933MO++8E48++qhtfgT9PoGKFvQFCxZE//79bX3NxXjTpk22vAwq+yvu1i9bAWF33nln5EMJ1zj77LOtN1ann356zJo1yzW9atlPASnrSEHPnQrnyAdO69ats0X269cvcq1xjYEDB1p7MnHixLjmmmtc04sxY8bEuHHjbHlDhw6NqVOn2vIGDRoUM2fOtOXlOTxt2jRbHkE6gSFDhljXVfcxk/NzHjPsoOvHjDvhUNtBT8EcPny4G2Np88aPHx/5hqZrnHHGGfHSSy+54uLEE0+MN99805aXmxb5oNI1WrZsGdu3b3fFkfMHCCDo+0FB0Bv+OYKgl6vHCLreDwRdY4iga/yKqEbQi6BK5sEQQNAPhlblfRZB13qGoGv8alKNoCPoNTlOGsxnEPRytRJB1/uBoGsMEXSNXxHVCHoRVMk8GAII+sHQqrzPIuhazxB0jV9NqhF0BL0mx0mD+QyCXq5WIuh6PxB0jSGCrvErohpBL4IqmQdDAEE/GFqV91kEXesZgq7xq0k1go6g1+Q4aTCfQdDL1UoEXe8Hgq4xRNA1fkVUI+hFUCXzYAgg6AdDq/I+i6BrPUPQNX41qUbQEfSaHCcN5jMIerlaiaDr/UDQNYYIusaviGoEvQiqZB4MAQT9YGhV3mcRdK1nCLrGrybVCDqCXpPjpMF8BkEvVysRdL0fCLrGEEHX+BVRjaAXQZXMgyGAoB8Mrcr7LIKu9QxB1/jVpBpBR9Brcpw0mM8g6OVqJYKu9wNB1xgi6Bq/IqoR9CKoknkwBBD0g6FVeZ9F0LWeIegav5pUI+gIek2OkwbzGQS9XK1E0PV+IOgaQwRd41dENYJeBFUyD4YAgn4wtCrvswi61jMEXeNXk2oEHUGvyXHSYD6DoJerlQi63g8EXWOIoGv8iqhG0IugSubBEEDQD4ZW5X0WQdd6hqBr/GpSjaAj6DU5ThrMZxD0crUSQdf7gaBrDBF0jV8R1Qh6EVTJPBgCCPrB0Kq8zyLoWs8QdI1fTaoRdAS9JsdJg/kMgl6uViLoej8QdI0hgq7xK6IaQS+CKpkHQwBBPxhalfdZBF3rGYKu8atJNYKOoNfkOGkwn0HQy9VKBF3vB4KuMUTQNX5FVCPoRVAl82AIIOgHQ6vyPougaz1D0DV+NalG0BH0mhwnDeYzCHq5Womg6/1A0DWGCLrGr4hqBL0IqmQeDAEE/WBoVd5nEXStZwi6xq8m1Qg6gl6T46TBfAZBL1crEXS9Hwi6xhBB1/gVUY2gF0GVzIMhgKAfDK3K+yyCrvUMQdf41aQaQUfQa3KcNJjPIOjlaiWCrvcDQdcYIugavyKqEfQiqJJ5MAQQ9IOhVXmfRdC1niHoGr+aVCPoCHpNjpMG8xkEvVytRND1fiDoGkMEXeNXRDWCXgRVMg+GAIJ+MLQq77MIutYzBF3jV5NqBB1Br8lx0mA+g6CXq5UIut4PBF1jiKBr/IqoRtCLoErmwRBA0A+GVuV9FkHXeoaga/xqUo2gI+g1OU4azGcQ9HK1EkHX+4GgawwRdI1fEdUIehFUyTwYAgj6wdCqvM8i6FrPEHSNX02qK1rQ77vvvnj88cdr8j1r9Jnf/va3cd1119XoszX90Ny5c+PYY4+t6ccP+LmVK1fGxz/+8QN+rqYf6NmzZ1x++eU1/Xidf27MmDHRpk0b28994IEHIr+zayxevDiuvPJKV1xMnTo1hg4dasvbuHFj/PCHP7TluYNS0JNfjx49bNE/+9nPrD3etWtXnHPOObb5/ehHP4p+/frZ8tq3bx9PPPGELS/l6LDDDrPlLVmyJK644gpb3s9//vPo1q2bLe+dd96J+fPn2/IqIegrX/mK9ZybPXt2DBw40PbVn3vuuTjrrLNsedOmTYs8rl1j+vTpkQ92nGPs2LHOOGtWrll5nrjGq6++GqeddporLubMmROnnnqqLS+D5s2bZz1HVq1aFf3797fN8cEHH7TOr3v37vHjH//YNj930O233x5NmjSxxT7//PNxxhln2PLc1yX3veUbb7wRJ510ku37vvXWW/bjxX1dyjV68ODBtu9c10EVLegLFiywLnht27aNLVu2WHuwffv2yCdNrtGnT59YunSpKy5mzJhhvRGyTeyDoFGjRkU+6XSNlN+UYNe49tprIy+UjPIQ6Ny5c6xZs8Y2oddeey1OPvlkW96wYcMiBcQ1yr6DPnr06LjnnntcXzceffTRuOyyy2x5h+IOej703LZtm41hr169Yvny5ba8FGq3ANsmdwgG5XqV65Zr5APKvH9zjY4dO8a6detccdU5+/bti0aNGtkyu3btGm+//bYtL2Xf+WDx6aefjuHDh9vm5w665ZZbIt+AdI0bbrghcpPPNR577LG4+OKLXXHV9+V5f34ojdxs2Lx5s+0r5/nh3AyxTayGQQj6fqAQ9BoeNXX4MQS9DmE3kB+FoGuNzLdWxo0bp4XsV42g21DaghB0G8pDIghB19uMoGsMEXSNXyVUI+i/2yUEHUFnB11YudhBF+AVVIqga2ARdI1fJVQj6JXQpfLMEUHXe4GgawwRdI1fJVQj6Aj6Rx6n7KCX7xRmB718PSn7jBB0rUMIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQd+PAL+Dri1b7KBr/IqoRtA1qgi6xq8SqhH0SuhSeeaIoOu9QNA1hgi6xq8SqhF0BB1BR9BtaxWCbkNpC0LQNZQIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQUfQbWsVgm5DaQtC0DWUCLrGrxKqEfRK6FJ55oig671A0DWGCLrGrxKqEXQEHUFH0G1rFYJuQ2kLQtA1lAi6xq8SqhH0SuhSeeaIoOu9QNA1hgi6xq8SqhF0BB1BR9BtaxWCbkNpC0LQNZQIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQUfQbWsVgm5DaQtC0DWUCLrGrxKqEfRK6FJ55oig671A0DWGCLrGrxKqEXQEHUFH0G1rFYJuQ2kLQtA1lAi6xq8SqhH0SuhSeeaIoOu9QNA1hgi6xq8SqhF0BB1BR9BtaxWCbkNpC0LQNZQIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQUfQbWsVgm5DaQtC0DWUCLrGrxKqEfRK6FJ55oig671A0DWGCLrGrxKqEXQEHUFH0G1rFYJuQ2kLQtA1lAi6xq8SqhH0SuhSeeaIoOu9QNA1hgi6xq8SqhF0BB1BR9BtaxWCbkNpC0LQNZQIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQUfQbWsVgm5DaQtC0DWUCLrGrxKqEfRK6FJ55oig671A0DWGCLrGrxKqEXQEHUFH0G1rFYJuQ2kLQtA1lAi6xq8SqhH0SuhSeeaIoOu9QNA1hgi6xq8SqhF0BB1BR9BtaxWCbkNpC0LQNZQIusavEqoR9EroUnnmiKDrvUDQNYYIusavEqoRdAQdQUfQbWsVgm5DaQtC0DWUCLrGrxKqEfRK6FJ55oig671A0DWGCLrGrxKqEfQGJOhjx46NKVOm2I67zZs3R7du3SJv8B1j9erV8fnPfz6aNm3qiKvOuPXWW6NHjx62vLPPPjuSY1nH9ddfH7t377ZN7/nnn49evXrZ8lauXBnf/va3bXnr1q2Lb33rW7a8Z599NtasWWPLy6Bhw4bZzhHrxD4IO//886NDhw626MMPPzwGDRpky3vwwQejX79+tryePXtaj8HRo0fHtm3bbPNbsmRJXH311ba8//k//2dkT1xj+fLlcfvtt7viYsWKFdZ+LFu2LH7zm9/Y5pfXuQkTJkTeDLlG9rhPnz6uuGjevHlcfvnltrwXXnghInZ/KQAAIABJREFU/uzP/syWt2vXrshrU1nHpEmTrFP71a9+FU2aNLFlTp8+3XoNKeLe7ZJLLrF935xfrvtHHHGELXPDhg1xxhln2PKGDBliXadvu+02673qj3/84zj22GNt37fs16W8F5w/f77t+7qD8rr561//2hZbxHUp7wW//vWv2+ZY10GNqqqqqur6h7p+3oIFC6J///6uuGjVqpX1xjQn1qJFi9i5c6dtjt27d4+UQteYMWNGnHXWWa640ufceeedkU9iXSMfcOTNhmsMHz48nn76aVdczJkzJwYOHGjLO+qooyIvHIfSOOWUU+L111+3feV8qHjuuefa8soedO+998aNN95om2Y+LJk5c6Ytb8CAATF37lxbXt++fWPhwoW2vHxYkjvezpEC7Lwu5UPPvGFzDXdP3MfMPffcE/kgq6wjBX3EiBG26eV65dwMyfU011XXSPHduHGjK646p3HjxuG8Pc45vvvuu7Y5zp49O0499VRbnjto5MiR1Q8CXeOOO+6Im2++2RUX48ePj1GjRtny8mHJSy+9ZMvL+/K8Py/r2LFjR7UzOUfmbd++3RaZDzicmyG2idUwCEHfD1S7du1i06ZNNURXs4/lAZcHsmsg6BpJBF3j16VLl1i1apUWUmHVCLrWMARd41eEoLtvhBB0rcfu6kNN0I8++ujINxadwy3oOUfn22wIutZtBF3jV4Sgt23bNrZs2aJNbL9qBN2G8uCD3DvoCPrB96DSKhB0rWMIusYvq9lB1xi6d0Pdu7WVsIOOoGvHIDvoGj/3DjqCrvWjiGp20DWqh+IOOoL+u8cMO+j78UDQtQWlEqoRdK1LCLrGD0HX+SHoOkMEXWOIoGv8EHSNX1azg64xZAdd48cOusavJtUIOoLO76DX5Ez5iM8car+DjqALB8sHpeygawwRdI1fViPoGkMEXeOHoGv8EHSdH4KuMUTQNX41qUbQEXQEvSZnCoJeTQBBFw4WBF2HF1H9F/X5I3EaSgRd44ega/wQdI0fgq7zQ9A1hgi6xq8m1Qg6go6g1+RMQdARdOE42b+UHXQNJIKu8ctqBF1jiKBr/BB0jR+CrvND0DWGCLrGrybVCDqCjqDX5ExB0BF04ThB0Pln1mp7+PBX3GtL7v/XuR/qIOhaTxB0jR+CrvND0DWGCLrGrybVCDqCjqDX5ExB0BF04ThB0BH02h4+CHptySHoOrn/SCj7v4POX3F3ddqXw19x11jyV9w1flnNP7OmM6x1Av/MWq3R/b/CGTNmIOgCRv5InACvQkr5d9C1RvHvoGv8EHSNX1azg64xRNA1flnNv4OuMbzjjjvi5ptv1kL2q2YHXUPJDrrGrybV7KDvR4l/Zq0mh0xlf4Z/Zk3rH38kTuOX1fwOusbQLVv8O+haP7K6V69esXz5cj3ogwR3T9zHDK+4a63mFXeNX1bzz6xpDBF0jR+CrvGrSTWCjqCzg16TM+UjPsMOugCvQkrZQdcaxQ66xo8ddI1fViPoGkN20DV+Wc0OusaQHXSNn7saQXcT/f08BB1BR9CF8wxBF+BVSCmCrjUKQdf4IegaPwRd54eg6wwRdI0hgq7xc1cj6G6iCPofJcor7sUfcPX9E3jFXesAr7hr/LKaV9w1hu7dUPfr1H379o2FCxdqX3K/agRdR+k+ZnjFXesJr7hr/LKaV9w1hrzirvFD0DV+NalmB50ddHbQa3KmfMRn2EEX4FVIKTvoWqPYQdf4Iegav6xG0DWG7KBr/LKaHXSNITvoGj93NYLuJvr7eQg6go6gC+cZgi7Aq5BSBF1rFIKu8UPQNX4Ius4PQdcZIugaQwRd4+euRtDdRBH0P0qUV9yLP+Dq+yfwirvWAV5x1/hlNa+4awzdu6G84q71I6v5K+46Q2fCpEmTYsSIEbZIBF1HiaBrDBF0jZ+7GkF3E0XQEfT/RIB/B107ydhB1/hVQjU76FqX2EHX+LGDrvHLavdDHX4HXesJv4Ou8ctqfgddY8jvoGv8EHSNX02qecV9P0rsoNfkkKnsz7CDrvWPHXSNX1azg64xdMsWO+haP7KaHXSdoTOBHXSdZuPGjaOqqkoP+iCBHXQNJTvoGj93NYLuJvr7eRUv6F/60peiQ4cOFlLLly+Pu+++25L1YciNN94YvXv3tmRu3Lgx1q1bF0ceeaQlL0Py1bVc+FwjbwyOOeYYV1w89dRT8bnPfc6W91d/9VfRunVrW96KFSuiR48etrwNGzbED37wA1ve4sWL46GHHrKdI2vWrIk5c+bY5pffd9asWba8rVu3Vp8jrnMu8/7mb/7Gxi+/6ODBg+3rjA1gRPyP//E/omfPnrbI22+/Pfbs2WPLc59zv/3tb63fd/369fHd73432rRpY/nOixYtiu9///u2vDym9+3bF3369LHML69Lu3btsq778+bNq/6jWq6R1/Z8iOAaKW+33XabKy7yGPzyl79sy8v7on//93+35eWa+sMf/tCW9/zzz8czzzxjO6bffffdeOWVV2zrdB7TZ555pm1+ec6l7LuuSzm/+++/P0499VRbTyZMmBBdu3a15eV9Zdu2bW15ucZ861vfsuXl/Hbv3m3Lc1+XtmzZEgsWLLDNzx2Ugv6nf/qn1nNu9erV0alTJ8tU85zLe49LL73UklcfIRUt6HmAtGzZ0sZt8+bN0b59e1teBm3fvt06x1xA3377bdsc3a+4X3zxxfHYY4/Z5jdkyJCYNm2aLW/o0KExdepUW577FffTTz/dKqzHH398qRf5vGnJm13nyHN406ZNtsju3bvHypUrbXll30EfM2ZMjBs3zvZ93edc2XfQO3fuHHmj4RrNmzePnTt3uuKqc/L8yDfGXOO9996LFi1auOIi1/3p06fb8tzHjPu6dNNNN8XYsWNt39e9g37aaafFq6++apufe0094ogjIqXVOdw76MuWLbM+JMpffXFuNpx33nnVD01c48EHH4xrr73WFRe33HJL5BuQruE+h88444x46aWXXNOr/uPNeX9e1pEPed33bnk8pzO5Rv4qzUknneSKq/OcihZ0N61KEPTc9Vi6dKntqyPoGsqyC3o+Yc/fVSvrKELQO3bsWL2L7hr9+vWzPuRA0LXOuGXL/Yq7+3XvVq1aWW9akn7uLjhv7rWO/n41gq4RLbugu9fUfNvC+VAs6bsFfdWqVZG/IlbWgaBrnTnUBF2j9Yerc3MlPcw15s+fH7nWVOpA0PfrHIKuH8bsoGsM3TvoCLrWj6x230wi6FpPEHSNH4Ku83Pvvh1qO+juNRVB149pBF1jiKBr/LIaQf9dhgg6gm79d9ARdG2RQtA1flnNDrrGkFfcNX7soGv8spoddI0hO+gav6xmB11jyCvuGr+yv+Kufbs/XI2gI+gfeVyxg66fcgi6xhBB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGD0HX+GU1r7jrDEuTgKDrrUDQNYYIusYPQdf5IegaQwRd44eg6/wQdJ0hgq4xRNA1fgi6xg9B1/mVKgFB19uBoGsMEXSNH4Ku80PQNYYIusYPQdf5Ieg6QwRdY4iga/wQdI0fgq7zK1UCgq63A0HXGCLoGj8EXeeHoGsMEXSNH4Ku80PQdYYIusYQQdf4IegaPwRd51eqBARdbweCrjFE0DV+CLrOD0HXGCLoGj8EXeeHoOsMEXSNIYKu8UPQNX4Ius6vVAkIut4OBF1jiKBr/BB0nR+CrjFE0DV+CLrOD0HXGSLoGkMEXeOHoGv8EHSdX6kSEHS9HQi6xhBB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGD0HX+CHoOr9SJSDoejsQdI0hgq7xQ9B1fgi6xhBB1/gh6Do/BF1niKBrDBF0jR+CrvFD0HV+pUpA0PV2IOgaQwRd44eg6/wQdI0hgq7xQ9B1fgi6zhBB1xgi6Bo/BF3jh6Dr/EqVgKDr7UDQNYYIusYPQdf5IegaQwRd44eg6/wQdJ0hgq4xRNA1fgi6xg9B1/mVKgFB19uBoGsMEXSNH4Ku80PQNYYIusYPQdf5Ieg6QwRdY4iga/wQdI0fgq7zK1UCgq63A0HXGCLoGj8EXeeHoGsMEXSNH4Ku80PQdYYIusYQQdf4IegaPwRd51eqBARdbweCrjFE0DV+CLrOD0HXGCLoGj8EXeeHoOsMEXSNIYKu8UPQNX4Ius6vVAkIut4OBF1jiKBr/BB0nR+CrjFE0DV+CLrOD0HXGSLoGkMEXeOHoGv8EHSdX6kSEHS9HQi6xhBB1/gh6Do/BF1jiKBr/BB0nR+CrjNE0DWGCLrGD0HX+CHoOj8p4f/8n/8T77//vpSxf/G2bdvikksuic6dO1syV69eHUuWLImWLVta8nJ+J510UjRp0sSSt2PHjvjbv/3bGDlypCUvQ/7kT/7E2pMtW7bE0UcfbZtf5rn45aR27doV7du3t80vj+edO3dG69atLZmZ9dOf/tSSlSErVqyIo446ynZMr1u3Li6//PI48sgjLXPMYzqHKy/PucMOOyxatGhhmV/m/dVf/VWceeaZlrwMeeKJJ+KCCy6w5f3X//pfY+PGjba8fPDZrl07W96mTZus59yePXsib8Zd59z27dujqqrKlrdhw4bIOTqvS4899pjtHM5zLq91vXv3tvQ486666qrq88411q9fX71uuUauWx07dnTFRYcOHeIHP/iBLS+zZs2aZcvL62Yeg657mbxuHn744bZzJNfVRYsW2b5vHs8nnnii7TqSx3Reh1O6HCO/7+LFix1R/y/jmmuuseaddtpp8dWvftWWeccdd8TSpUtteXmNc13Xc1Lvvfde9XnsGnm+Pfzww664WLZsWfW9tOsczonlNcl1Xcq8Y4891nZtz3Pk6aefjn79+tkY1nVQo6q8k6jQsWDBgujfv79t9s2bN68+yZwjTzKnEHbt2jXefvtt2xR/8YtfxGc+8xlbnnsH/b777osbbrjBNr8777wzbrnlFlvetddeG/mk2DXyZtwllzmnOXPmxMCBA13TiyOOOCLeffddW16jRo2qZcY5Vq5cGd26dbNF5kLvkrec1Cc/+cl45ZVXbPMbPHhwzJgxw5Y3dOjQmDp1amnzBg0aFDNnzrTNb8CAAfHGG2/Y8vIa4rzx2717dzRt2tQ2vwxq1qxZZK5r5E1aSo1r9OnTp/rhtmu4j5khQ4bEtGnTXNOLc845J37zm9/Y8lK2Jk6caMv7+c9/Hpdeeqkt77jjjrMKtW1i+wW5d9BTkPLtGtdI8Vi4cKErrlpmhg8fbsvLjZ8JEybY8lLQb775Zlve+PHjY9SoUba8iy66KCZPnmzLe/HFFyOv7a7RqVOnWLt2rSuu2muc1xDbxPYLygdjzgcSRczxj2Ui6PvRyV2e3J0p88gbF+dTxLyxdz3VTW4IermOHreg5xPYNWvW2L5kEYKeouB868L2ZT8IGjZsWDz33HO2WARdQ3n22WdbZUubTd1Ut2nTJvLBk2u4X+vPhyZz5851TS8QdA1lrle5brnGySefHK+99porrpAct6CvWrUqunTpYptrPnjP67trIOgaybILevfu3SM3L1wjxTffFmMURwBBR9ARdOH8cu+gC1P5g6UIupuonoegawzdsoWga/3IagRdY1j2HXQEXetvViPoGkN20DV+CLrGrz6qEXQEHUEXzjwEXYAXEeyga/yymh10jSGCrvFD0HV+CLrO0J3ADrpGlFfcNX7uV9wRdK0f9VGNoCPoCLpw5iHoAjwEXYP3QTWCrmFE0DV+CLrOD0HXGboTEHSNKIKu8UPQNX4NoRpBR9ARdOFMRtAFeAi6Bg9Bt/BD0HWMvOKuMUTQNX5FVCPoGlUEXeOHoGv8GkI1go6gI+jCmYygC/AQdA0egm7hh6DrGBF0jSGCrvErohpB16gi6Bo/BF3j1xCqEXQEHUEXzmQEXYCHoGvwEHQLPwRdx4igawwRdI1fEdUIukYVQdf4Iegav4ZQjaAj6Ai6cCYj6AI8BF2Dh6Bb+CHoOkYEXWOIoGv8iqhG0DWqCLrGD0HX+DWEagQdQUfQhTMZQRfgIegaPATdwg9B1zEi6BpDBF3jV0Q1gq5RRdA1fgi6xq8hVCPoCDqCLpzJCLoAD0HX4CHoFn4Iuo4RQdcYIugavyKqEXSNKoKu8UPQNX4NoRpBR9ARdOFMRtAFeAi6Bg9Bt/BD0HWMCLrGEEHX+BVRjaBrVBF0jR+CrvFrCNUIOoKOoAtnMoIuwEPQNXgIuoUfgq5jRNA1hgi6xq+IagRdo4qga/wQdI1fQ6hG0BF0BF04kxF0AR6CrsFD0C38EHQdI4KuMUTQNX5FVCPoGlUEXeOHoGv8GkI1go6gI+jCmYygC/AQdA0egm7hh6DrGBF0jSGCrvErohpB16gi6Bo/BF3j1xCqEXQEHUEXzmQEXYCHoGvwEHQLPwRdx4igawwRdI1fEdUIukYVQdf4Iegav4ZQjaAj6Ai6cCYj6AI8BF2Dh6Bb+CHoOkYEXWOIoGv8iqhG0DWqCLrGD0HX+DWEagQdQUfQhTMZQRfgIegaPATdwg9B1zEi6BpDBF3jV0Q1gq5RRdA1fgi6xq8hVCPoCDqCLpzJCLoAD0HX4CHoFn4Iuo4RQdcYIugavyKqEXSNKoKu8UPQNX4NobriBf20006LVq1aWXrx/vvvx/r16y1ZGbJt27bYvn27LS+zzjzzzNi3b58lc8+ePfF3f/d3cdFFF1nyMuQzn/lMrFmzxpZ3/vnnx2233WbLu/rqq2P27Nm2vD/5kz+JH//4x9GkSRNL5ptvvhn9+/e35l166aXRtGlTy/z27t0b7733XrRs2dKSt2PHjuocZ95LL70U2Zcyjt27d8fAgQNt50iew926dYt169bZvm7btm1jy5Yttrzsba6FrtGsWbPqPNc516FDh5g7d67tHFmxYkXkHJ0jr3GtW7e2RCa77t272/jlOeyaW37BPEfatWtXvc44RuYdeeSRsXXrVkdcdUaLFi2s1/bk9+Fa6JjkGWecEU8++aQjqjrjueeei8svv9x2jrRp0yYWLVpkm18RQTlH171l3rvNmDEjOnfubJvqoEGDbMdMniMPPfRQDB8+3Da/Cy64IGbOnGnNu//++215V111VUyZMsWWl9fhvPdw3Wv98z//c3z1q1+1rdONGjWqPl5cx/SuXbti/vz5Nn4ZdPjhh1ev1Yz/IFDRgp4yfdRRR5W6l3my5k20a+SNy+bNm11x0bt371i6dKktb/LkyVbhHzVqVIwfP942v6FDh8bUqVNteQMGDKi+uXeNlHPnonf88cfHggULXNOLhQsXRr9+/Wx5edGoqqqy5WXQkiVLqo/rso5PfvKT8corr9imd/fdd8fXvvY1W5476N57740bb7zRFnvCCSfEvHnzbHl5Dr/xxhu2vBSPPO9cI68heTPkHPkAIW/KXSOP59NPP90VV32Ny2uda6xduzY6derkiqs+nvO4do2bbropxo4d64qz52zcuDHyQRaj9gTywafzIVFuNJx66qm1n9B/qtywYcMhJUd5X5n3l64xZMgQ673lL3/5y/jsZz/rml4cccQRkeexazhl/8M5bdq0ybruu75rfeVUtKDXF7SD+bn5tMr5ZDx3PlauXHkwU/ijnz3xxBMjd21d41AT9Hw9dvr06S581Te5s2b93/bePHrLus7/f4GyyCL7IgoqxOJCCIRLIoimgTrNFIjDNI2hNnOmEp1GO2kSmkunmkXJlpmjeDKdI4ZZk4ZlGSCNIKEoqaBCCIKssijIJvzO6/PD80XTPnc8n/eH677vx/sc/+p+P3lfj9f7Wh7X64Lm2fLyBu78YmDVqlVx5JFH2tZXDkFfvXp1dOnSxbZGd9CoUaPqOlKuMWXKlBg/frwrzp7jFvTsHDk7M+5P3N2Cnt1V54N9Fji7g86vGvLFXe/eve17p6iBtSboRa1DJa0r75t5/3QNt6C71lUpOW5Bzy9R8/nXNdyfuGfTIpsXroGgu0h+cA6CXmbGCLoGuOgddARdqy+CrvHL2Qi6xhBB1/jlbARdY1j0Drp2dMxOAgh6sfYBgq7VA0HX+JUyG0EvhZLwGwRdgBdR9wlSkT9xR9C1+iLoGj8EXeeHoOsMEXSNIYKu8auE2Qh6saqEoGv1QNA1fqXMRtBLoST8BkEX4CHoGryIur+jxifuMkZrAJ+4azj5xF3jl7P5xF1jyCfuGr9anI2gF6vqCLpWDwRd41fKbAS9FErCbxB0AR6CrsFD0GV+5QhA0DWqCLrGD0HX+SHoOsNaS0DQi1VxBF2rB4Ku8StlNoJeCiXhNwi6AA9B1+Ah6DK/cgQg6BpVBF3jh6Dr/BB0nWGtJSDoxao4gq7VA0HX+JUyG0EvhZLwGwRdgIega/AQdJlfOQIQdI0qgq7xQ9B1fgi6zrDWEhD0YlUcQdfqgaBr/EqZjaCXQkn4DYIuwEPQNXgIusyvHAEIukYVQdf4Ieg6PwRdZ1hrCQh6sSqOoGv1QNA1fqXMRtBLoST8BkEX4CHoGjwEXeZXjgAEXaOKoGv8EHSdH4KuM6y1BAS9WBVH0LV6IOgav1JmI+ilUBJ+g6AL8BB0DR6CLvMrRwCCrlFF0DV+CLrOD0HXGdZaAoJerIoj6Fo9EHSNXymzEfRSKAm/QdAFeAi6Bg9Bl/mVIwBB16gi6Bo/BF3nh6DrDGstAUEvVsURdK0eCLrGr5TZCHoplITfIOgCPARdg4egy/zKEYCga1QRdI0fgq7zQ9B1hrWWgKAXq+IIulYPBF3jV8psBL0USsJvEHQBHoKuwUPQZX7lCEDQNaoIusYPQdf5Ieg6w1pLQNCLVXEEXasHgq7xK2U2gl4KJeE3CLoAD0HX4CHoMr9yBCDoGlUEXeOHoOv8EHSdYa0lIOjFqjiCrtUDQdf4lTIbQS+FkvAbBF2Ah6Br8BB0mV85AhB0jSqCrvFD0HV+CLrOsNYSEPRiVRxB1+qBoGv8SpmNoJdCSfgNgi7AQ9A1eAi6zK8cAQi6RhVB1/gh6Do/BF1nWGsJCHqxKo6ga/VA0DV+pcxG0EuhJPwGQRfgIegaPARd5leOAARdo4qga/wQdJ0fgq4zrLUEBL1YFUfQtXog6Bq/UmYj6KVQEn6DoAvwEHQNHoIu8ytHAIKuUUXQNX4Ius4PQdcZ1loCgl6siiPoWj0QdI1fKbMR9P0obd++PZo3b14Kt5J/07Rp09i1a1fJv6/vh23atInNmzfX97OS//cuXbrEmjVrSv59fT+cNm1ajB49ur6flfy/f+pTn4oHH3yw5N/X98MBAwbEM888U9/PSv7fP/zhD8ezzz5b8u/r+2G/fv1i0aJF9f2s5P/9iCOOiFWrVpX8+/p+mFn5oOEajRo1ir1797ri6nKWLFkSPXv2tGWuXr06unbtass75ZRT4sknn7Tl/ed//mdceeWVtrzly5dHjx49bHmTJ0+OFBrXOOGEE+K5555zxcXRRx8dy5Yts+W99NJL0adPH1te3kN27Nhhy8ugZs2axc6dO22ZuZ+HDBliyyt60JgxY+KBBx6wLfPTn/503HPPPbY89zXLtrB9Qe71vfnmm9GqVSv3Mq15hx9+eLzxxhu2zF/+8pdx7rnn2vI2bNgQHTp0sOW99tprkc8frvHKK6/UXatd46abboqJEye64mLYsGExc+ZMW95PfvIT67N0u3bt4vXXX7etLz2kbdu2trwMShfp3LmzNbOSwxD0/aqXGy43sXM0adLEKui///3vY+DAgZYlphitWLHC+jC+fv366NSpk2V9GeKWt6uvvjr+4z/+w7a+7IY+9NBDtrz77rsv8mHNNTp27BhZE9do3bp1bNq0yRUXe/bsiTxHnKNFixaRb3ddIx9a8uHFNVLeXnzxRVdcnHjiifGHP/zBlud+ifXJT34y8sWda6xdu9Z6E3/88cfjzDPPdC0v2rdvH+vWrbPlZVC+yHKOfBHtFPQ5c+bEySef7FyiNeuss86KGTNm2DLdL2bHjRsX9957r219d911V1x66aW2vHz5MnfuXFveggULYtCgQbY89zXftrD9gvJryq1bt9qi3ff2vn37xuLFi23rK/p9yX0OH3fccfHCCy/Y+KWo5r3ONfIF0ZYtW1xxdc9tzuZjLixf3CHo/69ECPp+27Ucb4TcNw53d9B2tlZIkPutaT4E3XHHHbajf/jhh+OCCy6w5bk78t26dYuVK1fa1pcvYBo3bmzLy6B8QeQUJPeDy+DBg2P+/Pm2Yx46dGjMnj3bljd8+HBrJ2DChAlx22232dbnDkpBz+6Ha/Tu3dv6Asa1rv1z8kVbdh1dI1845XEXdVx00UVx//3325aX+2XWrFm2vGuuuSZuueUWW14K+iWXXGLLGzlyZEyfPt2Wl4LuajSU45qfme6vu9xfK7rv7fkSZt68ebYaF/2+5D6HTz311MgXla7Rv3//WLhwoSsuunfvXteQcw232+S6svmTXwkz/n8CCPp+OwFBr/7TAkHXaoyga/xyNoKuM3QmIOg6TQRdY4iga/zcL2URdK0eORtB1xgi6Bq/apiNoCPo1bCPSz4GBL1kVO/7QwRd44eg6/zcCQi6ThRB1xgi6Bo/BF3jl7PpoGsM6aBr/HI2HfR3M0TQEXT9rKqgBARdKxaCrvFD0HV+7gQEXSeKoGsMEXSNH4Ku8UPQdX4Ius4QQUfQP3AX8Ym7foIVPQFB1yqEoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgGdytTcAAAgAElEQVRB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNARdARdP48qNgFB10qHoGv8EHSdnzsBQdeJIugaQwRd44ega/wQdJ0fgq4zRNAR9AYV9CZNmsSuXbv0nbsvYcmSJdGzZ09bnjto3bp1kTfLog63oJ933nnx8MMP2w73vvvui3HjxtnyevToEcuXL7fltW7dOrZs2WLL27NnTxxyyCG2vAxq2bJlbN261ZbZvXv3WLFihS2vT58+kULjGv3794+FCxe64mL48OExc+ZMW96ECRPitttus+W5gx555JEYNWqULbZDhw6xfv16W145gpo1axY7d+60RT/55JMxZMgQW5476Kyzzorf/va3tthhw4bFrFmzbHluQb/11lvjX/7lX2zrO/nkk2Pu3Lm2vP/7v/+L008/3Zbnvubnwho1ahR79+61rbFVq1bx5ptv2vLc9/Z+/frFokWLbOsr+n3JfQ4ff/zx8fzzz9v45XP+0qVLbXlt2rSJzZs32/LcbpMLW7NmTXTu3Nm2xm3btkWLFi1seQ0d1Giv8wrU0Ks3/3n5UOV8aMmL8e7du6Nt27a2lWZeXphd48QTT4yNGzda4pLdD37wgxg9erQlL0NSVp0PQvlgmietawwaNCh++tOfRtOmTS2RKTI33HCDLS+PN2XLtb48yJT0/M8x8oZx1FFH1Um1Y6SYZz2OO+44R1zd9SBv5K7rQubkDSPPY8fIvM997nNxxRVXOOLqMh599NE455xzbHn5gii7C0Udd999d1x55ZW2cyQf7HMfus6RHTt2WIX/jTfeiKOPPtp2vHmsP//5z+PMM88sZInzHHG/MLnwwgsjX5a7Rl4Tss6u0a5du1i1apWtxvnSff78+ba8//3f/43PfvaztrxsgmSdnfeRrIUzL+/Fhx56qKXEeayZ53p8z7x8oXPppZfa1jd+/Ph44YUXLHkZ4hZCd16+mM1mjetZ6yc/+Ul87Wtfs+W9/fbbsX37dtuezqzMdJ4jya558+aWPZN7evbs2dG3b19L3sEIQdAPBvUC/ZlDhw6N3/3ud7YVTZs2zSrol19+edx+++229bm7g3lDu+OOO2zrywv8BRdcYMvLFwj5YFXUkQ8YjRs3ti5v9erV0aVLF1vmwIEDY8GCBba8wYMHW2syZcqUyIchxoERcH/ifswxx8SyZcsObDHvMys7bynVzpEvD5zdvKJ/4u5kl1n5Qmzy5Mm22Hy5MWPGDFte3tfz4dQ1Ro4cGdOnT3fF1V1P87rqGnzirpN86KGH4vzzz9eD9iVcdtllceedd9ry3M9u7g56Nqby+dc18rk8z2PXcH8JmC8Vnc2uPM7DDz/c+oVmfhGCoLt2EDkNTgBB15Aj6Bo/BF3jl7MRdI0hgq7xy9kIusYQQdf4Iegav5yNoGsMEXSNH4L+p/zooOt7qqITEHStfAi6xg9B1/gh6Do/BF1niKBrDBF0jR+CrvFD0HV+CLrOkA76uxki6PqequgEBF0rH4Ku8UPQNX4Ius4PQdcZIugaQwRd44ega/wQdJ0fgq4zRNARdH0XVVECgq4VE0HX+CHoGj8EXeeHoOsMEXSNIYKu8UPQNX4Ius4PQdcZIugIur6LqigBQdeKiaBr/BB0jR+CrvND0HWGCLrGEEHX+CHoGj8EXeeHoOsMEXQEXd9FVZSAoGvFRNA1fgi6xg9B1/kh6DpDBF1jiKBr/BB0jR+CrvND0HWGCDqCru+iKkpA0LViIugaPwRd44eg6/wQdJ0hgq4xRNA1fgi6xg9B1/kh6DpDBB1B13dRFSUg6FoxEXSNH4Ku8UPQdX4Ius4QQdcYIugaPwRd44eg6/wQdJ0hgo6g67uoihIQdK2YCLrGD0HX+CHoOj8EXWeIoGsMEXSNH4Ku8UPQdX4Ius4QQUfQ9V1URQkIulZMBF3jh6Br/BB0nR+CrjNE0DWGCLrGD0HX+CHoOj8EXWeIoCPo+i6qogQEXSsmgq7xQ9A1fgi6zg9B1xki6BpDBF3jh6Br/BB0nR+CrjNE0BF0fRdVUQKCrhUTQdf4IegaPwRd54eg6wwRdI0hgq7xQ9A1fgi6zg9B1xki6Ai6vouqKAFB14qJoGv8EHSNH4Ku80PQdYYIusYQQdf4IegaPwRd54eg6wwRdARd30VVlICga8VE0DV+CLrGD0HX+SHoOkMEXWOIoGv8EHSNH4Ku80PQdYYIOoKu76IqSkDQtWIi6Bo/BF3jh6Dr/BB0nSGCrjFE0DV+CLrGD0HX+SHoOkMEHUHXd1EVJSDoWjERdI0fgq7xQ9B1fgi6zhBB1xgi6Bo/BF3jh6Dr/BB0nSGCjqDru6iKEhB0rZgIusYPQdf4Ieg6PwRdZ4igawwRdI0fgq7xQ9B1fgi6zhBBR9A/cBft3r07Dj30UH2X7Zewa9euaNKkiS1z+/bt0bx5c1te//794w9/+IMtb9q0aZEXKte4/PLL4/bbb3fFxfDhw2PmzJm2PLeg33fffTFu3Djb+vr06ROLFy+25e3cuTOaNm1qy9uzZ08ccsghtrwMWrFiRRx11FG2zN69e8fLL79sy8uapNC4xpQpU2L8+PGuuJrLcQt6165dY/Xq1TaOeb7t2LHDlpdBzZo1izyXXePZZ5+NvJfUyrjiiiti8uTJtsMtuqCffPLJMXfuXNvxLliwIAYOHGjLa9u2bWzatMmWl0GNGjWKfIHsGu3atYuNGze64qJHjx6xfPlyW95DDz0U559/vi3vsssuizvvvNOW5352GzZsWMyaNcu2vqILeufOnWPt2rW2402vSb9xjpYtW8bWrVttkXmdGTBggC2voYMa7XVegRp69eY/b/PmzZE3olatWlmSX3vttTrhz7e7jvHmm29GCk2+ZXKMzMuXEk7hOvvss61C/Ytf/CLygdc1rrzyyli3bp0rLrZs2WJdXz5kZE1atGhhWWO+0Mn6uvZ0ri/3tCsvRSZFoWPHjpbj3bZtW91DlfOcy/W5Xorl+vJG/vd///eW482QZHjeeefZ8txB1157bfzyl7+0xbZv3z4effRRW96cOXOs18AlS5bEEUccYTuHc8+MHTu2LtMx8r6U50hKjWPk+r7//e9b9+Bpp51mfYGQLz2vuuoqx+HWZbjvS3l+fPzjH7et78EHH4xPfvKTtrwMyheLruv+008/bZXfhQsXxte+9jXbObd+/fo6Qe/QoYOFYZ4juT7XS4nMSznv16+fZX0Zks+DKa2uccopp9Q9H7lG3tNvvfVWV1x86UtfijVr1tjy3M+C+VL27rvvtq0v73Pf/OY3befI66+/Hj/60Y+sz24XXnih9b703e9+Ny644AIbw4YOQtD3I56C7npoeSc2RSsvpq7RvXv3ug6ha2TXI29uruHuoLvW9U7OTTfdFBMnTrTFnnHGGZEdONcYMmRIzJs3zxVXdwNftGiRLa9Lly7Wm5q7S5EH6v7csW/fvtavEKZPnx4jR4601aToQdlpzI6ja6SsTp061RVX+Jx8cG7durV1ne5OhfsT9xQF53U190vuG0Z1EkjRcr7IT0rue9PKlSujW7du1VmA9zkqdwf9xhtvjOuuu87GL7/MzC80XePUU0+NlGDXOP3002P27NmuuFi6dGn06tXLlpdu4+x258LSv9LDXCOfffP5rVIHgr5f5RB0fRsj6BpDBF3jh6Dr/NwJCLpGFEHX+OVsBF1nWOQEBL141UHQtZog6Bq/nI2g6wwLk4Cg66VA0DWGCLrGD0HX+bkTEHSNKIKu8UPQdX5FT0DQi1chBF2rCYKu8UPQdX6FSkDQ9XIg6BpDBF3jh6Dr/NwJCLpGFEHX+CHoOr+iJyDoxasQgq7VBEHX+CHoOr9CJSDoejkQdI0hgq7xQ9B1fu4EBF0jiqBr/BB0nV/RExD04lUIQddqgqBr/BB0nV+hEhB0vRwIusYQQdf4Ieg6P3cCgq4RRdA1fgi6zq/oCQh68SqEoGs1QdA1fgi6zq9QCQi6Xg4EXWOIoGv8EHSdnzsBQdeIIugaPwRd51f0BAS9eBVC0LWaIOgaPwRd51eoBARdLweCrjFE0DV+CLrOz52AoGtEEXSNH4Ku8yt6AoJevAoh6FpNEHSNH4Ku8ytUAoKulwNB1xgi6Bo/BF3n505A0DWiCLrGD0HX+RU9AUEvXoUQdK0mCLrGD0HX+RUqAUHXy4GgawwRdI0fgq7zcycg6BpRBF3jh6Dr/IqegKAXr0IIulYTBF3jh6Dr/AqVgKDr5UDQNYYIusYPQdf5uRMQdI0ogq7xQ9B1fkVPQNCLVyEEXasJgq7xQ9B1foVKQND1ciDoGkMEXeOHoOv83AkIukYUQdf4Ieg6v6InIOjFqxCCrtUEQdf4Ieg6v0IlIOh6ORB0jSGCrvFD0HV+7gQEXSOKoGv8EHSdX9ETEPTiVQhB12qCoGv8EHSdX6ESEHS9HAi6xhBB1/gh6Do/dwKCrhFF0DV+CLrOr+gJCHrxKoSgazVB0DV+CLrOr1AJCLpeDgRdY4iga/wQdJ2fOwFB14gi6Bo/BF3nV/QEBL14FULQtZog6Bo/BF3nV6gEBF0vB4KuMUTQNX4Ius7PnYCga0QRdI0fgq7zK3oCgl68CiHoWk0QdI0fgq7zK1QCgq6XA0HXGCLoGj8EXefnTkDQNaIIusYPQdf5FT0BQS9ehRB0rSYIusYPQdf5FSoBQdfLgaBrDBF0jR+CrvNzJyDoGlEEXeOHoOv8ip6AoBevQgi6VhMEXeOHoOv8pIRt27ZFixYtpIz9J5dD0Js3bx7bt2+3rbFz586xdu1aW17//v1j4cKFtjwEXUPpFvQePXrE8uXLtUXtN7tdu3axceNGW16jRo1i7969trwMatu2bWzatMmW2b1791ixYoUtb/r06TFy5EhbXtGDak3Q8/zI88Q1Mq99+/auuLoc931p3rx58ZGPfMS2xmHDhsXjjz9uy5s6dWqMHTvWlkdQsQi8+uqrkddp53Dfm/74xz/GMcccY1tivrhr1aqVLc8dhKBrRIsu6HkPeeutt7SDfM/s3M9bt261ZS5YsCAGDBhgy2vooEZ73U/HDXgEixcvjs9+9rPRoUMHy5+6dOnS+Pa3v23JypA33ngjdu7caVtf5l1zzTXRqVMn2xrzpnHsscfa8gYNGhQ/+MEPbHmf+cxnrEKYN7Wrr77atr6HHnooLrjgAlve7bffHhs2bLDlrVq1Klq3bl33n2Nkp+Kb3/ymLS9fit144422vDxH8r9u3bo5Drcuy5mXi/roRz8at956q2V9lRByzz33WIV19erVcemll9oO/ctf/nL88Ic/tOXlNaZZs2bRpEkTS2bmNW7cOA499FBL3u7du2PXrl1x2GGH2fIy8/DDD7fkZciePXus96V169bFCSecYFtfvkidNGmSLe/zn/+89UVqy5YtI19K1Mr49a9/HZdffrn1PpKPxq49nfeQ3NO9e/e2lCSfEfLZ8ogjjrDkZUi+yM8X+q7x3HPPWZ9VV65cGQMHDnQtL5599tno2rWrLW/ZsmXWFzC5Z1544QXb+ubOnRvr16+35WXQqaeeavOb3NP5ktflN8nv+uuvj4suush6zA0ZVvGC3q9fPxuvNm3aWDtvtoXtF9SrV6/IFwmuUfQOet50U1pd47rrrqsTwqKOhx9+2Cr8eX4sWrTIdrgpvnmjdI18CEr5cI68wOcDuWv07ds38mWga9RaB93FrVw53/3ud+OLX/yiLT5FOoXVNdydPNe69s9xH3Pe55YsWWJb6mmnnRZPPPGELe+2226LCRMm2PK+8Y1vxLXXXmvLGz9+fEyZMsWWV/SgSvjEvUuXLpHrdA33vd399d7QoUNj9uzZrsON4cOHx8yZM215+ZXOrFmzbHkpq3PmzLHluTvotoWVMSi/fsymjWvks28+v1XqQND3qxyCrm9j9yfuCLpWE/dNHEHX6pGzEXSdoTMBQddpIugaQwRd44ega/xyNoKuMUTQNX45G0F/N0MEHUEv9N9BR9C1ix6CrvHL2XTQdYZFTkDQ9eog6BpDBF3jh6Br/BB0nR+CrjNE0BH0D9xFdND1E4wOusaQT9w1fjmbT9x1hrWUgKDr1UbQNYYIusYPQdf4Ieg6PwRdZ4igI+gI+n4E+Dvo+kXFmYCg6zQRdJ1hLSUg6Hq1EXSNIYKu8UPQNX4Ius4PQdcZIugIOoKOoOtXkjIlIOg6WARdZ1hLCQi6Xm0EXWOIoGv8EHSNH4Ku80PQdYYIOoKOoCPo+pWkTAkIug4WQdcZ1lICgq5XG0HXGCLoGj8EXeOHoOv8EHSdIYKOoCPoCLp+JSlTAoKug0XQdYa1lICg69VG0DWGCLrGD0HX+CHoOj8EXWeIoCPoCDqCrl9JypSAoOtgEXSdYS0lIOh6tRF0jSGCrvFD0DV+CLrOD0HXGSLoCDqCjqDrV5IyJSDoOlgEXWdYSwkIul5tBF1jiKBr/BB0jR+CrvND0HWGCDqCjqAj6PqVpEwJCLoOFkHXGdZSAoKuVxtB1xgi6Bo/BF3jh6Dr/BB0nSGCjqAj6Ai6fiUpUwKCroNF0HWGtZSAoOvVRtA1hgi6xg9B1/gh6Do/BF1niKAj6Ag6gq5fScqUgKDrYBF0nWEtJSDoerURdI0hgq7xQ9A1fgi6zg9B1xki6Ag6go6g61eSMiUg6DpYBF1nWEsJCLpebQRdY4iga/wQdI0fgq7zQ9B1hgg6go6gI+j6laRMCQi6DhZB1xnWUgKCrlcbQdcYIugaPwRd44eg6/wQdJ0hgo6gI+gIun4lKVMCgq6DRdB1hrWUgKDr1UbQNYYIusYPQdf4Ieg6PwRdZ4igI+gIOoKuX0nKlICg62ARdJ1hLSUg6Hq1EXSNIYKu8UPQNX4Ius4PQdcZIugIOoKOoOtXkjIlIOg6WARdZ1hLCQi6Xm0EXWOIoGv8EHSNH4Ku80PQdYYIOoL+gbuoZcuW8eabb+q7rIwJvXr1iqVLl9r+hP79+8fChQttedOmTYvRo0fb8i6//PK4/fbbbXnXXXdd3HjjjbY8d1DRBb1jx46xbt0622Hv3bs3GjdubMvLoLzIb9q0yZbZt2/fWLx4sS1v+vTpMXLkSFseQRoBBF3jl7MRdI0hgq7xe/XVV6N79+5ayHtmN2rUKPL+5Brt2rWLjRs3uuKiX79+sWjRIlvekCFDYt68eba8oUOHxuzZs215w4cPj5kzZ9ryhg0bFrNmzbLl1Zqg79mzx/7s1qpVq9i6dautJnl+5PNbpY5Ge51XoAam8Oijj8by5cutf+p5550XRxxxhDXTGTZp0qTo0aOHLfI3v/lNnH322ba85s2bx6c//Wlb3s033xxdu3a15aW4/eu//qst74YbbogVK1bY8vKG0aVLF1term3ixIm2vOT32GOP2c6R1157LT71qU/Z1rd58+a466676iTdNV544YU47rjjXHExYMAA60unSy+9NPJh0jXat28f3/rWt1xx8dWvfjWyw+UaO3fujLvvvtsVF/mg9swzz9jy3nrrLes1cO3atdGsWbNo06aNZY15juzYsSM6d+5sycuQH/3oR5EvuF1j9+7dMWjQIFdc7Nq1Kz73uc/Z8nIP/vM//7Mtb/LkyVZ+9913Xxx99NG29T333HNxwgkn2PL+8Ic/xIknnmjLy6bF+vXrredILs55zn384x+vk2rX+NrXvhY9e/Z0xdXJvnN9GzZssD5r/fznP4+/+qu/sh2vO+/OO++se1HpGvns5nw2z2e3bKC5xrZt2+LCCy+0PguOGjUqDjvsMNcS61zpnHPOseU1dFBFC3pDw+LPg8B7CeRF+bLLLrOBcb/Fzofc+fPn29a3atWqOPLII215KZb5JtY58oWOUwiL3kG/5JJL6l5KuMYtt9wS11xzjSsuUj6uuOIKW97YsWNj6tSptjx3B71JkyaRAldLo2nTpnUS7Br5pdiSJUtccXX7JfdNrYy8HuR1wTU+8pGPxO9//3tXXF1Xy/lVkvuvNeWBujvoK1eujG7dutkYDh48OJ566ilbnvvZ44477oh8eVwr44EHHogxY8bYDtf9dWu+zHFeU1PQnS9lE1y+RHC9FLMV4iAGIegHET5/dOUTQNC1GiLoGr+cjaBrDBF0jV/ORtB1hs4EBF2niaBrDBF0jR+CrvGrhtkIejVUkWM4aAQQdA09gq7xQ9B1fgi6zhBB1xk6ExB0nSaCrjFE0DV+CLrGrxpmI+jVUEWO4aARQNA19Ai6xg9B1/kh6DpDBF1n6ExA0HWaCLrGEEHX+CHoGr9qmI2gV0MVOYaDRgBB19Aj6Bo/BF3nh6DrDBF0naEzAUHXaSLoGkMEXeOHoGv8qmE2gl4NVeQYDhoBBF1Dj6Br/BB0nR+CrjNE0HWGzgQEXaeJoGsMEXSNH4Ku8auG2Qh6NVSRYzhoBBB0DT2CrvFD0HV+CLrOEEHXGToTEHSdJoKuMUTQNX4IusavGmYj6NVQRY7hoBFA0DX0CLrGD0HX+SHoOkMEXWfoTEDQdZoIusYQQdf4Iegav2qYjaBXQxU5hoNGAEHX0CPoGj8EXeeHoOsMEXSdoTMBQddpIugaQwRd44ega/yqYTaCXg1V5BgOGgEEXUOPoGv8EHSdH4KuM0TQdYbOBARdp4mgawwRdI0fgq7xq4bZCHo1VJFjOGgEEHQNPYKu8UPQdX4Ius4QQdcZOhMQdJ0mgq4xRNA1fgi6xq8aZiPo1VBFjuGgEUDQNfQIusYPQdf5Ieg6QwRdZ+hMQNB1mgi6xhBB1/gh6Bq/apiNoFdDFTmGg0YAQdfQI+gaPwRd54eg6wwRdJ2hMwFB12ki6BpDBF3jh6Br/KphNoJeDVXkGA4aAQRdQ4+ga/wQdJ0fgq4zRNB1hs4EBF2niaBrDBF0jR+CrvGrhtkIejVUkWM4aAQQdA09gq7xQ9B1fgi6zhBB1xk6ExB0nSaCrjFE0DV+CLrGrxpmI+jVUEWO4aARQNA19Ai6xg9B1/kh6DpDBF1n6ExA0HWaCLrGEEHX+CHoGr9qmI2gV0MVOYaDRgBB19Aj6Bo/BF3nh6DrDBF0naEzAUHXaSLoGkMEXeOHoGv8qmE2gl4NVeQYDhoBBF1Dj6Br/BB0nR+CrjNE0HWGzgQEXaeJoGsMEXSNH4Ku8auG2Qh6NVSRYzhoBBB0DT2CrvFD0HV+CLrOEEHXGToTEHSdJoKuMUTQNX4IusavGmZXvKDfcMMN1VAHjqGBCAwfPjzOPPNM25/2N3/zN7Fp0yZb3jPPPBMDBgyw5a1YsSL+7u/+zpb3xBNPxO7du215mzdvjp/+9Kdx9NFH2zL79OkT3bp1s+Xt3bs3zjnnHFver3/96zjjjDNsebNnz46zzz7blveb3/wmhg4dast7/PHH42Mf+5gtb+3atTF58mRb3kc/+tF44YUXbHk7duyIbdu22fIee+yxmDRpki0vg/K+edZZZ9kyW7RoEc2aNbPlNW/ePL7whS/Y8tzn3MyZMyPvJa6R15ibbrrJFVd3fmzZssWWd++990aXLl1seXlfGj9+vC3v6aefjoEDB9ry8p6ee6Zt27a2zDzeiy++2JZ35ZVXRseOHW153//+96N37962vGSX93bXmDhxoiuqLmf79u3x7W9/25Y5bty4eO2112x57nNk9erVcfvtt9vWt3jx4vjxj39sy8ugT33qU3H88cdbMys5rKIFfcaMGTFixIhK5s/aG5jA9ddfb33YLXoHvV+/frFo0SIb5XxIW7NmjS2vEjro+fB30kkn2Y551KhR8cgjj9jypkyZYn3Yvfbaa+Mb3/iGbX0TJkyI2267zZbnDip6Bz3Pt65du1oPOx/WnMLl7qDng6RT0C+66KK4//77bQyHDRsWs2bNsuVdc801ccstt9jy3EF5vcrrlmvk9TSvq0UejRs3jnxx4horV660vjh2reudnAsuuCAefvhhW6y7g56C7nyJ9cUvfjG+853v2I73gQceiDFjxtjyTj/99MiX74zaJYCg127ta/LIEXSt7Ai6xi9nI+g6Q2cCgq7TRNA1hgi6xq8csxF0jSqCrvFD0DV+1TAbQa+GKnIMJRNA0EtG9b4/RNA1fgi6zs+dgKDrRBF0jSGCrvErx2wEXaOKoGv8EHSNXzXMRtCroYocQ8kEEPSSUSHo+wjwibu2Z/jEXePHJ+4av5zNJ+4aQz5x1/jlbD5x1xjyibvGj9mVRwBBr7yasWKBAIIuwIuo+3ur/B10jSGfuGv83LPpoOtE6aBrDOmga/zKMZsOukaVDrrGjw66xq8aZiPo1VBFjqFkAgh6yajooNNB1zbLvtl00DWMdNA1fjmbDrrGkA66xi9n00HXGNJB1/gxu/IIIOiVVzNWLBBA0AV4dNA1ePtm00G3YLSF0EHXUdJB1xjSQdf4lWM2HXSNKh10jR8ddI1fNcxG0KuhihxDyQQQ9JJRve8P+cRd45ezEXSdoaEZcsgAACAASURBVDMBQddpIugaQwRd41eO2Qi6RhVB1/gh6Bq/apiNoFdDFTmGkgkg6CWjQtD3EeAfidP2DJ+4a/z4xF3jl7P5xF1jyCfuGr+czSfuGkM+cdf4MbvyCCDolVczViwQQNAFeHzirsHbN5sOugWjLYQOuo6SDrrGkA66xq8cs+mga1TpoGv86KBr/KphNoJeDVXkGEomgKCXjOp9f8gn7hq/nI2g6wydCQi6ThNB1xgi6Bq/csxG0DWqCLrGD0HX+FXDbAS9GqrIMZRMAEEvGRWCvo8An7hre4ZP3DV+fOKu8cvZfOKuMeQTd41fzuYTd40hn7hr/JhdeQQQ9MqrGSsWCCDoAjw+cdfg7ZtNB92C0RZCB11HSQddY0gHXeNXjtl00DWqdNA1fnTQNX7VMBtBr4YqcgwlE0DQS0b1vj/kE3eNX85G0HWGzgQEXaeJoGsMEXSNXzlmI+gaVQRd44ega/yqYTaCXg1V5BhKJoCgl4wKQd9HgE/ctT3DJ+4aPz5x1/jlbD5x1xjyibvGL2fzibvGkE/cNX7MrjwCCHrl1YwVCwQQdAEen7hr8PbNpoNuwWgLoYOuo6SDrjGkg67xK8dsOugaVTroGj866Bq/apiNoFdDFTmGkgkg6CWjet8f8om7xi9nI+g6Q2cCgq7TRNA1hgi6xq8csxF0jSqCrvFD0DV+1TAbQa+GKnIMJRNA0EtGhaDvI8An7tqe4RN3jR+fuGv8cjafuGsM+cRd45ez+cRdY8gn7ho/ZlceAQS98mrGigUCCLoAj0/cNXj7ZtNBt2C0hdBB11HSQdcY0kHX+JVjNh10jSoddI0fHXSNXzXMRtDfU8UUOEZxCMyYMSPyP9dwC/rNN98cq1evdi0v5s+fH4MHD7bl/epXv4quXbva8lasWBHnn3++LS+7Cj/5yU9seRnUp0+fOOKII2yZzZo1i0984hO2vH//93+PHj162PLat28fDz74oC3vqquuih07dtjyGjVqFJMnT7bl3XXXXbF161Zb3re+9a3YsmWLLS/ZvfXWW7a87KDnw1rLli0tmcnud7/7XeRfV3GNww47LPI8cY3c01/60pdccXH//ffHgAEDbHlPPvlknHzyyba8ZPdv//Zvtrz//u//jp07d9ryZs+eHZ06dbLlPfXUUzFu3Dhb3iuvvBLf/va3bXl5X8r7XJs2bWyZl1xySVx88cW2PHfQSSedZD3edu3axU9/+lPbMq+++urYvn27LS+DvvOd79jysr5Lliyx5WXWV77yFVve3Llz45RTTrHlZdAnP/nJOPLII62ZhP0/Agj6e3bD3r172R8FInDDDTeE86WJW9ALhOp9l5IPQk7h79atW92nekUe+UIipcY1+vbtG4sXL3bF1dUjX8S4xpQpU2L8+PGuuMLnpGzlJ8uu0atXL+uDVZMmTaxylKKVHWrncGe6O+jumkydOjXGjh3rRFjorHyJlcLgGiNHjozp06e74mLBggUxcOBAW16+bHK+KM+F0UHXyuPuoGurKf/sBx54IMaMGWP7g/r37x8LFy605XXv3j2yweIazZs3t76Idq2rmnIQ9PdUE0Ev1vZG0LV6IOgav5yNoOsMnQm1JuhOduXKQtDLRfbAchH0A+O2/ywEXWOIoGv8ii7oLVq0sH7JptGqztkIOoJe6J2NoGvlQdA1fgi6zs+dgKC7iep5CLrO0JmAoOs0EXSNIYKu8UPQNX7VMBtBR9ALvY8RdK08CLrGD0HX+bkTEHQ3UT0PQdcZOhMQdJ0mgq4xRNA1fgi6xq8aZiPoCHqh9zGCrpUHQdf4Ieg6P3cCgu4mquch6DpDZwKCrtNE0DWGCLrGD0HX+FXDbAQdQS/0PkbQtfIg6Bo/BF3n505A0N1E9TwEXWfoTEDQdZoIusYQQdf4Iegav2qYjaAj6IXexwi6Vh4EXeOHoOv83AkIupuonoeg6wydCQi6ThNB1xgi6Bo/BF3jVw2zEXQEvdD7GEHXyoOga/wQdJ2fOwFBdxPV8xB0naEzAUHXaSLoGkMEXeOHoGv8qmE2go6gF3ofI+haeRB0jR+CrvNzJyDobqJ6HoKuM3QmIOg6TQRdY4iga/wQdI1fNcxG0BH0Qu9jBF0rD4Ku8UPQdX7uBATdTVTPQ9B1hs4EBF2niaBrDBF0jR+CrvGrhtkIOoJe6H2MoGvlQdA1fgi6zs+dgKC7iep5CLrO0JmAoOs0EXSNIYKu8UPQNX7VMBtBR9ALvY8RdK08CLrGD0HX+bkTEHQ3UT0PQdcZOhMQdJ0mgq4xRNA1fgi6xq8aZiPoCHqh9zGCrpUHQdf4Ieg6P3cCgu4mquch6DpDZwKCrtNE0DWGCLrGD0HX+FXDbAQdQS/0PkbQtfIg6Bo/BF3n505A0N1E9TwEXWfoTEDQdZoIusYQQdf4Iegav2qYjaAj6IXexwi6Vh4EXeOHoOv83AkIupuonoeg6wydCQi6ThNB1xgi6Bo/BF3jVw2zEXQEvdD7GEHXyoOga/wQdJ2fOwFBdxPV8xB0naEzAUHXaSLoGkMEXeOHoGv8qmE2go6gF3ofI+haeRB0jR+CrvNzJyDobqJ6HoKuM3QmIOg6TQRdY4iga/wQdI1fNcxG0BH0Qu9jBF0rD4Ku8UPQdX7uBATdTVTPQ9B1hs4EBF2niaBrDBF0jR+CrvGrhtkIOoJe6H2MoGvlQdA1fgi6zs+dgKC7iep5CLrO0JmAoOs0EXSNIYKu8UPQNX7VMBtBL6Ogr1u3LmbNmlUN++QvOobRo0f/Rb//cz8uuqDfe++90bx5c9vxrl69Or7whS/Y8h588MHINbrGyy+/HBMnTnTFxebNm6NRo0Zx+OGHWzK3bNkSuWeOOOIIS16G/PGPf4xjjz3Wlrd27doYOHCgLa9JkyYxZswYW960adMKnTd58uTYuXOn7XgXLlxYtwddY8eOHXHxxRdH27ZtLZFr1qyJZs2a2fI2bdoU27dvj65du1rWl3k//OEP69boHCeeeKItLl8gTJgwwZZX9HMk17dr1y7b8WbWz372M1ve17/+9Zg+fbot77XXXotJkybZ7iO5sO985zvRsWNHyxrzWfAzn/lMtGvXzpKXIb/97W9jxIgRtryrrrrKdk14Z1GZ6Rruc+4Xv/hFnHfeea7l1e2XvPa7Rj5rfehDH3LFRe7BI488Mjp16mTJzDznNTUXNXjw4DjmmGMs66uGEAT9PVXcu3evta6HHXZY3cNQrYy8iNaSoN95551x2WWX2cp7/vnnx0MPPWTLcwetWrWq7iLvGilG7nMuxSilwTX69u0bixcvdsXVPZiOHDnSlnfJJZdEdsxcY/jw4TFz5kxXXLjzTjvttHjiiSds6+vVq1csWbLElnfIIYfE22+/bcsrxzliW9y+IPcxu2vi3jPDhg2zvnw/88wzY8aMGbayjB8/PqZMmWLLcwctWLDA+pLSfc3P492zZ4/1xV2K0fr1620o+/XrF4sWLbLlDRkyJObNm2fLGzp0aMyePduW576PuM/hU089NebMmWM7XncHvVu3brFy5Urb+rZt2xYtW7a05WVQPre1adPGmlnJYQj6e6rnloXcwLmRa2Ug6FqlEXSNX87OB6F8u+saCLpG0v1g5ZYttwweeuihsXv3bg3afrMrQdDdx+yuiXvPuB/uEXTtdHFf88sh6PliO19wuwaCrpF0n8NFF/SePXtaX0Qj6Nr+K2U2go6gl7JPSv4Ngl4yqvf9IYKu8UPQdX5uoXbnuWXLLYNuWUXQ9T3t3jPuh3sEXasxgq7xy9l00DWGCLrGL2fTQX83QwQdQdfPqv0SEHQNJ4Ku8UPQdX5uoXbnuWULQdf3jPulhLsm7j2DoGt7xv2JO4Ku1QNB1/kh6DpDBB1B/7O7iE/ctZMMQdf4IegaPwRd5+cWaneeW7bcMuiWVTro+p527xkEXasJgq7xy9l84q4xdJ/DCLpWDzrof8qPDjoddP2sooNuY4ig6yjd3RT+DrpWEwRd44ega/xyNoKuM3QmIOg6TQRdY4iga/z4O+gav1JmI+gIein7pOTf0EEvGdX7/hBB1/jRQdf5uYXaneeWLTro+p5xfzXgrol7z7gf7vk76NoedL+UzdW4/xV3/pE4rcbu+4j7HKaDrtWXDjod9Hp3EJ+414voz/4AQdf4IegaPwRd5+d+EHLnuWXLLYNuWaWDru9p955xP9wj6FqNEXSNX87mH4nTGCLoGj8EHUGvdwch6PUiQtD3I8D/D7q2X8ohH+6HNT5x12qMoGv8ynGOaCv609nulxLulyYIurviWh6fuGv8cjafuGsM3S/ZEHStHgg6gl7vDkLQ60WEoCPo2ibZb3Y55ANB18rjFmp3nlu23DLoltVynCPaDkHQ3Q/3dNC1Hem+5udq+MRdq8nQoUNj9uzZWsh+s933Efc5jKDrpeZfcX83Q/4O+nv2FIKunWR84q7x4xN3jV/Odj+s0UHXauJ+sELQtXqUY7b7pYT7pYl7z7gf7hF0bVe6r/kIulaPnI2gawz79+8fCxcu1EL2m92zZ89YsmSJLY9/JM6G8gODEHQE3brLEHQNJ4Ku8UPQdX5uoXbnuWXLLYNuWaWDru9p955B0LWa8Im7xi9n84m7xtB9DtNB1+qRs+mg00H/s7uIDrp2kiHoGj8EXeOHoOv83ELtznPLFoKu7xn3Swl3Tdx7xv1wTwdd24N00DV+OZt/JE5jiKBr/BD0P+VHB50Oun5W7ZeAoGs4EXSNH4Ku83MLtTvPLVtuGXTLKh10fU+79wyCrtWEDrrGjw66zs99DiPoek3ooNNBp4Oun0cfmICga3ARdI0fgq7zcwu1O88tWwi6vmfcLyXcNXHvGffDPR10bQ/SQdf40UHX+SHoOkMEHUFH0PXzCEHfR4D/mzVtM5WjO+h+WOMfidNqjKBr/Mpxjmgr+tPZCLpGFEHX+Lmv+bka/hV3rSb8I3EaP/6ROI1fNczmE/f3VJG/g65tazroGj866Bo/Oug6P7dQu/Pc3VB3t9Ytqwi6vqfde4YOulYTPnHX+OVs/pE4jaH7HKaDrtUjZ9NBp4NOB10/j+ig00G37KJyyIe7m0IHXSs1gq7xK8c5oq2IDrr74Z4OurYj3dd8OuhaPXI2HXSNIR10jV81zKaDTgfduo/poGs46aBr/Oig6/zcQu3Oc3dD6aDre8b91YC7Ju49g6Bre4YOusaPDrrOz30O00HXa0IHnQ46HXT9PKKDvo/AP/zDP0Q+bLjG1q1b4+6773bF1eWceOKJ0aZNG0vmjBkz4tJLL42WLVta8tauXVuX07lzZ0te8su/O3jkkUda8jZv3lyX4+KXWYMHD46xY8da1pchV111VRxyyCG2vGXLlkX79u1teVljV31zUVu2bImjjjrKtr7Vq1dH/udiuGPHjkhhdeVt3749GjduXPefY+T5kf81b97cERdvv/127N69O5o1a2bL69q1a+R/rrF+/fro2LGjKy7WrVsX2bV1jXwwbdu2rSsuOnToED/72c9seY899pitvrmoX/3qV3HPPffY7iNZ38MOO8yWl9es733ve9GlSxcLw7wv5X2zXbt2try8zjiv0xs3boyjjz7asr4McZ9zb7zxRrRu3dq2vldffTUOP/xwW14+K3Tv3t2Wl/e5/Cu3rmePN9980/qs+txzz8UnPvEJ2zmX58h1110Xffr0sTFs0aJFDBw40JbX0EF00N9DnL+Drm1BOugaP/f/F2le7BYvXqwtar/ZKW/HHnusLa8cn++uWLHCKnB5Y2vVqpXtmEeNGhWPPPKILW/KlCkxfvx4W961114b3/jGN2x57g56vtyYOnWqbX0rV660vdDJReWDrutB/J0854N4Zr7++uv2NTqP2V0T22ap0aC8XuV1yzVSZPI67Rr5ciNfcjiH+96U50deG1yjR48esXz5cldcPPTQQ5Ff8NXKuP322+Pyyy+3He7o0aMjn39dI4XfJeeuNe2fk89FzhcmmZ2NnxR111i0aFHkX1Gs1IGgv6dyCLq2lRF0jZ9b0AcNGhTz58/XFrXf7FWrVlllxv0QlEvNbqir82EDt18Qgq5RdQu6tpryzy7Hg1B2o5wvncpPgT/hYBJwC7r73/Uox99Bd9+b8p60Zs0aWxnd/0gcgq6Vxi3o2mrKP3vbtm227vk7q80vGvLLAddA0F0kDyAnP7cdMWLEAcz84CkIuoYTQdf4IegaPwRd51drHXSdWHkTEPTy8iW9fgIIev2M6vsFgl4foYb934veQW9YGn/5n4ag/+XM/tIZdNDfQwxB/0u30Lt/j6Br/BB0jR+CrvND0HWGzgQE3UmTrAMhgKAfCLV3z0HQdYbOBARdo4mga/xKmY2gI+il7JOSf4Ogl4zqfX+IoGv8EHSdH4KuM3QmIOhOmmQdCAEE/UCoIeg6tfIlIOgaWwRd41fKbAQdQS9ln5T8GwS9ZFQIekS4/54fgq7tv5yNoOsMnQkIupMmWQdCAEE/EGoIuk6tfAkIusYWQdf4lTIbQUfQS9knJf8GQS8ZFYKOoGubZd9s/hV3C8bChiDohS1NzSwMQddLzSfuOkNnAoKu0UTQNX6lzEbQEfRS9knJv0HQS0aFoCPo2mZB0C38ih6CoBe9QtW/PgRdrzGCrjN0JiDoGk0EXeNXymwEHUEvZZ+U/BsEvWRUCDqCrm0WBN3Cr+ghCHrRK1T960PQ9Roj6DpDZwKCrtFE0DV+pcxG0BH0UvZJyb9B0EtGhaAj6NpmQdAt/IoegqAXvULVvz4EXa8xgq4zdCYg6BpNBF3jV8psBB1BL2WflPwbBL1kVAg6gq5tFgTdwq/oIQh60StU/etD0PUaI+g6Q2cCgq7RRNA1fqXMRtAR9FL2Scm/QdBLRoWgI+jaZkHQLfyKHoKgF71C1b8+BF2vMYKuM3QmIOgaTQRd41fKbAQdQS9ln5T8GwS9ZFQIOoKubRYE3cKv6CEIetErVP3rQ9D1GiPoOkNnAoKu0UTQNX6lzEbQEfRS9knJv0HQS0aFoCPo2mZB0C38ih6CoBe9QtW/PgRdrzGCrjN0JiDoGk0EXeNXymwEHUEvZZ+U/BsEvWRUCDqCrm0WBN3Cr+ghCHrRK1T960PQ9Roj6DpDZwKCrtFE0DV+pcxG0BH0UvZJyb9B0EtGhaAj6NpmQdAt/IoegqAXvULVvz4EXa8xgq4zdCYg6BpNBF3jV8psBB1BL2WflPwbBL1kVAg6gq5tFgTdwq/oIQh60StU/etD0PUaI+g6Q2cCgq7RRNA1fqXMRtAR9FL2Scm/QdBLRoWgI+jaZkHQLfyKHoKgF71C1b8+BF2vMYKuM3QmIOgaTQRd41fKbAQdQS9ln5T8GwS9ZFQIOoKubRYE3cKv6CEIetErVP3rQ9D1GiPoOkNnAoKu0UTQNX6lzEbQEfRS9knJv6k1QZ80aVLcc889JfOp74d79uyJvPA1a9asvp+W9L83atQo8uHKlbdw4cK4+OKL4/DDDy/pz6/vR1u2bInGjRtHu3bt6vtpSf/7xo0b6463Y8eOJf2+lB/t2LEjjjvuuFJ+Wu9vtm/fHqecckrkOl3jwx/+cEyePNkVF1/+8pfj6aeftuVl0K5du2x5rVu3jrzOuPb0Cy+8ED179rTlLV++PPJh3LW+FStWxF//9V9bz7mf/exn0b17d1tN9u7dG7169bLlOYPynFu1apUzMjZs2BBDhgyxZhY5LK8vN998s21P57G+/fbb0bx5c8thZ43feust630k78XO+9xhhx0WLVq0sB3vIYccYsnKkLzHfe5zn4tLLrnElrlp06YYNGiQLc8d9JWvfCV+/OMf22JzrzzwwAO2vEWLFsWxxx5rPee6detmO+deeeWVGDBggPUcyT3dsmVLC8Pc0z/60Y/i3HPPteQdjBAE/T3U80HDOXKzpXDVyqg1QXfX9b777otx48bZYnv06BEpDK7RqlWryI6ea+QLhHwQco4UOOcaU1ZPOukk2xJT0J988klb3tChQ2P27Nm2vFtuuSWuueYaW14+3F9xxRW2vBNOOCGee+45W16K5ZIlS2x5Xbt2jdWrV9vymjZtGjt37rTlZZA788UXX4zevXtb1+gMO+uss+K3v/2tLfK2226LCRMm2PKKHpQiM3bsWNsyP/ShD8VLL71ky8vrfd6bnCNfHDufBxcvXhx9+vSxLbFv376R551r9OvXL1IKXeOOO+6ISy+91BVX+JyU8zFjxtjWmS+Nly5dastLOV+5cqUtb/PmzdG2bVtbXgblS+0Ua9dYsGBB3UuESh0I+nsq57wgZzSCrp0aN9xwQ1x//fVayH6zMyu73kUdDz/8cFxwwQW25blvuu7P9Moh6ClIa9assTF0C/qoUaPquvyuUWuCftppp8UTTzzhwhf5BcKzzz5ryzvmmGNi2bJltry8h2zdutWW9859yZlZdEG/6KKL4v7777cxrDVBd3/ini883V/p2Iq7L8gt6ClHKUmuMXjw4HjqqadccXVfhMybN8+Wh6BrKPv37x/5xaJrpPA7X0TziburMh+cg6Aj6NZdRgddw4mga/xyNoKuMSx6Bx1B1+qLoOv8EHSNIYKu8cvZCLrO0Jng7qAj6Hp18ouQ/NKkUgeCjqBb9y6CruFE0DV+CLrOD0HXGNJB1/iVYzYddI0qHXSNX86mg64zLHICgq5XJ/8ef/67RK6BoLtIHkDOjBkzYsSIEQcw84On8Im7hhNB1/gh6Bo/BF3nh6BrDBF0jV85ZiPoGlUEXeOHoOv8ip6AoOsVQtDfzZAO+nv2FIKunWQIusYPQdf4Ieg6PwRdY4iga/zKMRtB16gi6Bo/BF3nV/QEBF2vEIKOoP/ZXYSgaycZgq7xQ9A1fgi6zg9B1xgi6Bq/csxG0DWqCLrGD0HX+RU9AUHXK4SgI+gIun4efWACgq7BRdA1fgi6zg9B1xgi6Bq/csxG0DWqCLrGD0HX+RU9AUHXK4SgI+gIun4eIehlYoig62D5V9w1hgi6xg9B1/iVYzaCrlFF0DV+CLrOr+gJCLpeIQQdQUfQ9fMIQS8TQwRdB4ugawwRdI0fgq7xK8dsBF2jiqBr/BB0nV/RExB0vUIIOoKOoOvnEYJeJoYIug4WQdcYIugaPwRd41eO2Qi6RhVB1/gh6Dq/oicg6HqFEHQEHUHXzyMEvUwMEXQdLIKuMUTQNX4IusavHLMRdI0qgq7xQ9B1fkVPQND1CiHoCDqCrp9HCHqZGCLoOlgEXWOIoGv8EHSNXzlmI+gaVQRd44eg6/yKnoCg6xVC0BF0BF0/jxD0MjFE0HWwCLrGEEHX+CHoGr9yzEbQNaoIusYPQdf5FT0BQdcrhKAj6Ai6fh4h6GViiKDrYBF0jSGCrvFD0DV+5ZiNoGtUEXSNH4Ku8yt6AoKuVwhBR9ARdP08QtDLxBBB18Ei6BpDBF3jh6Br/MoxG0HXqCLoGj8EXedX9AQEXa8Qgo6gI+j6eYSgl4khgq6DRdA1hgi6xg9B1/iVYzaCrlFF0DV+CLrOr+gJCLpeIQQdQUfQ9fMIQS8TQwRdB4ugawwRdI0fgq7xK8dsBF2jiqBr/BB0nV/RExB0vUIIOoKOoOvnEYJeJoYIug4WQdcYIugaPwRd41eO2Qi6RhVB1/gh6Dq/oicg6HqFEHQEvcEFXd+2lZGwZ8+euOeee2L06NG2Bd9www1x/fXX2/Iya9KkSbY8d9Ddd98d//RP/xSNGze2RLdq1So2btwYTZo0seRlzrZt22x5u3fvjjfffNOWt2vXrujcuXPs3LnTcryZ9+Mf/zjOOeccS16eI4MGDYqVK1fa8nr16hV//OMfLXkZcu6558a9995ry/ve974XEydOtO3pPN6XXnrJltemTZu6c8R1zrVo0SI2b95s29ONGjWq28+uczj3tCsrN0nm/fKXv4xTTjnFsmfyHFm2bFn07NnTkpch559/fjz55JOWvFzfjTfeGJ///OcteRnyyiuvxHHHHWfLS355HXSNRx99NP72b//Wdo60a9cuXnzxRdfy6nJyT7v2de7pvHceeuihljVm3pw5c6Jfv362vBNOOKHuuuUYuad79OgRr776qiMuMu+rX/1qfOlLX7LkZchjjz0WZ511li0v75lHH320bU9PmTIlrr76alteyuqmTZtseS1btqy7b7rOkQ0bNkTei1155bgv5Z4ZOnSobc80dFCjvXv37m3oP9T1582YMSNGjBjhiqvLqWAcVg4HGrZu3bro1KnTgU7/k3m1Juj33XdfjBs3zsavY8eOsX79eltePrSkULtGykfezJ2jdevW1jW6Gfbp08f6cNq/f/9YuHChDeHw4cNj5syZtrx8YTdt2jRb3tq1a63y4e4OdujQwXrOvf7665GZztG0aVPbS6xcV9u2beseJl0jH/yWLFniioupU6fG2LFjbXljxoyJ7Ji5xjXXXBP55Ypr3HXXXXHJJZe44uLkk0+OuXPn2vIWLFgQAwcOtOXlfemNN96w5WVQvrBzPg+6753u+1K+PFi0aJGNYdHvSwMGDIhnnnnGdrzHH398PP/887a8fEG5dOlSW16+iM4Xx66RYp5S7RzNmjWLHTt22CKfeOKJOPXUU215DR2EoL+HuPOC3NDFrMY/r9YEnU/c9V3s/sS9b9++sXjxYn1h+xIGDx4c8+fPt+XlG+LZs2fb8tyCPmHChLjtttts63MHPf744zFs2DBbbO/eva0vYPKFWL50co7spmzdutUW6f6s/8Mf/nA8++yztvW5Bf2KK66IyZMn29ZXdEEfOXJkTJ8+3Xa8bkHv0qVLrF692ra+cgh6rnHNmjW2NbqFesiQITFv3jzb+op+X8pr/qxZs2zHmyKYX0m4hvsFR/fu3WPFihWu5UV+KZZfUzoHn7i/myaCjqA7zy97FoKuIXXfxN0PGeXooCPo2p5B0DV+CLrG656rngAAD99JREFUL2cj6BpDdwcdQdfqkbPd9073vR1B12qMoGv8cjaCjqD/2V1EB10/yZwJCLpG030Tdz9kIOhafXN20TsVdNC1GtNB1/jlbDroGkMEXeOHoOv83C+O6aBrNaGDrvErZTYddDropeyTg/YbBF1Dj6Br/HI2n7hrDBF0jR+CrvFD0HV+CLrO0P1y231vp4Ou1ZgOusaPDvqf8kPQEXT9rCpjAoKuwXXfxN0PGXTQtfrSQdf58XfQdYb8HXSNIX8HXePH30HX+OVsBF1jiKBr/BB0BL3eHcQn7vUiatAfIOgabgRd40cHXedHB11jSAdd40cHXedHB11n6H657b63I+hajRF0jR+CjqDXu4MQ9HoRNegPEHQNt/sm7n7IoIOu1ZcOus6PDrrOkA66xpAOusaPDrrGjw66zg9B1xnyj8S9myGfuL9nTyHo+knmTEDQNZoIusaPDrrOjw66xpAOusaPDrrOjw66ztD9ctt9b6eDrtUYQdf40UGng17vDkLQ60XUoD9A0DXc7pu4+yGDDrpWXzroOj866DpDOugaQzroGj866Bo/Oug6PwRdZ0gHnQ76n91FCLp+kjkTEHSNJoKu8aODrvOjg64xpIOu8aODrvOjg64zdL/cdt/b6aBrNUbQNX500Omg17uDEPR6ETXoDxB0Dbf7Ju5+yKCDrtWXDrrOjw66zpAOusaQDrrGjw66xo8Ous4PQdcZ0kGng04HXT+PGiwBQddQI+gaPzroOj866BpDOugaPzroOj866DpD98tt972dDrpWYwRd40cHnQ56vTuIDnq9iBr0Bwi6htt9E3c/ZNBB1+pLB13nRwddZ0gHXWNIB13jRwdd40cHXeeHoOsM6aDTQaeDrp9HDZaAoGuoEXSNHx10nR8ddI0hHXSNHx10nR8ddJ2h++W2+95OB12rMYKu8aODTge93h1EB71eRA36AwRdw+2+ibsfMuiga/Wlg67zo4OuM6SDrjGkg67xo4Ou8aODrvND0HWGdNDpoNNB18+jBktA0DXUCLrGjw66zo8OusaQDrrGjw66zo8Ous7Q/XLbfW+ng67VGEHX+NFBp4Ne7w6ig14vogb9AYKu4XbfxN0PGXTQtfrSQdf50UHXGdJB1xjSQdf40UHX+NFB1/kh6DpDOuh00Omg6+dRgyUg6BpqBF3jRwdd50cHXWNIB13jRwdd50cHXWfofrntvrfTQddqjKBr/Oig00GvdwfRQa8XUYP+AEHXcPfo0SOWL1+uhew3u1WrVpHC4Brl6KC3bt3ausaOHTvG+vXrXYccffr0iRdffNGW179//1i4cKEtb/jw4TFz5kxb3ujRo2PatGm2PHfQI488EqNGjbLFdujQwbpfyiHoTZs2jZ07d9qOuW3btrFp0yZbXq9evWLJkiW2vKlTp8bYsWNteWPGjIkHHnjAllf0DvrJJ58cc+fOtR3vggULYuDAgba8vC+98cYbtrwMaty4cTifB933Tve93S38Rb8vDRgwIJ555hnbnjn++OPj+eeft+X17Nkzli5dastr06ZNbN682ZbXpEmT2LVrly0vg5o1axY7duywZS5atCj69u1ry2vooEZ7nVegBl79jBkzYsSIEQ38p/LHVTKB66+/PiZNmlTYQ0jRyhuva6Scd+vWLQ499FBL5IYNGyIfNPJC6hqZ1bx5c0vc9u3brRf4vFkMGjTI9vD39ttvR8qM62Ey866++uq48sorLfwy5De/+U2cffbZtrz/+q//iq9//eu2PZidnkcffdS2vv/5n/+Jf/zHf4xDDjnEktmyZcu6ByvXnt6yZYtVFHJPpwCngDhG5uU1wfWwlnv61ltvjQsvvNCxvLqM119/PY499lhL3u7du2Pr1q2WrHdC8jqdQuMaL7/8cnTq1MkVF7/+9a/j4osvtp3D+Smr86XiOw/3rnOuHPeR4447LnLvOEaeI1OmTImPf/zjjrjIvMWLF0dKpmNk3iuvvBIpma6RL9jmzJnjiovu3bvHsmXLbNf9rl27xsqVK2157du3j9///ve2c27NmjV1zx6uZ7e8BuZzpSsv7yP5X14bXCNfbh955JGuuAbPQdAbHDl/4MEkUHRBP5hs+LPfn0B2erLj4xqDBw+O+fPnu+LqHtTGjx9vy3MHTZ48Oa644gpbbD6oZUfUNYr+d9Bdx7l/jvsrE/ffQZ81a1acccYZ5Th0Mg+AgPsrk5NOOimefvrpA1hJ5U5JUVi1apXtAPIeki+Pa2VMnDgxbrrpJtvhDhs2LPI64xruT9xPP/30mD17tmt55FQgAQS9AovGkg+cAIJ+4OxqdSaCrlUeQdf4lWM2gl4OqtWbiaDrtUXQNYYIusaP2ZVHAEGvvJqxYoEAgi7Aq9GpCLpWeARd41eO2Qh6OahWbyaCrtcWQdcYIugaP2ZXHgEEvfJqxooFAgi6AK9GpyLoWuERdI1fOWYj6OWgWr2ZCLpeWwRdY4iga/yYXXkEEPTKqxkrFggg6AK8Gp2KoGuFR9A1fuWYjaCXg2r1ZiLoem0RdI0hgq7xY3blEUDQK69mrFgggKAL8Gp0KoKuFR5B1/iVYzaCXg6q1ZuJoOu1RdA1hgi6xo/ZlUcAQa+8mrFigQCCLsCr0akIulZ4BF3jV47ZCHo5qFZvJoKu1xZB1xgi6Bo/ZlceAQS98mrGigUCCLoAr0anIuha4RF0jV85ZiPo5aBavZkIul5bBF1jiKBr/JhdeQQQ9MqrGSsWCCDoArwanYqga4VH0DV+5ZiNoJeDavVmIuh6bRF0jSGCrvFjduURQNArr2asWCCAoAvwanQqgq4VHkHX+JVjNoJeDqrVm4mg67VF0DWGCLrGj9mVRwBBr7yasWKBAIIuwKvRqQi6VngEXeNXjtkIejmoVm8mgq7XFkHXGCLoGj9mVx4BBL3yasaKBQIIugCvRqci6FrhEXSNXzlmI+jloFq9mQi6XlsEXWOIoGv8mF15BBD0yqsZKxYIIOgCvBqdiqBrhUfQNX7lmI2gl4Nq9WYi6HptEXSNIYKu8WN25RFA0CuvZqxYIICgC/BqdCqCrhUeQdf4lWM2gl4OqtWbiaDrtUXQNYYIusaP2ZVHAEGvvJqxYoEAgi7Aq9GpCLpWeARd41eO2Qh6OahWbyaCrtcWQdcYIugaP2ZXHgEEvfJqxooFAgi6AK9GpyLoWuERdI1fOWYj6OWgWr2ZCLpeWwRdY4iga/yYXXkEEPTKqxkrFggg6AK8Gp2KoGuFR9A1fuWYjaCXg2r1ZiLoem0RdI0hgq7xY3blEUDQK69mrFgggKAL8Gp0KoKuFR5B1/iVYzaCXg6q1ZuJoOu1RdA1hgi6xo/ZlUcAQa+8mrFigQCCLsCr0am9e/eOl19+2Xb0ffr0iRdffNGWN2XKlBg/frwtzx3kFvQRI0bEY489Zlvm448/HsOGDbPlHXXUUbFixQpbXjmCmjVrFjt37rRFd+3aNVavXm3LmzVrVpxxxhm2PII0Am5B/9CHPhQvvfSStqgKm92pU6dYv369bdXz58+PQYMG2fKKHlR0QT/++OPj+eeft2E88cQTY+HChbY8giqPQEULeuKeMWNG5VFnxQeVwJlnnnlQ/3z+8MohsG3bNqtsbd++PRYvXhwDBgywQUgZ/NjHPmbLcwc98cQT0b59e2ts9+7do0WLFpbMp556Klq2bGnJeifEuT7rwiLqJGHDhg222NzTKefHHHOMLXPr1q01JR82cGUKmjNnTrRr186aXuRzxHqgEeG+j+T68gVb//793UstbF6+tOvSpYttfb/61a/i3HPPteXlC5MTTjghmjdvbsuspXPEBq2Kgipe0KuoFhwKBCAAAQhAAAIQgAAEIAABCNQwAQS9hovPoUMAAhCAAAQgAAEIQAACEIBAcQgg6MWpBSuBAAQgAAEIQAACEIAABCAAgRomgKDXcPE5dAhAAAIQgAAEIAABCEAAAhAoDgEEvTi1YCUQgAAEIAABCEAAAhCAAAQgUMMEEPQaLj6HDgEIQAACEIAABCAAAQhAAALFIYCgF6cWrAQCEIAABCAAAQhAAAIQgAAEapgAgl7DxefQIQABCEAAAhCAAAQgAAEIQKA4BBD04tSClUAAAhCAAAQgAAEIQAACEIBADRNA0Gu4+Bw6BCAAAQhAAAIQgAAEIAABCBSHAIJenFqwEghAAAIQgAAEIAABCEAAAhCoYQIIeg0Xn0OHAAQgAAEIQAACEIAABCAAgeIQQNCLUwtWAgEIQAACEIAABCAAAQhAAAI1TABBr+Hic+gQgAAEIAABCEAAAhCAAAQgUBwCCHpxasFKIAABCEAAAhCAAAQgAAEIQKCGCSDoNVx8Dh0CEIAABCAAAQhAAAIQgAAEikMAQS9OLVgJBCAAAQhAAAIQgAAEIAABCNQwAQS9hovPoUMAAhCAAAQgAAEIQAACEIBAcQgg6MWpBSuBAAQgAAEIQAACEIAABCAAgRomgKDXcPE5dAhAAAIQgAAEIAABCEAAAhAoDgEEvTi1YCUQgAAEIAABCEAAAhCAAAQgUMMEEPQaLj6HDgEIQAACEIAABCAAAQhAAALFIYCgF6cWrAQCEIAABCAAAQhAAAIQgAAEapgAgl7DxefQIQABCEAAAhCAAAQgAAEIQKA4BBD04tSClUAAAhCAAAQgAAEIQAACEIBADRNA0Gu4+Bw6BCAAAQhAAAIQgAAEIAABCBSHAIJenFqwEghAAAIQgAAEIAABCEAAAhCoYQIIeg0Xn0OHAAQgAAEIQAACEIAABCAAgeIQQNCLUwtWAgEIQAACEIAABCAAAQhAAAI1TABBr+Hic+gQgAAEIAABCEAAAhCAAAQgUBwCCHpxasFKIAABCEAAAhCAAAQgAAEIQKCGCSDoNVx8Dh0CEIAABCAAAQhAAAIQgAAEikMAQS9OLVgJBCAAAQhAAAIQgAAEIAABCNQwAQS9hovPoUMAAhCAAAQgAAEIQAACEIBAcQgg6MWpBSuBAAQgAAEIQAACEIAABCAAgRomgKDXcPE5dAhAAAIQgAAEIAABCEAAAhAoDgEEvTi1YCUQgAAEIAABCEAAAhCAAAQgUMMEEPQaLj6HDgEIQAACEIAABCAAAQhAAALFIYCgF6cWrAQCEIAABCAAAQhAAAIQgAAEapgAgl7DxefQIQABCEAAAhCAAAQgAAEIQKA4BBD04tSClUAAAhCAAAQgAAEIQAACEIBADRNA0Gu4+Bw6BCAAAQhAAAIQgAAEIAABCBSHAIJenFqwEghAAAIQgAAEIAABCEAAAhCoYQIIeg0Xn0OHAAQgAAEIQAACEIAABCAAgeIQQNCLUwtWAgEIQAACEIAABCAAAQhAAAI1TABBr+Hic+gQgAAEIAABCEAAAhCAAAQgUBwCCHpxasFKIAABCEAAAhCAAAQgAAEIQKCGCfx/JD9tfjGtjEwAAAAASUVORK5CYII="/>
+</defs>
+</svg>
diff --git a/Chapter4/Subspaces_of_Rn.md b/Chapter4/Subspaces_of_Rn.md
index 6c028ac..bbfa626 100644
--- a/Chapter4/Subspaces_of_Rn.md
+++ b/Chapter4/Subspaces_of_Rn.md
@@ -122,8 +122,9 @@ $$
::::{figure} Images/Fig-Subspaces-Lines.svg
:name: Fig:Subspaces:Lines
+:class: dark-light
-A line is a subspace of $\R^2$ if and only if it goes through (0,0)
+A line is a subspace of $\R^2$ if and only if it goes through $(0,0)$.
::::
::::::
@@ -144,6 +145,7 @@ A disk $D$ specified by the inequality $x^2 + y^2 \leq a^2$, where $a$ is some p
::::{figure} Images/Fig-Subspaces-Disk.svg
:name: Fig:Subspaces:SubspacesDisk
+:class: dark-light
A disk is not a subspace of $\R^2$.
::::
@@ -166,9 +168,21 @@ Also give a set with only the properties i and iii.
</li>
</ol>
+```{applet}
+:url: subspaces_in_rn/the_game
+:fig: Images/Fig-TheGame.svg
+:name: Fig:Subspaces:SubspacesGame
+:class: dark-light
+
+A game to test your knowledge of subspaces. On the left, $\vect{u} + \vect{v}$ is computed; try to find a result where the sum is not in the subspace. On the right, a scalar multiple $c \cdot \vect{u}$ is computed; try to find a result that is not in the subspace. Confetti will appear if a solution is found that falls outside the set. Different sets can be selected with the dropdown menu.
+```
+
+<!-- You may get some inspriration from the applet below. -->
+
::::::
-::::::{dropdown} Solution to {numref}`Exc:Subspaces:NonSubspacesR2` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:Subspaces:NonSubspacesR2` (_click to show_)
+:class: solution, dropdown
We first give an example of a subset of $\R^2$ that only has properties i. and ii.
@@ -209,7 +223,8 @@ c_1\vect{u}+ c_2 \vect{v} \in S.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Subspaces:SpanClosed`
+:class: myproof
To show that a subspace satisfies property {eq}`Eq:Subspaces:SpanClosed`,
suppose that $S$ is a subspace, $\vect{u}$ and $\vect{v}$ are vectors in $S$ and
@@ -285,7 +300,8 @@ Recall {prf:ref}`Dfn:LinearCombinations:Span`: the span of zero vectors in $\R^n
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Subspaces:SpanIsSubspace`
+:class: myproof
If the number of vectors $r$ is equal to $0$, the span is equal to $\{\vect{0}\}$, the trivial subspace.
@@ -355,7 +371,8 @@ Once more we recall the convention that the span of zero vectors (i.e., when $r
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Subspaces:AllSubspacesR3`
+:class: myproof
of {prf:ref}`Prop:Subspaces:AllSubspacesR3`.
@@ -450,6 +467,7 @@ $$
The next proposition shows that the designation 'space' in the above definition is well justified.
::::::{prf:proposition}
+:label: Prop:Subspaces:AllSubspacesRn
Let $A$ be an $m\times n$ matrix.
@@ -471,7 +489,8 @@ The null space of $A$ is a subspace of $\R^n$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Subspaces:AllSubspacesRn`
+:class: myproof
Let $A$ be an $m\times n$ matrix.<ol type = "i">
@@ -597,7 +616,8 @@ Can you find a similar formula relating the null space of $AB$ to the null space
::::::
-::::::{dropdown} Solution to {numref}`Exc:Subspaces:ColABinColA` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:Subspaces:ColABinColA` (_click to show_)
+:class: solution, dropdown
Suppose that $A$ is an $m\times n$ and $B$ an $n \times p$ matrix.
Thus $AB$ is an $m\times p$ matrix.
@@ -649,7 +669,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:Subspaces:WhatIfAAeq0` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:Subspaces:WhatIfAAeq0` (_click to show_)
+:class: solution, dropdown
First we show that
@@ -676,6 +697,7 @@ So $A\vect{y} = A^2\vect{x} = \vect{0}$, and we may conclude that indeed $A^2 =
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/66b4134c-20e3-4a38-8f14-a32aa472aece?id=70616
:label: grasple_exercise_4_1_1
:dropdown:
@@ -684,6 +706,7 @@ So $A\vect{y} = A^2\vect{x} = \vect{0}$, and we may conclude that indeed $A^2 =
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/aa71ac1a-d82d-4c6d-a6af-7af0c25422b1?id=70617
:label: grasple_exercise_4_1_2
:dropdown:
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diff --git a/Chapter5/DetGeometric.md b/Chapter5/DetGeometric.md
deleted file mode 100644
index a877f16..0000000
--- a/Chapter5/DetGeometric.md
+++ /dev/null
@@ -1,527 +0,0 @@
-(Sec:DetGeometric)=
-
-# Determinants as areas or volumes
-
-## Introduction
-
-The word "determinant" already appeared in the section about invertibility: a $2\times2$ matrix
-$A = \left[\begin{array}{cc} a & b \\ c & d\end{array} \right]$ is invertible if and only if
-
-:::{math}
-:label: Eq:DetGeometric:DetNonzero
-
-ad-bc \neq 0.
-
-:::
-
-The expression $ad-bc$ we called the determinant of the matrix $A$. Formula {eq}`Eq:DetGeometric:DetNonzero` is also equivalent to the statement that the columns of the matrix $A$ are linearly independent.
-
-Likewise, by row reducing a general $3 \times 3$ matrix
-
-$$
-A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\
-a_{21} & a_{22} & a_{23}\\
-a_{31} & a_{32} & a_{33}\end{array} \right]
-$$
-
-we might end up with an expression containing all the entries $a_{ij}$ that tells us whether $A$ is invertible or not.
-In this section we will use a geometric approach to derive such an expression, and will again call this the determinant of the matrix $A$. Its formula, when looked at from the right perspective, shows an opportunity to generalize the concept to higher dimensions.
-We will follow that route in the second section ``. . . . . "
-
-We will start by introducing determinants as a way to compute areas (in the plane) and volumes (in the space $\R^3$).
-
-## Area, orientation, determinant
-
-We start with two vectors $\vect{u}$ and $\vect{v}$ (as always, starting from the origin) in $\R^2$. They 'span' a parallelogram $OACB$,
-where $A$ and $B$ are the end points of $\vect{u}$ and $\vect{v}$, and $C$ corresponds to the vector $\vect{u}+\vect{v}$. See {numref}`Figure %s <Fig:DetGeometric:PargramOACB>`
-
-::::{figure} Images/Fig-DetGeometric-PargramOACB.svg
-:name: Fig:DetGeometric:PargramOACB
-
-The parallelogram OACB
-::::
-
-::::::{prf:proposition}
-:label: Prop:DetGeometric:Area
-
-The area of the parallelogram $OACB$, spanned by the vectors $ \vect{u} =\left[\begin{array}{c} a \\ b \end{array}\right]$ and
-$\vect{v}=\left[\begin{array}{c} c \\ d \end{array}\right]$
-is given by $|ad-bc|$, i.e., the absolute value of $ad-bc$.
-
-::::::
-
-::::::{prf:proof}
-
-The quickest way to prove this is to translate it to the cross product. We introduce the vectors
-
-$$
-\tilde{\vect{u}} = \left[\begin{array}{c} a \\ b \\ 0 \end{array}\right]
- \quad \text{and} \quad
-\tilde{\vect{v}} = \left[\begin{array}{c} c \\ d \\ 0 \end{array}\right]
-$$
-
-::::{figure} Images/Fig-DetGeometric-OrientedArea1.svg
-:name: Fig:DetGeometric:OrientedArea1
-
-Oriented area
-::::
-
-See {numref}`Figure %s <Fig:DetGeometric:OrientedArea1>`
-We then have
-
-$$
-\tilde{\vect{u}} \times \tilde{\vect{v}} = \left[\begin{array}{c} 0 \\ 0 \\ ad-bc \end{array}\right]
-.
-$$
-
-The length of this cross product, which is given by $|ad-bc|$, gives the area of the parallelogram.
-
-::::::
-
-Now what about the sign here?
-
-::::::{prf:proposition}
-
-$ad - bc = \norm{\vect{u}}\norm{\vect{v}}\sin(\varphi)$, where $\varphi$ is the angle from $\vect{u}$ counterclockwise to $\vect{v}$.
-We will call this the **directed angle** from $\vect{u}$ to $\vect{v}$.
-
-::::::
-
-::::::{prf:proof}
-
-Again we can resort to properties of the cross product, but in fact it is not necessary to go up one dimension.
-By a small twist we can turn the determinant into an inner product:
-
-$$
-ad-bc = \left[\begin{array}{c} -b \\ a \end{array}\right]
- \ip \left[\begin{array}{c} c \\ d \end{array}\right]
- =
-\vect{u}^{\perp} \ip \vect{v},
-$$
-
-where $\vect{u}^{\perp}$ is the vector that is perpendicular to $\vect{u}$, points 'to the left' of $\vect{u}$, and has the same length as $\vect{u}$. See {numref}`Figure %s <Fig:DetGeometric:AreaPargram>`.
-
-::::{figure} Images/Fig-DetGeometric-AreaPargram.svg
-:name: Fig:DetGeometric:AreaPargram
-
-Area equals base length time height
-::::
-
-So
-
-$$
-ad-bc = \vect{u}^{\perp} \ip \vect{v} = \norm{\vect{u}^{\perp}} \norm{\vect{v}}\cos(\vartheta),
-$$
-
-where $\vartheta$ is the angle between $\vect{u}^{\perp}$ and $\vect{v}$.
-Here $\norm{\vect{v}}\cos(\vartheta)$ is $(\pm)$ the length of the projection of $\vect{v}$ onto the line perpendicular to $\vect{u}$, which can be interpreted as the height of the parallelogram. So then
-
-$$
-\norm{\vect{u}^{\perp}} \norm{\vect{v}}\cos(\vartheta) = \norm{\vect{u}} \norm{\vect{v}}\cos(\vartheta) = \pm \text{(base length)} \times \text{height} = \pm \text{area of } OACB.
-$$
-
-After some rewriting
-
-$$
-\begin{array}{ccl}
-ad - bc &=&
-\norm{\vect{u}^{\perp}}\norm{\vect{v}}\cos(\vartheta) \\
-&=& \norm{\vect{u}}\norm{\vect{v}}\cos(\tfrac12\pi-\varphi)\\
-&=& \norm{\vect{u}}\norm{\vect{v}}\sin(\varphi),
-\end{array}
-$$
-
-where $\varphi$ is the angle from $\vect{u}$ to the left (= counterclockwise) to $\vect{v}$.
-
-We see that $ad-bc$ is equal to the area of the parallelogram if the angle from $\vect{u}$ to $\vect{v}$ is less then $\pi$, and minus this area if the angle is between $\pi$ and $2\pi$.
-
-::::::
-
-::::::{prf:definition}
-:label: Dfn:DetGeometric:Orientation
-
-The **determinant** of the ordered set $(\vect{u},\vect{v})$ of two vectors $\vect{u} =\left[\begin{array}{c} a \\ b \end{array}\right]
-$ and $\vect{v}=\left[\begin{array}{c} c \\ d \end{array}\right]
-$ in $\R^2$ is defined as
-
-$$
-\det{(\vect{u},\,\vect{v})} = ad - bc.
-$$
-
-Alternatively, the determinant can be seen as an operator working on $2 \times 2$ matrices, coming with its own notation:
-
-$$
-\det{[ \,\vect{u} \,\, \vect{v}\, ]} = \det{\left[\begin{array}{cc} a & c \\ b & d \end{array}\right]} =
-\left|\begin{array}{cc} a & c \\ b & d \end{array}\right|= ad-bc.
-$$
-
-::::::
-
-Apart from the area, the determinant also says something about the relative position of the two vectors $\vect{u}$ and $\vect{v}$.
-In fact, we can use the determinant to define the orientation of two vectors in the plane (and later of $n$ vectors in $\R^n$).
-
-::::::{prf:definition}
-:label: Dfn:DetGeometric:Orientation2
-
-The ordered set $(\vect{u},\vect{v})$ of two linearly independent vectors $\vect{u}$ and $\vect{v}$ is said to be **positively oriented**
-if $\det{(\vect{u},\vect{v})} > 0$, and **negatively oriented**
-if $\det{(\vect{u},\vect{v})} < 0$.
-
-::::::
-
-::::::{prf:proposition}
-:label: Prop:DetGeometric:Properties2by2Det
-
-Two by two determinants obey the following rules:
-
-<ol type = "i">
-<li>
-
-$\det{(\vect{v},\vect{u})} = - \det{(\vect{u},\vect{v})}$.
-
-</li>
-<li>
-
-$\det{(\vect{u},\vect{v}+\vect{w})} = \det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})}$.
-
-</li>
-<li>
-
-$\det{(\vect{u},k\vect{v})} = k \det{(\vect{u},\vect{v})}$, $k \in \R$.
-
-</li>
-<li>
-
-$\det{(\vect{e_1},\vect{e_2})} = 1$.
-
-</li>
-</ol>
-
-These properties can also be expressed using matrices.
-
-Two by two determinants obey the following rules:
-
-<ol type = "i">
-<li>
-
-$ \begin{vmatrix} c & a \\ d & b \end{vmatrix} = - \begin{vmatrix} a & c \\ b & d \end{vmatrix}$.
-
-</li>
-<BR>
-<li>
-
-$\begin{vmatrix} a_{1} & b_{1}+ c_1\\ a_{2} & b_{2}+ c_2 \end{vmatrix} = \begin{vmatrix} a_{1} & b_{1}\\ a_{2} & b_{2}\end{vmatrix}+ \begin{vmatrix} a_{1} & c_1\\ a_{2} & c_2 \end{vmatrix}$.
-
-</li>
-<BR>
-<li>
-
-$\begin{vmatrix} a_{1} & k b_{1}\\ a_{2} & k b_{2}\end{vmatrix} = k \begin{vmatrix} a_{1} & b_{1}\\ a_{2} & b_{2}\end{vmatrix}$.
-
-</li>
-<BR>
-<li>
-
-$\begin{vmatrix} 1 & 0 \\ 0 & 1\end{vmatrix} = 1$.
-
-</li>
-</ol>
-
-::::::
-
-These properties are easily verified by applying the definition
-
-$$
-\left|\begin{array}{cc} a & c \\ b & d \end{array}\right| = ad-bc.
-$$
-
-::::::{exercise}
-:label: Exc:DetGeometric:Properties2by2Det
-
-Verify the four properties of {prf:ref}`Prop:DetGeometric:Properties2by2Det`
-
-::::::
-
-These preperties can also be looked at from the geometric interpretation 'signed area'.
-If you are interested, you may have a look at the proof.
-
-::::::{prf:proof}
-
-Three of the four properties are rather obvious:
-
-<ol type = "i">
-<li>
-
-Interchanging $\vect{u}$ and $\vect{v}$ changes the orientation. The signed area changes sign.
-
-</li>
-<li>
-
-$\det{(\vect{e_1},\vect{e_2})} $ is the area of the unit square.
-
-</li>
-<li>
-
-$\det{(\vect{u},k\vect{v})} = k\times\det{(\vect{u},\vect{v})}$, $k \in \R$.
-
-Giving one of the vectors a factor $k$ changes the area with a factor $|k|$. If $k > 0$, the orientation of the two vector does not change, so the determinant gets a factor $|k|$ which in this case is equal to $k$. If however $k < 0$, then the orientation does change, so the determinant gets a factor $-|k|$, which in this case is again equal to $k$.
-
-</li>
-</ol>
-
-The remaining property,
-
-<ol type = "i">
-<li>
-
-$\det{(\vect{u},\vect{v}+\vect{w})} = \det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})}$,
-
-</li>
-</ol>
-
-is the most interesting one. The two pictures of {numref}`Figure %s <Fig:DetGeometric:Linearity>` tell the story.
-
-::::{figure} Images/Fig-DetGeometric-SumRule.svg
-:name: Fig:DetGeometric:Linearity
-
-The sum rule in a picture. Note, **this is a 2D picture**.
-::::
-
-In the picture on the left, both $(\vect{u},\vect{v})$ and $(\vect{u},\vect{w})$ are positively oriented.
-So there
-
-$$
-\begin{array}{lcl}
-\det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})} &=&
-\text{area}(OABC) + \text{area}(CBDE) \\
-&=& \text{area}(OADE) = \det{(\vect{u},\vect{v}+\vect{w})},
-\end{array}
-$$
-
-since the two triangles $OCE$ and $ABD$ are congruent, so have equal areas.
-
-In the picture on the right, the orientation of $(\vect{u},\vect{v})$ is positive, the orientation of $(\vect{u},\vect{w})$ is negative, and the orientation of $(\vect{u},\vect{v}+\vect{w})$ is positive again.
-So there
-
-$$
-\begin{array}{lcl}
-\det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})} &=&
-\text{area}(OAFC) - \text{area}(OBDA) \\
-&=& \text{area}(OAFC) - \text{area}(CGEF) \\
-&=& \text{area}(OAEG) = \det{(\vect{u},\vect{v}+\vect{w})},
-\end{array}
-$$
-
-since now the areas of $OAFC$ and $CGEB$ add up to the area of $OABC$, owing to the
-congruence of the triangles $OGC$ and $AEF$.
-
-There are more pairwise orientations to consider, but the idea is hopefully clear to you.
-
-::::::
-
-::::::{exercise}
-
-Give a geometric argument for the following property of the determinant.
-
-Let $\vect{u}$ and $\vect{v}$ be two vectors in the plane, and $k$ a real number. Then
-
-$$
- \det{(\vect{u},\vect{v}+k \vect{u})} = \det{(\vect{u},\vect{v})}.
-$$
-
-::::::
-
-## $\, 3 \times 3$ determinants: volume and orientation
-
-Suppose $\vect{a}, \vect{b}, \vect{c}$ are three vectors in $\R^3$. For the moment, suppose they are linearly independent. So
-$\vect{a}, \vect{b}$ are not multiples of each other, and $\vect{c}$ is not in the plane spanned by $\vect{a}, \vect{b}$.
-Then the three vectors can be interpreted as three edges of a parallelepiped. See {numref}`Figure %s <Fig:DetGeometric:Paraped>`.
-
-::::{figure} Images/Fig-DetGeometric-Paraped.svg
-:name: Fig:DetGeometric:Paraped
-
-Volume equals base area time height
-::::
-
-::::::{prf:proof}
-
-Just as the area of a parallelogram can be computed as 'base length times height', the volume of a parallelepiped can be computed as 'base area times height'.
-As base region we can take the parallelogram spanned by $\vect{a}$ and $\vect{b}$, and then the base area becomes
-
-$$
-A = \norm{\vect{a} \times \vect{b}}.
-$$
-
-The height is found by projecting $\vect{c}$ onto the line through the origin that is perpendicular to the plane spanned by $\vect{a}$ and $\vect{b}$. A direction vector of this line is precisely given by $\vect{a} \times \vect{b}$.
-So
-
-$$
-h = \vect{c}\ip\vect{n}
-$$
-
-where $\vect{n}$ is the unit vector perpendicular to the 'base plane' that points to the same side as the vector $\vect{c}$. This unit vector is given by
-
-$$
-\vect{n} = \pm \frac{\vect{a} \times \vect{b}}{\norm{\vect{a} \times \vect{b}}}
-$$
-
-And then the formula 'height times base area' gives
-
-$$
-V = \vect{c}\ip\vect{n} = \pm \vect{c}\ip\frac{\vect{a} \times \vect{b}}{\norm{\vect{a} \times \vect{b}}}
-\cdot \norm{\vect{a} \times \vect{b}} = (\pm) \vect{c}\ip(\vect{a} \times \vect{b}).
-$$
-
-The statement about the sign follows immediately from the third defining property of the cross product.
-
-::::::
-
-Note that the expression $\vect{c}\ip(\vect{a} \times \vect{b})$ must be invariant under cyclic permutations of the three vectors.
-
-::::::{prf:proposition}
-:label: Prop:DetGeometric:CyclicPerm
-
-$$
-\vect{c}\ip(\vect{a} \times \vect{b}) = \vect{a}\ip(\vect{b} \times \vect{c})= \vect{b}\ip(\vect{c} \times \vect{a})
-$$
-
-and swapping two vectors gives a minus sign:
-
-$$
-\vect{c}\ip(\vect{a} \times \vect{b})= - \vect{c}\ip(\vect{b} \times \vect{a}).
-$$
-
-::::::
-
-::::::{prf:proof}
-
-The parallelepiped spanned by the three vectors does not change under any permutation, and the orientation remains the same under a cyclic permutation.
-
-::::::
-
-The last proposition sets the way to take the determinant one dimension higher.
-
-::::::{prf:definition}
-
-The **determinant** of the ordered set of three vectors $\vect{a}, \vect{b}$ and $ \vect{c}$ in $\R^3$ is defined by the expression
-
-$$
-\det{(\vect{a}, \vect{b},\vect{c})} = \vect{a}\ip(\vect{b} \times \vect{c}).
-$$
-
-Note that the determinant is a real number.
-
-Alternatively, we can define the determinant as a function working on $3 \times 3$ matrices. And that is the sole interpretation we will use from now on.
-
-If we put
-
-$$
-A = [ \vect{a}\quad \vect{b}\quad\vect{c} ] = \left[\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right] ,
-$$
-
-then
-
-$$
-\det{A} = \left|\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right|= \vect{a}\ip(\vect{b} \times \vect{c}).
-$$
-
-::::::
-
-::::::{prf:proposition}
-:label: Prop:DetGeometric:ColExpand
-
-$$
-\left|\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right|=
-a_1\left|\begin{array}{cc} b_2 & c_2\\ b_3 & c_3 \end{array} \right|-
-a_2 \left|\begin{array}{cc} b_1 & c_1 \\ b_3 & c_3 \end{array} \right|+
-a_3 \left|\begin{array}{cc} b_1 & c_1 \\ b_2 & c_2\end{array} \right|.
-$$
-
-This can be further evaluated as
-
-$$
-\left|\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right|= a_1b_2c_3 - a_1b_3c_2 - a_2b_1c_3 +a_2b_3c_1 + a_3b_1c_2 - a_3b_2c_1.
-$$
-
-::::::
-
-The expression involving the three $2 \times 2$ determinants will be the stepping stone to define the determinant of a general $n \times n$ matrix.
-
-::::::{prf:proof}
-
-The identities are verified by evaluating the triple product:
-
-$$
-\left[\begin{array}{c} a_1 \\ a_2\\ a_3 \end{array} \right]
- \ip
-\left( \left[\begin{array}{c} b_1 \\ b_2\\ b_3 \end{array} \right]
- \times \left[\begin{array}{c} c_1 \\ c_2\\ c_3 \end{array} \right]
-\right)
- =
-\left[\begin{array}{c} a_1 \\ a_2\\ a_3 \end{array} \right]
- \ip
-\left[\begin{array}{c} b_2c_3-b_3c_2 \\ b_3c_1-b_1c_3\\ b_1c_2-b_2c_1 \end{array} \right]
- =
-$$
-
-$$
-\begin{array}{cl}
-=& a_1(b_2c_3 - b_3c_2) - a_2(b_1c_3 -b_3c_1) + a_3(b_1c_2 - b_2c_1)\\
-=& a_1b_2c_3 - a_1b_3c_2 - a_2b_1c_3 + a_2b_3c_1) + a_3b_1c_2 - a_3b_2c_1).
-\end{array}
-$$
-
-::::::
-
-The next proposition summarizes the relevant properties of $3 \times 3$ determinants.
-
-::::::{prf:proposition}
-:label: Prop:DetGeometric:Summary
-
-For the determinant
-
-$$
-D = \det{A} = \det{\left[\vect{a}\quad \vect{b}\quad\vect{c} \right]
-} = \left|\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right|
-$$
-
-the following properties hold
-
-<ol type = "i">
-<li>
-
-$|D|$, i.e. the absolute value of the determinant, is equal to the volume of the parallelepiped with the edges $\vect{a},\vect{b}$ and $\vect{c}$.
-
-</li>
-<li>
-
-:::{latextable}
-\begin{tabular}[t]{lcl} $D=0 $ & $\iff $& the matrix $A$ is singular \\
-& $\iff $& the vectors $\{\vect{a}, \vect{b},\vect{c}\}$ are linearly dependent.
-\end{tabular}
-:::
-
-Equivalently:
-
-:::{latextable}
-\begin{tabular}[t]{lcl} $D\neq 0 $ & $\iff $& the matrix $A$ is invertible \\
-& $\iff $& the vectors $\{\vect{a}, \vect{b},\vect{c}\}$ are linearly independent.
-\end{tabular}
-:::
-
-</li>
-<li>
-
-$D > 0 \quad \iff \quad $ the ordered set $(\vect{a},\vect{b},\vect{c})$ is positively oriented.
-
-That is, oriented in the same way as the basis $(\vect{e}_1,\vect{e}_2,\vect{e}_3)$.
-
-</li>
-<li>
-
-$\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right| = 1$.
-
-</li>
-</ol>
-
-::::::
diff --git a/Chapter5/DeterminantsExtras.md b/Chapter5/DeterminantsExtras.md
index db5148a..6623a41 100644
--- a/Chapter5/DeterminantsExtras.md
+++ b/Chapter5/DeterminantsExtras.md
@@ -71,8 +71,9 @@ Note that if the vectors $\vect{v}_1, \ldots, \vect{v}_n$ in {prf:ref}`Dfn:DetEx
::::::{figure} Images/Fig-DetExtras-ParPed.svg
:name: Fig:DetExtras:Parped
+:class: dark-light
-The parallelepiped generated by two vectors
+The parallelepiped generated by two vectors.
::::::
::::::{prf:definition} Orientation in $\R^n$
@@ -106,7 +107,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetExras:DetAsScaleFactorR2`
+:class: myproof
If the matrix $A$ is not invertible, the range of $T$, which is given by $\text{span}\{\vect{a}_1, \vect{a}_2\}$, is contained in a line.
Each region $R$ is then mapped onto a subset $S$ that is contained in this line, so
@@ -119,8 +121,9 @@ Next suppose that $A$ is invertible. Then the unit grid is mapped onto a grid wi
::::{figure} Images/Fig-DetExtras-StandardGrid.svg
:name: Fig:DetExtras:Grid
+:class: dark-light
-The image of the standard grid
+The image of the standard grid.
::::
@@ -161,8 +164,9 @@ See {numref}`Figure %s <Fig:DetExtras:ImageOfSquare>`
::::{figure} Images/Fig-DetExtras-ImageOfSquare.svg
:name: Fig:DetExtras:ImageOfSquare
+:class: dark-light
-The image of a square with 'corner' $\vect{p}$ and side length $r$
+The image of a square with 'corner' $\vect{p}$ and side length $r$.
::::
For a general (reasonable) region $R$ we sketch the idea and omit the technical details.
@@ -173,6 +177,7 @@ The limit of the areas of these approximations when the grids get finer and fine
::::{figure} Images/Fig-DetExtras-Subdivision.svg
:name: Fig:DetExtras:Subdivision
+:class: dark-light
Approximating a region by smaller and smaller squares.
@@ -213,7 +218,8 @@ where $S$ is the image of $R$ under $T$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetExtras:ScaleFactorRn`
+:class: myproof
<BR>
@@ -260,7 +266,8 @@ $\det{A} > 0$ and **reverses** the orientation if $\det{A} < 0$.
If the determinant is 0, then the set $\{T(\vect{v}_1), \ldots,T(\vect{v}_n) \}$ will be linearly dependent, and for such a set the orientation is not defined.
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:DetExtras:ScaleFactorR2`
+:class: myproof
This too follows immediately from the product rule of determinants.
@@ -296,13 +303,16 @@ $$
{numref}`Figure %s <Fig:DetExtras:Orientation>` visualizes what is going on.
-:::{figure} Images/Fig-DetExtras-Orientation.svg
+```{applet}
+:url: detextras/orientation
+:fig: Images/Fig-DetExtras-Orientation.svg
:name: Fig:DetExtras:Orientation
+:class: dark-light
Images under transformations with negative and positive determinant.
-:::
+```
-The images of a unit vector that rotates counterclockwise under transformation $A$ move around clockwise, i.e., in the *opposite* orientation/direction. Under transformation $B$ the images will go around the origin counterclockwise, i.e., in the *same* direction as the original vectors.
+The images of a unit vector that rotates counterclockwise under transformation $A$ move around clockwise, i.e., in the _opposite_ orientation/direction. Under transformation $B$ the images will go around the origin counterclockwise, i.e., in the _same_ direction as the original vectors.
::::
@@ -322,10 +332,10 @@ Let $A$ be an $n\times n$ matrix, and $\vect{v}$ a vector in $\R^n$. Then $A^{(i
::::::{prf:example}
For the matrix $A = \begin{bmatrix} 1 & 3 & 1 \\ 1 & 4 & 2 \\ 3 & 1 & 5 \end{bmatrix}$
-and the vector $\vect{v} = \begin{bmatrix} \color{blue}6 \\ \color{blue}7 \\ \color{blue}8 \end{bmatrix}$ we have that
+and the vector $\vect{v} = \begin{bmatrix} \class{blue}6 \\ \class{blue}7 \\ \class{blue}8 \end{bmatrix}$ we have that
$$
- A^{(2)}(\vect{v}) = \begin{bmatrix} 1 & \color{blue}6 & 1 \\ 1 & \color{blue}7 & 2 \\ 3 & \color{blue}8 & 5 \end{bmatrix}.
+ A^{(2)}(\vect{v}) = \begin{bmatrix} 1 & \class{blue}6 & 1 \\ 1 & \class{blue}7 & 2 \\ 3 & \class{blue}8 & 5 \end{bmatrix}.
$$
::::::
@@ -404,10 +414,10 @@ $$
::::::
-The following proof of Cramer's rule rests rather nicely on properties of the determinant function. But feel free to skip it.
-
-::::::{dropdown} Proof of {prf:ref}`Thm:DetExtras:Cramer`
+The following proof of Cramer's rule rests rather nicely on properties of the determinant function. But feel free to skip it.
+::::::{admonition} Proof of {prf:ref}`Thm:DetExtras:Cramer`
+:class: myproof, dropdown
Suppose $\vect{x} = \vect{c} = \left[\begin{array}{c} c_1 \\ \vdots\\ c_n\end{array} \right]
$ is the unique solution of the linear system $A\vect{x} = \vect{b}$, with the invertible matrix $A = [ \vect{a}_1 \, \, \vect{a}_2 \, \ldots \,\vect{a}_n ]$.
@@ -441,7 +451,7 @@ By the linearity property (in all of the columns) of the determinant ({prf:ref}`
:::{math}
:label: Eq:DetExtras:ProofCramer
-c_1\det{(A)} + c_2\det{(A^{(1)}(\vect{a}_2))} + \ldots + c_n\det{(A^{(1)}(\vect{a}_n))} - \det{(A^{(1)}(\vect{b}))} = 0.
+c_1\det{(A)} + c_2\det{(A^{(1)}(\vect{a}\_2))} + \ldots + c_n\det{(A^{(1)}(\vect{a}\_n))} - \det{(A^{(1)}(\vect{b}))} = 0.
:::
@@ -547,8 +557,8 @@ C_{1n} &C_{2n} &C_{3n} & \ldots &C_{nn} \\
::::::
-::::::{prf:proof}
-<BR>
+::::::{admonition} Proof of {prf:ref}`Prop:DetExtras:Inverse`
+:class: myproof
The $j$th column $\vect{b}_j$ of $B = A^{-1}$ is the solution of the linear system $A\vect{x} = \vect{e}_j$.
@@ -617,7 +627,8 @@ For clarity we used dots to indicate products. Note that the first two products
The proof we think, is short and instructive.
-::::::{dropdown} Proof of {prf:ref}`Prop:DetExtras:AdjointProperty`
+::::::{admonition} Proof of {prf:ref}`Prop:DetExtras:AdjointProperty`
+:class: myproof, dropdown
For an invertible matrix the statement follows immediately from {prf:ref}`Prop:DetExtras:Inverse`.
@@ -726,15 +737,14 @@ Conversely, we can write the cross product in terms containing determinants.
:label: Eq:DetExtras:DetCrossProd
\begin{array}{rcl}
-\left[\begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array}\right] \times
-\left[\begin{array}{c}b_1 \\ b_2 \\ b_3 \end{array}\right]
-&=& \left[\begin{array}{c}a_2b_3-a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_2b_1-a_2b_1 \end{array}\right] \\
+\left[\begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array}\right] \times
+\left[\begin{array}{c}b_1 \\ b_2 \\ b_3 \end{array}\right]
+&=& \left[\begin{array}{c}a_2b_3-a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_2b_1-a_2b_1 \end{array}\right] \\
&=&
-\left|\begin{array}{cc} a_2 & b_2 \\a_3 & b_3 \end{array}\right|\vect{e}_1
-- \left|\begin{array}{cc} a_1 & b_1 \\ a_3 & b_3 \end{array}\right|\vect{e}_2
-+ \left|\begin{array}{cc} a_1 & b_1 \\a_2 & b_2 \end{array}\right|\vect{e}_3.
-\end{array}
-
+\left|\begin{array}{cc} a_2 & b_2 \\a_3 & b_3 \end{array}\right|\vect{e}_1
+- \left|\begin{array}{cc} a_1 & b_1 \\ a_3 & b_3 \end{array}\right|\vect{e}_2
++ \left|\begin{array}{cc} a_1 & b_1 \\a_2 & b_2 \end{array}\right|\vect{e}_3.
+ \end{array}
:::
@@ -894,13 +904,12 @@ $$
so property iv. is satisfied too.
::::::
-
We end the chapter with a proof of {prf:ref}`Prop:DetExtras:Properties-ndimCrossProd`.
<BR>
-So, if you are interested, push the button on the right.
-
-::::::{dropdown} Proof of {prf:ref}`Prop:DetExtras:Properties-ndimCrossProd`
+So, if you are interested, push the button on the right.
+::::::{admonition} Proof of {prf:ref}`Prop:DetExtras:Properties-ndimCrossProd`
+:class: myproof, dropdown
The properties follow from the observation that for each vector $\vect{v}$ in $\R^n$
@@ -1078,19 +1087,18 @@ $$
::::::
-
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ddb8daf3-3773-44c9-8df0-fe3084a6e7c4?id=93170
:label: grasple_exercise_5_4_1
:dropdown:
:description: To compute the area of a triangle with sides $\vect{u}$ and $\vect{v}$ in the plane.
::::::
-
-
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8d7a0672-6283-4bb1-9b43-b41a03067e40?id=93171
:label: grasple_exercise_5_4_2
:dropdown:
@@ -1098,8 +1106,8 @@ $$
::::::
-
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/62770bc4-da31-4212-a713-bb2843b0e580?id=93172
:label: grasple_exercise_5_4_3
:dropdown:
@@ -1108,6 +1116,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f787e084-9a77-40b4-b755-97890b98cfb6?id=93176
:label: grasple_exercise_5_4_4
:dropdown:
@@ -1115,9 +1124,8 @@ $$
::::::
-
-
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3add427a-88a3-4da0-8f0a-2bf8bb8781dd?id=93179
:label: grasple_exercise_5_4_5
:dropdown:
@@ -1125,9 +1133,8 @@ $$
::::::
-
-
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8d4a98f7-50ac-4705-8b34-680b7b8395d9?id=93181
:label: grasple_exercise_5_4_6
:dropdown:
@@ -1135,12 +1142,11 @@ $$
::::::
-
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bc3df113-95b3-470a-a730-3ad8faab08f5?id=93183
:label: grasple_exercise_5_4_7
:dropdown:
-:description: To compute the normal vector $N(\vect{a}_1,\vect{a}_2,\vect{a}_3)$ as in {numref}`Subsection %s <Subsec-DetExtras-DetAndCrossProd>`
+:description: To compute the normal vector $N(\vect{a}_1,\vect{a}_2,\vect{a}_3)$ as in {numref}`Subsection %s <Subsec-DetExtras-DetAndCrossProd>`
::::::
-
diff --git a/Chapter5/DeterminantsGeometric.md b/Chapter5/DeterminantsGeometric.md
index 236c2ad..e3c8c9b 100644
--- a/Chapter5/DeterminantsGeometric.md
+++ b/Chapter5/DeterminantsGeometric.md
@@ -38,8 +38,9 @@ where $A$ and $B$ are the end points of $\vect{u}$ and $\vect{v}$, and $C$ corre
::::{figure} Images/Fig-DetGeometric-PargramOACB.svg
:name: Fig:DetGeometric:PargramOACB
+:class: dark-light
-The parallelogram OACB
+The parallelogram $OACB$.
::::
::::::{prf:proposition}
@@ -51,7 +52,8 @@ is given by $|ad-bc|$, i.e., the absolute value of $ad-bc$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:Area`
+:class: myproof
The quickest way to prove this is to translate it to the cross product ({numref}`Sec:CrossProduct`). To make use of the cross product we introduce the vectors
@@ -67,8 +69,9 @@ Thus we embed the plane into $\R^3$ as the $x_1$-$x_2$-plane. See {numref}`Figur
:url: det_geometric/orientedarea1
:fig: Images/Fig-DetGeometric-OrientedArea1.svg
:name: Fig:DetGeometric:OrientedArea1
+:class: dark-light
-Oriented area
+Oriented area.
```
So we embed the plane $\R^2$ as the $x$-$y$-plane in $\R^3$.
@@ -86,13 +89,15 @@ The length of this cross product is equal to $|ad-bc|$. This gives the area of t
Now what about the sign here?
::::::{prf:proposition}
+:label: Prop:DetGeometric:DirectedAngle
$ad - bc = \norm{\vect{u}}\norm{\vect{v}}\sin(\varphi)$, where $\varphi$ is the angle from $\vect{u}$ counterclockwise to $\vect{v}$.
We will call this the **directed angle** from $\vect{u}$ to $\vect{v}$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:DirectedAngle`
+:class: myproof
Again we can resort to properties of the cross product, but in fact it is not necessary to go up one dimension.
By a small twist we can turn the determinant into an inner product:
@@ -108,8 +113,9 @@ where $\vect{u}^{\perp}$ is the vector that is perpendicular to $\vect{u}$, poin
::::{figure} Images/Fig-DetGeometric-AreaPargram.svg
:name: Fig:DetGeometric:AreaPargram
+:class: dark-light
-The parallelogram $OACB$ and the orthogonal vector $\vect{u}^{\perp}$
+The parallelogram $OACB$ and the orthogonal vector $\vect{u}^{\perp}$.
::::
So
@@ -252,7 +258,8 @@ Verify the four properties of {prf:ref}`Prop:DetGeometric:Properties2by2Det`
The properties have a clear geometric interpretation using the notion of signed area.
The following alternative proof uses this geometric viewpoint.
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:Properties2by2Det`
+:class: myproof
Three of the four properties are quickly settled.
@@ -293,16 +300,12 @@ $\det{(\vect{u},\vect{v}+\vect{w})} = \det{(\vect{u},\vect{v})} + \det{(\vect{u}
is the most interesting one. The two pictures of {numref}`Figure %s <Fig:DetGeometric:Linearity1>` and {numref}`Figure %s <Fig:DetGeometric:Linearity2>` tell the story.
-::::{figure}
-:name:
-
-::::
-
```{applet}
:url: det_geometric/linearity_one
:fig: Images/Fig-DetGeometric-SumRule.svg
:name: Fig:DetGeometric:Linearity1
:position: 2,2
+:class: dark-light
The sum rule in a picture with $(\vect{u},\vect{w})$ positively oriented. Note, this is a **2D picture**.
```
@@ -325,6 +328,7 @@ Since the two triangles $OCE$ and $ABD$ are congruent and they have equal areas.
:fig: Images/Fig-DetGeometric-SumRule.svg
:name: Fig:DetGeometric:Linearity2
:position: 2,2
+:class: dark-light
The sum rule in a picture with $(\vect{u},\vect{w})$ negatively oriented. Note, this is a **2D picture**.
```
@@ -355,8 +359,9 @@ Then the three vectors can be interpreted as three edges of a parallelepiped.
:url: det_geometric/paraped
:fig: Images/Fig-DetGeometric-Paraped.svg
:name: Fig:DetGeometric:Paraped
+:class: dark-light
-Volume equals base area times height
+Volume equals base area times height.
```
::::::{prf:proposition}
@@ -373,7 +378,8 @@ $$
::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:VolumeParped`
+:class: myproof
Just as the area of a parallelogram can be computed as 'base length times height', the volume of a parallelepiped can be computed as 'base area times height'. See {numref}`Figure %s <Fig:DetGeometric:Paraped>`.
As base region we can take the parallelogram spanned by $\vect{a}$ and $\vect{b}$, and then the base area becomes
@@ -421,7 +427,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:CyclicPerm`
+:class: myproof
The parallelepiped spanned by the three vectors does not change under any permutation, and the orientation remains the same under a cyclic permutation.
@@ -480,7 +487,8 @@ $$
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetGeometric:ColExpand`
+:class: myproof
The identities are verified by evaluating the triple product:
@@ -554,6 +562,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3126529d-db82-43e2-862d-7f013f39f619?id=93128
:label: grasple_exercise_5_1_1
:dropdown:
@@ -562,6 +571,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/13e76393-8f38-48aa-9685-2132208a0cc8?id=93131
:label: grasple_exercise_5_1_2
:dropdown:
@@ -570,6 +580,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fc111f55-9f43-4730-a1f9-e3b5f03069bd?id=93133
:label: grasple_exercise_5_1_3
:dropdown:
@@ -578,6 +589,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2d846d56-3729-468e-80d8-74ec6d348719?id=93134
:label: grasple_exercise_5_1_4
:dropdown:
@@ -586,6 +598,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1053b23-02ab-4ffb-bcda-70f808a9910a?id=74408
:label: grasple_exercise_5_1_5
:dropdown:
@@ -594,6 +607,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/edc50bb7-425e-4a76-927e-f20504128f5f?id=65686
:label: grasple_exercise_5_1_6
:dropdown:
@@ -602,6 +616,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/47e1fd08-2d8a-4c0d-816a-37e314707191?id=87501
:label: grasple_exercise_5_1_7
:dropdown:
@@ -610,6 +625,7 @@ $\det{I} = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/22a384c2-3434-4650-a52a-e954c470e08d?id=92929
:label: grasple_exercise_5_1_8
:dropdown:
diff --git a/Chapter5/DeterminantsViaCofactors.md b/Chapter5/DeterminantsViaCofactors.md
index 31570f5..1d812e4 100644
--- a/Chapter5/DeterminantsViaCofactors.md
+++ b/Chapter5/DeterminantsViaCofactors.md
@@ -192,11 +192,11 @@ $ along its third row.
$$
\left|\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
-\color{blue}a_{31} & \color{blue}a_{32} & \color{blue}a_{33}
+\class{blue}{a_{31}} & \class{blue}{a_{32}} & \class{blue}{a_{33}}
\end{array}\right|=
-{\color{blue}a_{31}}\left|\begin{array}{cc}a_{12} & a_{13}\\ a_{22} & a_{23}\end{array}\right|-
-{\color{blue}a_{32}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{21} & a_{23}\end{array}\right|+
-{\color{blue}a_{33}}\left|\begin{array}{cc}a_{11} & a_{12}\\ a_{21} & a_{22}\end{array}\right|
+\class{blue}{a_{31}}\left|\begin{array}{cc}a_{12} & a_{13}\\ a_{22} & a_{23}\end{array}\right|-
+\class{blue}{a_{32}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{21} & a_{23}\end{array}\right|+
+\class{blue}{a_{33}}\left|\begin{array}{cc}a_{11} & a_{12}\\ a_{21} & a_{22}\end{array}\right|
$$
$$
@@ -212,11 +212,11 @@ $$
Cofactor expansion along the second column yields
$$
-\left|\begin{array}{rrr} a_{11} & \color{blue}a_{12} & a_{13} \\
-a_{21} & \color{blue}a_{22} & a_{23} \\
-a_{31} & \color{blue}a_{32} & a_{33}
+\left|\begin{array}{rrr} a_{11} & \class{blue}{a_{12}} & a_{13} \\
+a_{21} & \class{blue}{a_{22}} & a_{23} \\
+a_{31} & \class{blue}{a_{32}} & a_{33}
\end{array}\right|=
--{\color{blue}a_{12}}\left|\begin{array}{cc}a_{21} & a_{23}\\ a_{31} & a_{33}\end{array}\right|+{\color{blue}a_{22}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{31} & a_{33}\end{array}\right|-{\color{blue}a_{32}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{21} & a_{23}\end{array}\right|
+-\class{blue}{a_{12}}\left|\begin{array}{cc}a_{21} & a_{23}\\ a_{31} & a_{33}\end{array}\right|+\class{blue}{a_{22}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{31} & a_{33}\end{array}\right|-\class{blue}{a_{32}}\left|\begin{array}{cc}a_{11} & a_{13}\\ a_{21} & a_{23}\end{array}\right|
$$
$$
@@ -388,7 +388,8 @@ For a triangular matrix the determinant is equal to the product of the entries o
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetCofactors:TriangularMatrices`
+:class: myproof
We can use the same strategy as in {prf:ref}`Ex:DetCofactors:Triangular`.
That is, for an upper triangular matrix expand along the columns from left to right, for a lower triangular matrix
@@ -407,7 +408,8 @@ A triangular matrix is invertible if and only if it has a non-zero determinant.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetCofactors:InvertibleTriangular`
+:class: myproof
Let us first consider the case of an $n \times n$ upper triangular matrix $U$, with entries $u_{ij}$. Such a matrix is an echelon matrix. It is invertible if and only if it has $n$ linearly independent columns, which is the case if all diagonal elements $u_{ii}$ are nonzero. And this last is equivalent to
@@ -482,6 +484,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/34eb983b-c7e7-40f3-a983-6bfb970f6836?id=93135
:label: grasple_exercise_5_2_1
:dropdown:
@@ -491,6 +494,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b86bd320-47cd-45cb-ab88-81b20a48c427?id=93136
:label: grasple_exercise_5_2_2
:dropdown:
@@ -500,6 +504,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2f993d71-6a19-435d-a449-cc0dbb8237d5?id=93137
:label: grasple_exercise_5_2_3
:dropdown:
@@ -508,6 +513,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/40de4737-1823-425b-b997-a07c53cb2f96?id=93138
:label: grasple_exercise_5_2_4
:dropdown:
@@ -517,6 +523,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/57cd522b-6096-416a-a739-fea5cbbc77c9?id=93139
:label: grasple_exercise_5_2_5
:dropdown:
@@ -525,6 +532,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/25c54a22-eb00-4cbf-ac0c-21b4974a48ff?id=92927
:label: grasple_exercise_5_2_6
:dropdown:
@@ -533,6 +541,7 @@ In a similar way, the property $\text{det}\big(A^T\big) = \text{det}(A)$ for $4
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/48fedcac-0039-4eaa-9f71-20fa86ed4536?id=93142
:label: grasple_exercise_5_2_7
:dropdown:
diff --git a/Chapter5/DeterminantsViaRowReduction.md b/Chapter5/DeterminantsViaRowReduction.md
index 00dc2df..0962a6d 100644
--- a/Chapter5/DeterminantsViaRowReduction.md
+++ b/Chapter5/DeterminantsViaRowReduction.md
@@ -249,7 +249,8 @@ If a matrix $A$ has two equal rows (or columns), then $\det{A} = 0$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Cor:DetRowReduction:EqualRows`
+:class: myproof
Suppose the $i$th and the $j$th row of $A$ are equal, and let $\det{A} = d$. Let $B$ be the matrix $A$ with the $i$th and $j$th row interchanged.
@@ -291,9 +292,8 @@ $$
The proof is -- we think -- quite instructive. (However, feel free to skip it.)
-
-::::::{dropdown} Proof of {prf:ref}`Thm:DetRowReduction:Invertibility`
-
+::::::{admonition} Proof of {prf:ref}`Thm:DetRowReduction:Invertibility`
+:class: myproof, dropdown
In the previous section we have already seen that the statement is true for triangular matrices.
@@ -349,7 +349,8 @@ $$
The idea of the proof is to break it down to products of the form $\det{(EA)} = \det{E}\cdot\det{A}$, where $E$ is an elementary matrix (Equation{eq}`Eq:DetRowReduction:ElementaryMatrices`). For more details you open the proof below.
-::::::{dropdown} Proof of {prf:ref}`Thm:DetRowReduction:ProductRule`.
+::::::{admonition} Proof of {prf:ref}`Thm:DetRowReduction:ProductRule`
+:class: myproof, dropdown
We already know that the identity holds if $A$ is an elementary matrix.
It will also hold if $A$ is not invertible, as in that case $AB$ is also not invertible,
@@ -394,7 +395,8 @@ If the matrix $A$ is invertible, then $\text{det}\big(A^{-1}\big)= \dfrac{1}{\de
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Cor:DetRowReduction:DetOfInverse`
+:class: myproof
We can combine the three properties
@@ -474,7 +476,8 @@ $$
::::::
-::::::{dropdown} Solution to {numref}`Exc:DetRowReduction:PropNonProp` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:DetRowReduction:PropNonProp`
+:class: solution, dropdown
We treat the statements one by
@@ -598,9 +601,8 @@ Then
Click on the symbol to the right below for the proof of {prf:ref}`Prop:DetRowReduction:RowOps` and {prf:ref}`Prop:DetRowReduction:SumofCols`.
-::::::{dropdown} Proof of {prf:ref}`Prop:DetRowReduction:RowOps` and {prf:ref}`Prop:DetRowReduction:SumofCols`.
-
-%::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:DetRowReduction:RowOps` and {prf:ref}`Prop:DetRowReduction:SumofCols`
+:class: myproof, dropdown
For typographical reasons we will prove the three rules stated as column operations.
For an $n \times n$ matrix
@@ -743,38 +745,38 @@ For instance, to interchange column $2$ and column $5$ in a $5 \times 5$ matrix
$$
\begin{array}{rl}
- \left|\begin{array}{ccccc} a_{11} & {\color{blue}a_{12}} & a_{13} & a_{14} & {\color{red}a_{15}} \\
- a_{21} & {\color{blue}a_{22}} & a_{23} & a_{24} & {\color{red}a_{25}} \\
- \vdots & {\color{blue}\vdots} & \vdots & \vdots & {\color{red}\vdots} \\
- \vdots & {\color{blue}\vdots} & \vdots & \vdots & {\color{red}\vdots} \\
- a_{51} & {\color{blue}a_{52}} & a_{53} & a_{54} & {\color{red}a_{55}}
+ \left|\begin{array}{ccccc} a_{11} & \class{blue}{a_{12}} & a_{13} & a_{14} & \class{red}{a_{15}} \\
+ a_{21} & \class{blue}{a_{22}} & a_{23} & a_{24} & \class{red}{a_{25}} \\
+ \vdots & \class{blue}{\vdots} & \vdots & \vdots & \class{red}{\vdots} \\
+ \vdots & \class{blue}{\vdots} & \vdots & \vdots & \class{red}{\vdots} \\
+ a_{51} & \class{blue}{a_{52}} & a_{53} & a_{54} & \class{red}{a_{55}}
\end{array}\right| = &
- - \left|\begin{array}{ccccc} a_{11} & a_{13} & {\color{blue}a_{12}} & a_{14} & {\color{red}a_{15}} \\
- a_{21} & a_{23} & {\color{blue}a_{22}} & a_{24} & {\color{red}a_{25}} \\
- \vdots & \vdots & {\color{blue}\vdots} & \vdots & {\color{red}\vdots} \\
- \vdots & \vdots & {\color{blue}\vdots} & \vdots & {\color{red}\vdots} \\
- a_{51} & a_{53} & {\color{blue}a_{52}} & a_{54} & {\color{red}a_{55}}
+ - \left|\begin{array}{ccccc} a_{11} & a_{13} & \class{blue}{a_{12}} & a_{14} & \class{red}{a_{15}} \\
+ a_{21} & a_{23} & \class{blue}{a_{22}} & a_{24} & \class{red}{a_{25}} \\
+ \vdots & \vdots & \class{blue}{\vdots} & \vdots & \class{red}{\vdots} \\
+ \vdots & \vdots & \class{blue}{\vdots} & \vdots & \class{red}{\vdots} \\
+ a_{51} & a_{53} & \class{blue}{a_{52}} & a_{54} & \class{red}{a_{55}}
\end{array}\right| = \\[2ex]
- + \left|\begin{array}{ccccc} a_{11} & a_{13} & a_{14} & {\color{blue}a_{12}} & {\color{red}a_{15}} \\
- a_{21} & a_{23} & a_{24} & {\color{blue}a_{22}} & {\color{red}a_{25}} \\
- \vdots & \vdots & \vdots & {\color{blue}\vdots} & {\color{red}\vdots} \\
- \vdots & \vdots & \vdots & {\color{blue}\vdots} & {\color{red}\vdots} \\
- a_{51} & a_{53} & a_{54} & {\color{blue}a_{52}} & {\color{red}a_{55}} \end{array}\right| = &
- - \left|\begin{array}{ccccc} a_{11}& a_{13} & a_{14} & {\color{red}a_{15}} & {\color{blue}a_{12}} \\
- a_{21} & a_{23} & a_{24} & {\color{red}a_{25}} & {\color{blue}a_{22}} \\
- \vdots & \vdots & \vdots & {\color{red}\vdots} & {\color{blue}\vdots} \\
- \vdots & \vdots & \vdots & {\color{red}\vdots} & {\color{blue}\vdots} \\
- a_{51} & a_{53} & a_{54} & {\color{red}a_{55}} & {\color{blue}a_{52}} \end{array}\right| = \\[2ex]
- + \left|\begin{array}{ccccc} a_{11} & a_{13} & {\color{red}a_{15}} & a_{14} & {\color{blue}a_{12}} \\
- a_{21} & a_{23} & {\color{red}a_{25}} & a_{24} & {\color{blue}a_{22}} \\
- \vdots & \vdots & {\color{red}\vdots} & \vdots & {\color{blue}\vdots} \\
- \vdots & \vdots & {\color{red}\vdots} & \vdots & {\color{blue}\vdots} \\
- a_{51} & a_{53} & {\color{red}a_{55}} & a_{54} & {\color{blue}a_{52}} \end{array}\right| =&
- -\left|\begin{array}{ccccc} a_{11} & {\color{red}a_{15}}& a_{13} & a_{14} & {\color{blue}a_{12}} \\
- a_{21} & {\color{red}a_{25}}& a_{23} & a_{24} & {\color{blue}a_{22}} \\
- \vdots & {\color{red}\vdots}& \vdots & \vdots & {\color{blue}\vdots} \\
- \vdots & {\color{red}\vdots}& \vdots & \vdots & {\color{blue}\vdots} \\
- a_{51} & {\color{red}a_{55}}& a_{53} & a_{54} & {\color{blue}a_{52}} \end{array}\right|
+ + \left|\begin{array}{ccccc} a_{11} & a_{13} & a_{14} & \class{blue}{a_{12}} & \class{red}{a_{15}} \\
+ a_{21} & a_{23} & a_{24} & \class{blue}{a_{22}} & \class{red}{a_{25}} \\
+ \vdots & \vdots & \vdots & \class{blue}{\vdots} & \class{red}{\vdots} \\
+ \vdots & \vdots & \vdots & \class{blue}{\vdots} & \class{red}{\vdots} \\
+ a_{51} & a_{53} & a_{54} & \class{blue}{a_{52}} & \class{red}{a_{55}} \end{array}\right| = &
+ - \left|\begin{array}{ccccc} a_{11}& a_{13} & a_{14} & \class{red}{a_{15}} & \class{blue}{a_{12}} \\
+ a_{21} & a_{23} & a_{24} & \class{red}{a_{25}} & \class{blue}{a_{22}} \\
+ \vdots & \vdots & \vdots & \class{red}{\vdots} & \class{blue}{\vdots} \\
+ \vdots & \vdots & \vdots & \class{red}{\vdots} & \class{blue}{\vdots} \\
+ a_{51} & a_{53} & a_{54} & \class{red}{a_{55}} & \class{blue}{a_{52}} \end{array}\right| = \\[2ex]
+ + \left|\begin{array}{ccccc} a_{11} & a_{13} & \class{red}{a_{15}} & a_{14} & \class{blue}{a_{12}} \\
+ a_{21} & a_{23} & \class{red}{a_{25}} & a_{24} & \class{blue}{a_{22}} \\
+ \vdots & \vdots & \class{red}{\vdots} & \vdots & \class{blue}{\vdots} \\
+ \vdots & \vdots & \class{red}{\vdots} & \vdots & \class{blue}{\vdots} \\
+ a_{51} & a_{53} & \class{red}{a_{55}} & a_{54} & \class{blue}{a_{52}} \end{array}\right| =&
+ -\left|\begin{array}{ccccc} a_{11} & \class{red}{a_{15}}& a_{13} & a_{14} & \class{blue}{a_{12}} \\
+ a_{21} & \class{red}{a_{25}}& a_{23} & a_{24} & \class{blue}{a_{22}} \\
+ \vdots & \class{red}{\vdots}& \vdots & \vdots & \class{blue}{\vdots} \\
+ \vdots & \class{red}{\vdots}& \vdots & \vdots & \class{blue}{\vdots} \\
+ a_{51} & \class{red}{a_{55}}& a_{53} & a_{54} & \class{blue}{a_{52}} \end{array}\right|
\end{array}
$$
@@ -814,6 +816,7 @@ This settles all matters.
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b34a791a-3f42-4d10-9952-f6f5699a68fb?id=104164
:label: grasple_exercise_5_3_1
:dropdown:
@@ -823,6 +826,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1d3924d9-ea34-4a89-8b7c-33e385d144ba?id=104312
:label: grasple_exercise_5_3_2
:dropdown:
@@ -831,6 +835,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1fcb337d-f906-423a-acd5-8d8c69d4d04b?id=93158
:label: grasple_exercise_5_3_3
:dropdown:
@@ -839,6 +844,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/cabb663b-7b86-4215-81aa-0a3da91a5688?id=103719
:label: grasple_exercise_5_3_4
:dropdown:
@@ -848,6 +854,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1354915d-4cf4-4559-8ac2-68573807199d?id=103702
:label: grasple_exercise_5_3_5
:dropdown:
@@ -858,6 +865,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1354915d-4cf4-4559-8ac2-68573807199d?id=103702
:label: grasple_exercise_5_3_6
:dropdown:
@@ -866,6 +874,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/882506bb-6a5e-479f-b095-bb5b95be2467?id=104166
:label: grasple_exercise_5_3_7
:dropdown:
@@ -876,6 +885,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/993b010f-3351-4b98-b9b7-1d04c1c959be?id=93143
:label: grasple_exercise_5_3_8
:dropdown:
@@ -885,6 +895,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2d51357d-e56d-4de5-a882-493a795fd222?id=93144
:label: grasple_exercise_5_3_9
:dropdown:
@@ -896,6 +907,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9974012a-1ac9-439f-919f-2647be1ba4ba?id=92965
:label: grasple_exercise_5_3_10
:dropdown:
@@ -905,6 +917,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4a01fc67-0acc-44aa-9ba2-18c1accae720?id=93145
:label: grasple_exercise_5_3_11
:dropdown:
@@ -912,6 +925,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f2e09cfe-9d88-4f7b-a295-bad7feda89e5?id=93150
:label: grasple_exercise_5_3_12
:dropdown:
@@ -923,6 +937,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/35bff21c-6434-4e4a-b154-965de08479c0?id=93146
:label: grasple_exercise_5_3_13
:dropdown:
@@ -932,6 +947,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7c4c18ba-96ba-432a-97b0-0a269a0a9f55?id=93147
:label: grasple_exercise_5_3_14
:dropdown:
@@ -940,6 +956,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5deab9d8-20f3-4b59-b54e-3b61c981c8c7?id=93148
:label: grasple_exercise_5_3_15
:dropdown:
@@ -949,6 +966,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c3f025b5-2ca4-48cb-a1f9-4e144c8bc258?id=93149
:label: grasple_exercise_5_3_16
:dropdown:
@@ -958,6 +976,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a5713d1f-696b-42e5-ab74-553eec26b00b?id=93151
:label: grasple_exercise_5_3_17
:dropdown:
@@ -968,6 +987,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9ab31fa4-6686-4865-8d43-602dc1fe670e?id=93152
:label: grasple_exercise_5_3_18
:dropdown:
@@ -978,6 +998,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9ae9228f-ab17-4853-9995-e38e16d87c22?id=93153
:label: grasple_exercise_5_3_19
:dropdown:
@@ -986,6 +1007,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8db6831f-2671-443a-af64-799d1d0d9179?id=93154
:label: grasple_exercise_5_3_20
:dropdown:
@@ -997,6 +1019,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/116e83e9-1db7-47ce-a2f3-ad398aee0201?id=93155
:label: grasple_exercise_5_3_21
:dropdown:
@@ -1005,6 +1028,7 @@ This settles all matters.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/821d81b1-2cec-4fa4-b4a7-b1f9c32d6e06?id=93156
:label: grasple_exercise_5_3_22
:dropdown:
@@ -1015,6 +1039,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5b89a008-2e3d-48a5-a764-0b1b6a3ec4dc?id=93157
:label: grasple_exercise_5_3_23
:dropdown:
@@ -1024,6 +1049,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e0bfbb0c-002f-485f-9b2f-5249938b6e40?id=93162
:label: grasple_exercise_5_3_24
:dropdown:
@@ -1035,6 +1061,7 @@ This settles all matters.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/41f5ca17-ab3e-4487-b5fa-ee325cae85aa?id=93164
:label: grasple_exercise_5_3_25
:dropdown:
@@ -1052,7 +1079,8 @@ Give an alternative proof of {prf:ref}`Cor:DetRowReduction:EqualRows` using Rule
::::::
-::::::{dropdown} Solution to {numref}`Exc:DetRowReduction:EqualRows` (*click to show*)
+::::::{admonition} Solution to {numref}`Exc:DetRowReduction:EqualRows`
+:class: solution, dropdown
Suppose $A$ is a matrix with two equal rows, say row $i$ and row $j$ are equal.
@@ -1062,14 +1090,14 @@ For instance, with a $4\times 4$ matrix with equal second and fourth row we woul
$$
\begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
- \color{blue}a_{21} & \color{blue}a_{22} & \color{blue}a_{23} & \color{blue}a_{24} \\
+ \class{blue}{a_{21}} & \class{blue}{a_{22}} & \class{blue}{a_{23}} & \class{blue}{a_{24}} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
- \color{blue}a_{21} & \color{blue}a_{22} & \color{blue}a_{23} & \color{blue}a_{24}
+ \class{blue}{a_{21}} & \class{blue}{a_{22}} & \class{blue}{a_{23}} & \class{blue}{a_{24}}
\end{vmatrix} =
\begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
- \color{blue}0 & \color{blue}0 &\color{blue} 0 & \color{blue}0
+ \class{blue}0 & \class{blue}0 &\class{blue} 0 & \class{blue}0
\end{vmatrix}.
$$
diff --git a/Chapter6/CharPolynomial.md b/Chapter6/CharPolynomial.md
index 7d3317d..b3d5207 100644
--- a/Chapter6/CharPolynomial.md
+++ b/Chapter6/CharPolynomial.md
@@ -16,7 +16,8 @@ Suppose $A$ is an $n\times n$ matrix. Then $\lambda$ is an eigenvalue of $A$ if
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Eigenvalues:DetAminusLambdaI`
+:class: myproof
There's not much new here.
@@ -135,7 +136,8 @@ For an $n\times n$ matrix $A$ the function det$(A - \lambda I)$ is a polynomial
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:EigenValues:CharPoly`
+:class: myproof
We have to dive into the hardware of determinants a bit. If the determinant of an $n\times n$ matrix $M$ is computed by iteratively expanding along the first rows, i.e., doing it the hard way, we end up with a sum of $n!$ terms. Each term is the product of $n$ entries of $M$, where each row and each column of $A$ is represented exactly once.
@@ -376,6 +378,7 @@ The eigenvalue $\lambda = -1$ has both algebraic multiplicity and geometric mult
The following exercise, which is meant to shed some more light on the concepts just introduced, requires few technicalities.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6e01d5c1-897b-44bf-bbb2-4653ff095f48?id=92498
:label: grasple_exercise_6_1_A
:dropdown:
@@ -416,8 +419,8 @@ $$
::::::
-::::::{dropdown} (Sketch of the) Proof of {prf:ref}`Prop:EigenValues:CharPolyTrace`
-%::::::{prf:proof}
+::::::{admonition} (Sketch of the) Proof of {prf:ref}`Prop:EigenValues:CharPolyTrace`
+:class: myproof, dropdown
For $n=2$ we have already seen that the characteristic polynomial of the most general $2 \times 2$ matrix
$A = \left[\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right]
@@ -517,8 +520,8 @@ Let $A$ an $n\times n$ matrix with $n$ eigenvalues $\lambda_1,\lambda_2, \ld
::::::
-
-::::::{dropdown} Proof of {prf:ref}`Prop:Eigenvalues:SumEigenvaluesAndTrace`
+::::::{admonition} Proof of {prf:ref}`Prop:Eigenvalues:SumEigenvaluesAndTrace`
+:class: myproof, dropdown
This is more a statement about algebra, in particular about polynomials, than about linear algebra. In {numref}`Section %s <Section:ComplexEV>` we will see that it also holds for matrices with complex eigenvalues.
@@ -615,6 +618,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b43cd5dc-3fff-432a-bdec-d56e38c39e89?id=91450
:label: grasple_exercise_6_2_1
:dropdown:
@@ -625,6 +629,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b89efbb3-c5cc-4fab-874b-8dd285644ab2?id=91452
:label: grasple_exercise_6_2_2
:dropdown:
@@ -634,6 +639,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/137aaf98-60d5-4aab-82f4-10ea40811a7b?id=91453
:label: grasple_exercise_6_2_3
:dropdown:
@@ -644,6 +650,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e3793a52-25f0-48cd-b47f-b59f872e3e1a?id=91454
:label: grasple_exercise_6_2_4
:dropdown:
@@ -653,6 +660,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3597a3e9-17b7-491c-89a7-4f33f1a4fb8c?id=91482
:label: grasple_exercise_6_2_5
:dropdown:
@@ -661,6 +669,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9e16e7ab-87f1-448c-a52a-a8d6c1470c4b?id=91483
:label: grasple_exercise_6_2_6
:dropdown:
@@ -674,6 +683,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2c30dad3-17a4-4f20-b277-fd13e0c93e9f?id=92384
:label: grasple_exercise_6_2_7
:dropdown:
@@ -683,6 +693,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e35e7cfb-2a21-4849-ae48-6c7a94e85707?id=92409
:label: grasple_exercise_6_2_8
:dropdown:
@@ -692,6 +703,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/eb3e8ea4-3a0b-4767-aa62-0d8b05e35dda?id=91484
:label: grasple_exercise_6_2_9
:dropdown:
@@ -701,6 +713,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e9044c04-4bfb-474e-8823-bff6449b92ab?id=92210
:label: grasple_exercise_6_2_10
:dropdown:
@@ -710,6 +723,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/82387c7b-49c8-4438-b72c-e1d023fb2780?id=92211
:label: grasple_exercise_6_2_11
:dropdown:
@@ -719,6 +733,7 @@ Every matrix $A$ is a zero of its characteristic polynomial.
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b846c6e6-f2c6-4d5b-bc40-adaed4cf6276?id=92490
:label: grasple_exercise_6_2_12
:dropdown:
diff --git a/Chapter6/ComplexEigenvalues.md b/Chapter6/ComplexEigenvalues.md
index 45761f7..2e9a15a 100644
--- a/Chapter6/ComplexEigenvalues.md
+++ b/Chapter6/ComplexEigenvalues.md
@@ -35,12 +35,12 @@ $$
=
\left[\begin{array}{cc|c} -1 - i & -2& 0 \\ 1 & 1-i & 0 \end{array}\right]
\sim
-\left[\begin{array}{cc|c} 0 & \color{blue}0 & 0 \\ 1 & 1-i & 0 \end{array}\right]
+\left[\begin{array}{cc|c} 0 & \class{blue}0 & 0 \\ 1 & 1-i & 0 \end{array}\right]
,
$$
where the row operation we invoke is: add the second row $(1+i)$ times to the first row.
-The blue {\color{blue}0} is the result of the evaluation of
+The blue \class{blue}{0} is the result of the evaluation of
$$
-2 + (1+i)(1-i).
@@ -140,7 +140,8 @@ then $\overline{AC} = \overline{A}$ $\overline{C}$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:ComplexEV:PropConjugate`
+:class: myproof
The statements follow immediately from the definitions of the sum and product of two matrices, and of the corresponding rules of complex arithmetic that say
@@ -176,7 +177,8 @@ $\overline{\vect{v}} = \vect{u}-i\vect{w}$ is an eigenvector for $\overline{\lam
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:ComplexEV:Conjugation`
+:class: myproof
Suppose $A(\vect{u}+i\vect{w}) = (\alpha + \beta i)(\vect{u}+i\vect{w})$.
@@ -262,7 +264,8 @@ $, for some $r > 0$ and angle $\varphi$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:ComplexEV:Rotation`
+:class: myproof
Both columns of $A$ have length $r = \sqrt{a^2 + (\pm b)^2} = \sqrt{a^2 + b^2}$.
@@ -319,7 +322,7 @@ $$
{prf:ref}`Prop:ComplexEV:Rotation` can be generalized as follows. If a real $n \times n$ matrix $A$ has a non-real eigenvalue, there is always a rotation 'hidden' in the transformation $T: \vect{x} \mapsto A\vect{x}$.
::::::{prf:proposition}
-
+:label: Prop:ComplexEV:Invariant
Suppose the real $n \times n$ matrix $A$ has a complex eigenvalue $\lambda = \alpha - \beta i$, with $\beta \neq 0$. Then there exist two linearly independent _real_ vectors $\vect{u}$ and $\vect{w}$ for which
$$
@@ -333,7 +336,8 @@ That means that the two-dimensional subspace $S = \Span{\vect{u},\vect{w}}$ is i
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:ComplexEV:Invariant`
+:class: myproof
Let $\vect{v}$ be an eigenvector for $\lambda=\alpha - \beta i$.
So, $\vect{v} = \vect{u}+i\vect{w}$, for real vectors $\vect{u}$ and $\vect{w}$.
@@ -568,6 +572,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bf1aef1c-7948-4b32-951e-d79940282bfb?id=91545
:label: grasple_exercise_6_4_1
:dropdown:
@@ -577,6 +582,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4e7c813d-ce37-42a2-be36-9dffb42d5f0b?id=91546
:label: grasple_exercise_6_4_2
:dropdown:
@@ -585,6 +591,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/66dcb059-1c37-477c-9f23-8bd1bb26fb44?id=92368
:label: grasple_exercise_6_4_3
:dropdown:
@@ -594,6 +601,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/507921c2-d568-44d9-a254-536440ca613e?id=91553
:label: grasple_exercise_6_4_4
:dropdown:
@@ -602,6 +610,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/593e9ddd-a7b8-4617-958e-95f328e28e80?id=91547
:label: grasple_exercise_6_4_5
:dropdown:
@@ -610,6 +619,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f60a57c2-b0b3-45de-b18c-e0da1bbef601?id=91555
:label: grasple_exercise_6_4_6
:dropdown:
@@ -617,6 +627,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/440c03f9-9a5a-40a6-aa1a-77660a588b20?id=91556
:label: grasple_exercise_6_4_7
:dropdown:
@@ -626,6 +637,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f3a6972e-f17b-4984-b0ee-38012ec542b3?id=91563
:label: grasple_exercise_6_4_8
:dropdown:
@@ -634,6 +646,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/79c7c433-fbcc-41b8-85d2-b1938a26d40e?id=91566
:label: grasple_exercise_6_4_9
:dropdown:
@@ -643,6 +656,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1720d806-7602-41b3-9beb-569967e74c84?id=92543
:label: grasple_exercise_6_4_10
:dropdown:
@@ -651,6 +665,7 @@ A matrix $A$ is complex diagonalizable if and only if for each eigenvalue the ge
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/510f78e6-717e-4550-8564-7bbf2c2cf673?id=55388
:label: grasple_exercise_6_4_11
:dropdown:
diff --git a/Chapter6/Diagonalizability.md b/Chapter6/Diagonalizability.md
index 866f363..2460245 100644
--- a/Chapter6/Diagonalizability.md
+++ b/Chapter6/Diagonalizability.md
@@ -55,7 +55,8 @@ Moreover, if $\vect{v}$ is an eigenvector of $B$, then $P\vect{v}$ is an eigenve
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Diagonalizable:SimilarEigenvalues`
+:class: myproof
Suppose $\lambda$ is an eigenvalue of $B$, and $\vect{v}$ is a corresponding eigenvector. We then see that
@@ -78,7 +79,8 @@ Similar matrices have the same characteristic polynomial.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Diagonalizable:SimilarCharpoly`
+:class: myproof
Suppose $A = PBP^{-1}$.
@@ -151,7 +153,8 @@ Using the properties of similar matrices we can prove the inequality
that holds for the geometric and the algebraic multiplicity of an eigenvalue
(cf. {prf:ref}`Prop:EigenValues:SmallerGeomMultiplicity`).
- ::::::{dropdown} Proof of {prf:ref}`Prop:EigenValues:SmallerGeomMultiplicity` (geom.mult. $\leq$ alg.mult.)
+::::::{admonition} Proof of {prf:ref}`Prop:EigenValues:SmallerGeomMultiplicity` (geom.mult. $\leq$ alg.mult.)
+:class: myproof, dropdown
Suppose the $n\times n$ matrix $A$ has the eigenvalue $\lambda_1$ of geometric multiplicity $k$. We have to show that the algebraic multiplicity of $\lambda_1$ is *at least* equal to $k$. We will do so by constructing a matrix $B$ that is similar to $A$ and for which the eigenvalue $\lambda_1$ will clearly have algebraic multiplicity at least equal to $k$. <BR>
Suppose $\vect{v}_1,\ldots,\vect{v}_k$ are $k$ linearly independent eigenvectors for $\lambda_1$. We can extend $\{\vect{v}_1,\ldots,\vect{v}_k,\}$ to a basis $\{\vect{v}_1,\ldots,\vect{v}_k, \ldots, \mathbf{v}_n \}$ of $\mathbb{R}^n$.
@@ -250,8 +253,8 @@ $A$ and $B$ have the same rank.
::::::
-::::::{dropdown} Proof of {prf:ref}`Prop:Eigenvalues:SimilarMatrices`
-
+::::::{admonition} Proof of {prf:ref}`Prop:Eigenvalues:SimilarMatrices`
+:class: myproof, dropdown
Suppose $A = PBP^{-1}$.
@@ -344,7 +347,8 @@ Such a set of eigenvectors then forms a basis for $\R^n$.
Since this proposition is such a pillar on which much of the theory of matrices rests, and diagonalizable matrices are important because they are in many respects easy to work with, we give two proofs.
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Eigenvalues:DiagbleVersusEigenvectors`
+:class: myproof
The first proof is algebraic. First we note that
@@ -382,9 +386,8 @@ Comparing $AP$ and $PD$ column by column we see that $A\vect{p}_i = d_i\vect{p}_
The second proof has a geometric flavour. Open it if you are interested.
-
-::::::{dropdown} Second proof of {prf:ref}`Prop:Eigenvalues:DiagbleVersusEigenvectors`
-
+::::::{admonition} Second proof of {prf:ref}`Prop:Eigenvalues:DiagbleVersusEigenvectors`
+:class: myproof, dropdown
First we show that diagonalizability implies the existence of $n$ linearly independent eigenvectors.
@@ -621,7 +624,8 @@ For each eigenvalue the geometric multiplicity is equal to the algebraic multipl
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Thm:Diagonalizable:ThirdCharacterization`
+:class: myproof
First we show that a diagonalizable matrix satisfies the two conditions.
@@ -632,8 +636,8 @@ The basic idea is that, since eigenvectors for different eigenvalues are automat
::::::
-
-::::::{dropdown} (More detailed) Proof of {prf:ref}`Thm:Diagonalizable:ThirdCharacterization`
+::::::{admonition} (More detailed) Proof of {prf:ref}`Thm:Diagonalizable:ThirdCharacterization`
+:class: myproof, dropdown
Suppose that the $n \times n$ matrix $A$ has only real eigenvalues, say $\lambda_1,\ldots,\lambda_k$, and that for each eigenvalue $\lambda_i$ the geometric multiplicity $m_i$ is equal to the algebraic multiplicity, so
@@ -970,6 +974,7 @@ $$
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bd1c8f7a-917f-431f-889b-463ab7a7c6f6?id=91486
:label: grasple_exercise_6_3_1
:dropdown:
@@ -978,6 +983,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5bcb24df-9cfd-4e4b-bcae-b550fb0fad63?id=91488
:label: grasple_exercise_6_3_2
:dropdown:
@@ -986,6 +992,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c0d56365-5434-45b0-9c82-805112428024?id=91489
:label: grasple_exercise_6_3_3
:dropdown:
@@ -994,6 +1001,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5a71e703-acd5-48b1-9b6d-8a51f4f8cf95?id=91501
:label: grasple_exercise_6_3_4
:dropdown:
@@ -1002,6 +1010,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/537a306b-47d1-422a-bc15-c7a75b81c24b?id=91496
:label: grasple_exercise_6_3_5
:dropdown:
@@ -1010,6 +1019,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5a71e703-acd5-48b1-9b6d-8a51f4f8cf95?id=91501
:label: grasple_exercise_6_3_6
:dropdown:
@@ -1018,6 +1028,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f61dfb8f-db65-4f17-80c7-b1702b0c2c07?id=104493
:label: grasple_exercise_6_3_7
:dropdown:
@@ -1027,6 +1038,7 @@ $$
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/70b5964e-b6c7-4a64-a2e3-d10dc915f324?id=91503
:label: grasple_exercise_6_3_8
:dropdown:
@@ -1036,6 +1048,7 @@ $$
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/534ce865-0960-403a-affc-0f23f2d14110?id=91521
:label: grasple_exercise_6_3_9
:dropdown:
@@ -1044,6 +1057,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f3cdb060-469a-4a30-be46-1ecc7197d66a?id=91522
:label: grasple_exercise_6_3_10
:dropdown:
@@ -1052,6 +1066,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d1cb7e54-6c99-4a01-b161-832b37d650d0?id=91523
:label: grasple_exercise_6_3_11
:dropdown:
@@ -1060,6 +1075,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4398c155-7971-42f6-b809-31ae507c0326?id=87331
:label: grasple_exercise_6_3_12
:dropdown:
@@ -1067,6 +1083,7 @@ $$
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9aca77fa-a7c8-4998-be00-a55c19e9fd70?id=62419
:label: grasple_exercise_6_3_13
:dropdown:
diff --git a/Chapter6/EigenvaluesEigenvectors.md b/Chapter6/EigenvaluesEigenvectors.md
index dfc2665..7e571aa 100644
--- a/Chapter6/EigenvaluesEigenvectors.md
+++ b/Chapter6/EigenvaluesEigenvectors.md
@@ -156,11 +156,14 @@ since such a $c$ should simultaneously satisfy $2c = -6$ and $(-2)c = 0$.
So $\vect{v} = \begin{bmatrix} 2\\-2 \end{bmatrix}$ is not an eigenvector of $A$.
See also {numref}`Figure %s <Fig:Eigenvalues:Eigenvector-no-Eigenvector>`
-::::{figure} Images/Fig-Eigenvalues-Eigenvector-no-Eigenvector.svg
+```{applet}
+:url: eigenvalue_eigenvector/no_eigenvector
+:fig: Images/Fig-Eigenvalues-Eigenvector-no-Eigenvector.svg
:name: Fig:Eigenvalues:Eigenvector-no-Eigenvector
+:class: dark-light
-To be or not to be (an eigenvector)
-::::
+To be or not to be (an eigenvector).
+```
::::::
@@ -297,6 +300,7 @@ Moreover, these non-trivial solutions are exactly the corresponding eigenvectors
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/0a053b62-1e2c-4994-93eb-10e8f99a88dc?id=93701
:label: grasple_exercise_6_1_T1
:dropdown:
@@ -420,7 +424,8 @@ Then $S$ is a subspace of $\R^n$.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:EigenValues:Subspace`
+:class: myproof
We can proceed in two ways.
@@ -533,6 +538,7 @@ This is a matrix of rank 2, and $\begin{bmatrix} 1 \\1\\1 \end{bmatrix}$ can be
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/363143ee-08c2-4905-9801-474ed10f59e9?id=93697
:label: grasple_exercise_6_1_T2
:dropdown:
@@ -548,7 +554,8 @@ Then $\{ \vect{v}_1, \ldots, \vect{v}_k \}$ is a linearly independent set.
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:Eigenvalues:IndepEigenvectors`
+:class: myproof
<BR>
@@ -678,8 +685,9 @@ See Figure {numref}`Figure %s <Fig:Eigenvalues:Eigenvector>`
::::{figure} Images/Fig-Eigenvalues-Rotation.svg
:name: Fig:Eigenvalues:Eigenvector
+:class: dark-light
-A rotation has no eigenvectors
+A rotation has no eigenvectors.
::::
::::::
@@ -723,7 +731,8 @@ Equivalently: a matrix $A$ is singular (non-invertible) if and only if 0 is an e
::::::
-::::::{prf:proof}
+::::::{admonition} Proof of {prf:ref}`Prop:EigenValues:SingularMatrix`
+:class: myproof
We prove the second statement.
@@ -789,6 +798,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
## Grasple Exercises
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c99f6e1b-cec6-4be8-828f-7f93fde00a3b?id=91537
:label: grasple_exercise_6_1_1
:dropdown:
@@ -798,6 +808,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d858b381-992c-4af3-972d-62a39c4b7a09?id=91538
:label: grasple_exercise_6_1_2
:dropdown:
@@ -806,6 +817,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/51a282db-b59f-4bd3-b4b8-fa1f38e402cc?id=91539
:label: grasple_exercise_6_1_3
:dropdown:
@@ -814,6 +826,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fcb91395-9bf7-4fbe-8cb5-100e2c2ad010?id=91540
:label: grasple_exercise_6_1_4
:dropdown:
@@ -822,6 +835,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/860849b2-5787-47d1-9ae8-a663123a86d6?id=91541
:label: grasple_exercise_6_1_5
:dropdown:
@@ -830,6 +844,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fa1470cd-81c9-4926-a8a4-587939f4d891?id=91542
:label: grasple_exercise_6_1_6
:dropdown:
@@ -838,6 +853,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2faed848-2a76-4853-a998-4167399c1f68?id=91543
:label: grasple_exercise_6_1_7
:dropdown:
@@ -846,6 +862,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bb20d21d-7eb6-4e3e-8104-05d276883162?id=91544
:label: grasple_exercise_6_1_8
:dropdown:
@@ -855,6 +872,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/79980409-fec9-4ab0-9fe1-d1a3d334bb0a?id=92492
:label: grasple_exercise_6_1_9
:dropdown:
@@ -863,6 +881,7 @@ In a later section we will study matrices $A$ for which such a basis of eigenvec
::::::
::::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a467e69f-6a78-4595-a22f-5b68314c04d4?id=92494
:label: grasple_exercise_6_1_10
:dropdown:
@@ -884,7 +903,8 @@ Moreover, if $\vect{v}$ is an eigenvector of $A$ for eigenvalue $\lambda$, the
::::::
-::::::{dropdown} Solution to {numref}`Exc:EigenValues:EigenvaluesInverse` (_click to show_)
+::::::{admonition} Solution to {numref}`Exc:EigenValues:EigenvaluesInverse`
+:class: solution, dropdown
Suppose the nonzero vector $\vect{v}$ is an eigenvector for the eigenvalue $\lambda$ of the invertible matrix $A$. From {prf:ref}`Prop:EigenValues:SingularMatrix` we know that $\lambda \neq 0$. From
diff --git a/Chapter7/GramSchmidt.md b/Chapter7/GramSchmidt.md
index a8fa0b1..70263db 100644
--- a/Chapter7/GramSchmidt.md
+++ b/Chapter7/GramSchmidt.md
@@ -22,7 +22,6 @@ As a second basis vector we can take
:::{math}
:label: Eq:GramSchmidt:Step2
-
\vect{b}_2 = \vect{a}_2 - \text{proj}_{\vect{a}_1}(\vect{a}_2) =
\vect{a}_2 - \dfrac{\vect{a}_2\ip\vect{a}_1}{\vect{a}_1\ip\vect{a}_1} \vect{a}_1.
@@ -139,8 +138,8 @@ Check for yourself that the vectors $\vect{b}_1,\vect{b}_2, \vect{b}_3$ are inde
::::
-::::{prf:proof}
-Of {prf:ref}`Thm:GramSchmidt:GramSchmidt`
+::::{admonition} Proof of {prf:ref}`Thm:GramSchmidt:GramSchmidt`
+:class: myproof
Let $W_j$ be the subspace spanned by the first $j$ vectors $\vect{a}_1, \ldots, \vect{a}_j$, for $j = 1,2\ldots,m$.
@@ -166,8 +165,9 @@ $(\vect{b}_1, \ldots, \vect{b}_j)$ of $W_j$. See {numref}`Figure %s <Fig:GramSch
:::{figure} Images/Fig-GramSchmidt-GS-step123.svg
:name: Fig:GramSchmidt:GS-step123
+:class: dark-light
-First steps in the Gram-Schmidt process
+First steps in the Gram-Schmidt process.
:::
That is,
@@ -194,6 +194,7 @@ so are the vectors $\vect{b}_1, \ldots, \vect{b}_j, \vect{b}_{j+1}$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/bb336bca-f300-48ba-8744-e38ad3a7bcd0?id=87814
:label: grasple_exercise_7_3_A
:dropdown:
@@ -307,6 +308,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/40c42331-9c9f-45ca-b498-83a8ba884a57?id=87816
:label: grasple_exercise_7_3_B
:dropdown:
@@ -337,7 +339,8 @@ The matrix $Q$ is found by applying the Gram-Schmidt process to the (linearly in
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Thm:GramSchmidt:QR-decomp`
+:class: myproof
One way to see this, is to look at the creation of the orthonormal set $\{\vect{q}_1,\ldots,\vect{q}_m\}$
from the linearly independent set $\{\vect{a}_1,\ldots,\vect{a}_m\}$.
@@ -428,7 +431,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:GramSchmidt:QR-quick`
+:class: myproof
We know that for the matrix $Q$ as specified the decomposition
@@ -537,6 +541,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b07d879b-8d64-401d-9175-c346d4cbab9e?id=87823
:label: grasple_exercise_7_3_1
:dropdown:
@@ -545,6 +550,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6fd2eb99-c2be-4bc2-887b-1c0bfccbdcf9?id=87837
:label: grasple_exercise_7_3_2
:dropdown:
@@ -553,6 +559,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/15070d4c-f241-423d-a0af-3f1b4b57a397?id=87828
:label: grasple_exercise_7_3_3
:dropdown:
@@ -561,6 +568,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/eb46b234-d281-4d02-a92c-d0c4575ffe9c?id=87825
:label: grasple_exercise_7_3_4
:dropdown:
@@ -569,6 +577,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/63090a52-ba10-4881-b9c2-35ae64e79ffd?id=87827
:label: grasple_exercise_7_3_5
:dropdown:
@@ -577,6 +586,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/81e5b0f8-9d1a-4ff7-973f-e8b8cb84d42f?id=87838
:label: grasple_exercise_7_3_6
:dropdown:
@@ -585,6 +595,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6d39de25-aeaa-4ed0-8abd-7e98f4a0ef15?id=87705
:label: grasple_exercise_7_3_7
:dropdown:
@@ -593,6 +604,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/11b3cc56-0c2d-4ea8-b5ef-e5f26d58f362?id=87741
:label: grasple_exercise_7_3_8
:dropdown:
@@ -601,6 +613,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/319f882b-8498-492b-a97c-c1ce346a66f8?id=57189
:label: grasple_exercise_7_3_9
:dropdown:
@@ -609,6 +622,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/98d48efb-0be6-4dee-a905-55dff061ce17?id=90209
:label: grasple_exercise_7_3_10
:dropdown:
@@ -617,6 +631,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d4466fcf-11be-4578-ac9e-3a570710a154?id=87629
:label: grasple_exercise_7_3_11
:dropdown:
@@ -625,6 +640,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b7be453d-4fd1-4c41-a3be-72e909ed0220?id=87646
:label: grasple_exercise_7_3_12
:dropdown:
@@ -632,6 +648,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/2f957bf7-b1f1-424f-ad39-57af14cd1d86?id=87820
:label: grasple_exercise_7_3_13
:dropdown:
@@ -640,6 +657,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/fb9adc06-f068-4c5b-96ac-575415620c82?id=87821
:label: grasple_exercise_7_3_14
:dropdown:
@@ -648,6 +666,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8a85a120-ef0c-41ec-9f60-ca492a35865a?id=87822
:label: grasple_exercise_7_3_15
:dropdown:
@@ -656,6 +675,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c02e84cd-aa51-4d2a-8585-0cd56fa08ec6?id=87824
:label: grasple_exercise_7_3_16
:dropdown:
@@ -664,6 +684,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/
:label: grasple_exercise_7_3_17
:dropdown:
@@ -672,6 +693,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/12bda5c2-8ec2-44b2-b5c6-ec2cc4bb71d0?id=87841
:label: grasple_exercise_7_3_18
:dropdown:
@@ -680,6 +702,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a8741f26-64c7-4245-839a-8c5131bea496?id=87742
:label: grasple_exercise_7_3_19
:dropdown:
@@ -688,6 +711,7 @@ Warning: the columns of $Q$ being orthonormal is equivalent to $Q^TQ = I$. Howev
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/24a2c96f-0618-4a90-8229-ce04b1dd4640?id=87747
:label: grasple_exercise_7_3_20
:dropdown:
diff --git a/Chapter7/LeastSquares.md b/Chapter7/LeastSquares.md
index d4061cc..f01f2a4 100644
--- a/Chapter7/LeastSquares.md
+++ b/Chapter7/LeastSquares.md
@@ -12,9 +12,9 @@ In this section we will reconsider the inconsistent situation and ask ourselves
One common situation where an inconsistent linear system arises quite naturally is fitting a line through a set of points.
Suppose $n \geq 3$ points $(x_1,y_1), \ldots, (x_n,y_n)$ in the plane are given.
-Which line $\ell: y = ax + b$ best fits this set of points?
+Which line $\ell: y = ax + b$ best fits this set of points?
-There are different ways to define what is the _best_ line. For instance, we may mean the line that minimizes the sum of the distances of the points to the line. From a purely geometric point of view that seems the most natural way.
+There are different ways to define what is the _best_ line. For instance, we may mean the line that minimizes the sum of the (perpendicular) distances of the points to the line. From a purely geometric point of view that seems the most natural way.
Or, we can take the line for which the sum of vertical distances from the points to the line, i.e.,
$$
@@ -24,11 +24,14 @@ $$
is minimal. This approach makes sense in a physicial context where typically the $x$-variable may be an input variable over which a researcher has control, and $y$ is some output variable which may be liable to fluctuations and/or uncertainties.
See {numref}`Figure %s <Fig:LeastSquares:BestLines>` for both interpretations of 'best' line.
-:::{figure} Images/Fig-LeastSquares-BestLines.svg
+```{applet}
+:url: leastsquares/bestlines_split_canvas
+:fig: Images/Fig-LeastSquares-BestLines.svg
:name: Fig:LeastSquares:BestLines
+:class: dark-light
-What is the best best line?
-:::
+What is the best best line? Can you get the total distance in the left picture below 6.5? Can you get the total distance in the right picture below 9.0 ?
+```
Both are sensible ideas. However, to turn any of these two ideas into an algorithm to find the best line is not as straightforward as the computations that come up if we put the question into the realm of linear algebra.
And there it will turn out to be the problem of an inconsistent linear system.
@@ -72,7 +75,7 @@ is consistent. Which generally is not the case.
We will come back to this question in {numref}`Subsection %s <SubSec:LeastSquares:LinearModels>`.
-(SubSec:LeastSquares:NormalEquations)=
+(SubSec:LeastSquares:LS-solutions)=
## Least Squares Solutions
@@ -247,10 +250,12 @@ For each linear system $A\vect{x} = \vect{b}$, where $A$ is an $m \times n$ matr
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LeastSquares:Existence`
+:class: myproof
+
In {prf:ref}`Rem:LeastSquares:BestLinComb` it was noted that a least squares solution corresponds to the vector in Col $A$ that is closest to $\vect{b}$.
-The vector in Col $A$ that is closest to $\vect{b}$ is precisely the orthogonal projection of $\vect{x}$ onto Col $A$. (See {prf:ref}`Prop:Orthogonality:BestApprox`.)
+The vector in Col $A$ that is closest to $\vect{b}$ is precisely the orthogonal projection of $\vect{x}$ onto Col $A$. (See {prf:ref}`Prop:Orthogonality:BestApprox`.)
This projection, a linear combination of the colums of $A$, always exists.
@@ -262,7 +267,8 @@ Lastly, these coefficients are unique if and only if the columns of $A$ are line
::::{margin}
-:::{admonition} {eq}`Eq:OrthoBase:OrthoProj`.
+:::{admonition} {prf:ref}`Thm:OrthoBase:OrthoDecomp`.
+:class: theorem
Let $V$ be a subspace of $\R^{n}$ and let $\vect{v}_{1},...,\vect{v}_{k}$ be an orthogonal basis for $V$. For any $\vect{w}$ in $\R^{n}$, the _orthogonal projection_ of $\vect{w}$ on $V$ is given by
@@ -313,7 +319,7 @@ $$
::::
In {prf:ref}`Ex:LeastSquares:OrthogExample` the coefficients of the orthogonal projection were quickly found due to the fact that the vectors $\vect{a}_1$ and $\vect{a}_2$ were orthogonal.
-In {numref}`Section %s <Sec:Gram-Schmidt>` we saw how we can construct an orthogonal basis from an arbitrary basis. And then we can use the projection formula {eq}`Eq:OrthoBase:OrthoProj` to find the orthogonal projection. However, we will see that this is an unnecessary detour.
+In {numref}`Section %s <Sec:Gram-Schmidt>` we saw how we can construct an orthogonal basis from an arbitrary basis. And then we can use the projection formula {eq}`Eq:OrthoBase:OrthoProj` to find the orthogonal projection. However, we will see that this is an unnecessary detour.
(SubSec:LeastSquares:NormalEquations)=
@@ -403,12 +409,11 @@ where the norm of the error vector was found to be $\sqrt{15}$.
::::
-
In the proof properties of the orthogonal projection are combined in a clever way.
As usual, feel free to skip it.
-::::{dropdown} Proof of {prf:ref}`Thm:LeastSquares:NormalEquations`
-
+::::{admonition} Proof of {prf:ref}`Thm:LeastSquares:NormalEquations`
+:class: myproof, dropdown
As usual we denote the columns of the $m \times n$ matrix $A$ by $\vect{a}_1, \ldots, \vect{a}_n$.
@@ -470,7 +475,7 @@ $$
So, to find the least squares solution(s) of the linear system $A\vect{x} = \vect{b}$, we have to solve the normal equations
:::{math}
-:name: Eq:LeastSquares:NormalEquations
+:name: Eq:LeastSquares:NormalEquations2
A^TA \vect{x} = A^T\vect{b}.
:::
@@ -503,7 +508,8 @@ the matrix $A^TA$ is invertible.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:LeastSquares:InvertibleATA`
+:class: myproof
In fact, something stronger holds:
@@ -537,7 +543,8 @@ For any $m \times n$ matrix $A$, if $A^TA$ is invertible, then the columns of $A
::::
-::::{dropdown} Solution to {numref}`Exc:LeastSquares:InvertibleATA` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:LeastSquares:InvertibleATA`
+:class: solution, dropdown
Suppose that $A$ is an $m \times n$ matrix $A$ for which $A^TA$ is invertible.
To prove that $A$ has linearly independent columns we have to show that the equation
@@ -688,7 +695,8 @@ $$
Also explain this simpler formula by interpreting the $QR$ decomposition in a suitable way.
::::
-::::{dropdown} Solution to {numref}`Exc:LeastSquares:QR` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:LeastSquares:QR`
+:class: solution, dropdown
This involves some elementary matrix operations. <BR>
Suppose $A = QR$, where $Q^TQ = I$, and $R$ is an upper triangular matrix with a positive diagonal. So $R$ is invertible. <BR>
@@ -783,8 +791,9 @@ For this low-dimensional problem we can draw a picture.
:::{figure} Images/Fig-LeastSquares-SmallestLS.svg
:name: Fig:LeastSquares:SmallestLS
+:class: dark-light
-Multiple least squares solutions
+Multiple least squares solutions.
:::
The least squares solutions are depicted as the line $\ell: \vect{x} = \hat{\vect{x}}_0 + c\left[\begin{array}{c} 2 \\ 1 \end{array}\right]$.
@@ -872,6 +881,7 @@ $$
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9666528f-1e81-4273-b571-8dae64a7972c?id=91139
:label: grasple_exercise_7_4_1
:dropdown:
@@ -880,6 +890,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c96c0359-9599-4fd4-b5b2-bd7f9d2da463?id=91141
:label: grasple_exercise_7_4_2
:dropdown:
@@ -888,6 +899,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6068f5ac-3eb3-40a1-8686-ddbf05f172b2?id=91165
:label: grasple_exercise_7_4_3
:dropdown:
@@ -896,6 +908,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a0878702-dcc0-4216-b2d6-0b1d7d1b046e?id=91142
:label: grasple_exercise_7_4_4
:dropdown:
@@ -904,6 +917,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6a0628e8-065d-4390-b0c4-8ff131761de4?id=91161
:label: grasple_exercise_7_4_5
:dropdown:
@@ -912,6 +926,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/1d9a943a-b51f-48c9-99a9-691b80b8df60?id=91159
:label: grasple_exercise_7_4_6
:dropdown:
@@ -920,6 +935,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/d7480f19-afdb-474d-8542-299fc21a1952?id=91908
:label: grasple_exercise_7_4_7
:dropdown:
@@ -928,6 +944,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c6a52270-1c43-46a2-9dda-f1ab8b366066?id=91146
:label: grasple_exercise_7_4_8
:dropdown:
@@ -936,6 +953,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/743d744c-1bcb-460a-973e-3e693e86e20d?id=91157
:label: grasple_exercise_7_4_9
:dropdown:
@@ -944,6 +962,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b081f76a-0e03-48cc-b27b-afab51ac2c91?id=91155
:label: grasple_exercise_7_4_10
:dropdown:
@@ -952,6 +971,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/679d1581-08bc-416b-89bf-766faad9f118?id=91394
:label: grasple_exercise_7_4_11
:dropdown:
@@ -960,6 +980,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3d0a7884-09ee-4f89-a2e5-1c1476d7e2a3?id=91448
:label: grasple_exercise_7_4_12
:dropdown:
@@ -968,6 +989,7 @@ $$
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/396fd7a0-10cf-4b49-bb9e-fbc4acc2a06a?id=91148
:label: grasple_exercise_7_4_13
:dropdown:
@@ -1012,8 +1034,9 @@ The points and the 'first guess' are depicted in {numref}`Figure %s <Fig:LeastSq
:::{figure} Images/Fig-LeastSquares-FirstGuess.svg
:name: Fig:LeastSquares:FirstGuess
+:class: dark-light
-First guess for best line
+First guess for best line.
:::
For this line the sum of the squares of the errors
@@ -1065,8 +1088,9 @@ give a unique least squares solution, and it is $\hat{a} = 1.6$, $\hat{b} = 0.3$
:::{figure} Images/Fig-LeastSquares-LSLine.svg
:name: Fig:LeastSquares:LSline
+:class: dark-light
-Least squares line
+Least squares line.
:::
@@ -1280,7 +1304,7 @@ Several generalizations are possible. We mention two.
\begin{array}{c} (x_{11},x_{12},\ldots,x_{1k},y_1) \\
(x_{21},x_{22},\ldots,x_{2k},y_2)\\
\vdots \quad\quad \vdots \quad\quad \vdots\\
- (x_{n1},x_{n2},\ldots,x_{nk},y_2),
+ (x_{n1},x_{n2},\ldots,x_{nk},y_n),
\end{array}
:::
@@ -1324,9 +1348,8 @@ get different weights $w_i$. When building a statistical model this may be desir
Then the expression we want to minimize is given by
-
$$
- \sum_{i=1}^{n} {\color{blue}w_i} \big(y_i - \beta_1f_1(x_{i1},\ldots, x_{ik}) \,-\, \ldots \,-\,
+ \sum_{i=1}^{n} \class{blue}{w_i} \big(y_i - \beta_1f_1(x_{i1},\ldots, x_{ik}) \,-\, \ldots \,-\,
\beta_{\ell}f_{\ell}(x_{i1},\ldots, x_{ik})\big)^2.
$$
@@ -1410,8 +1433,9 @@ $(\ln(x_i), \ln(y_i))$, and the last two plots give the points with the two fits
:::{figure} Images/Fig-LeastSquares-PowerFit.svg
:name: Fig:LeastSquares:Powerfit
+:class: dark-light
-Least squares fitting via logarithmic scale
+Least squares fitting via logarithmic scale.
:::
::::
@@ -1419,6 +1443,7 @@ Least squares fitting via logarithmic scale
## Grasple Exercises (for Linear Models)
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6a2b9aeb-3c59-4b8f-8d7c-e51f651998fd?id=91883
:label: grasple_exercise_7_4_14
:dropdown:
@@ -1427,6 +1452,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/254298e0-f091-4a9d-8e8a-31cc3cf11f16?id=91884
:label: grasple_exercise_7_4_15
:dropdown:
@@ -1434,6 +1460,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/dad8ca0a-ef17-4757-803c-26b8ae9804de?id=91886
:label: grasple_exercise_7_4_16
:dropdown:
@@ -1441,6 +1468,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9ad7148d-ef39-46ef-8f14-20e97511655d?id=91889
:label: grasple_exercise_7_4_17
:dropdown:
@@ -1448,6 +1476,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/222ee704-85f5-470a-b48f-f88d9900a8d1?id=91890
:label: grasple_exercise_7_4_18
:dropdown:
@@ -1455,6 +1484,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ff6329bd-f5b3-41ce-828f-2086cf651181?id=91898
:label: grasple_exercise_7_4_19
:dropdown:
@@ -1462,6 +1492,7 @@ Least squares fitting via logarithmic scale
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f75014bf-0e90-43e7-acf4-216cb38ffd11?id=91903
:label: grasple_exercise_7_4_20
:dropdown:
diff --git a/Chapter7/OrthoBase.md b/Chapter7/OrthoBase.md
index 234ff15..4af00a4 100644
--- a/Chapter7/OrthoBase.md
+++ b/Chapter7/OrthoBase.md
@@ -30,7 +30,8 @@ An orthogonal set $S$ which does not contain $\vect{0}$ is linearly independent.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoBase:OrthoSetLinInd`
+:class: myproof
Assume $S$ is linearly dependent. Then there are vectors $\vect{v}_{1},...,\vect{v}_{n}$ in $S$ and scalars $c_{1},...,c_{n}$, not all zero, such that $\vect{0}=c_{1}\vect{v}_{1}+\cdots +c_{n}\vect{v}_{n}.$ But then, for any $i$:
@@ -82,7 +83,8 @@ $$\vect{v}=(\vect{v}\ip\vect{v}_{1})\vect{v}_{1}+\cdots +(\vect{v}\ip\vect{v}_{k
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:OrthoBase:WeightsOrthoBase`
+:class: myproof
Since $\vect{v}_{1},...,\vect{v}_{k}$ is a basis for $V$ and $\vect{v}$ is in $V$, there are scalars $c_{1},...,c_{k}$ such that $\vect{v}=c_{1}\vect{v}_{1}+\cdots +c_{k}\vect{v}_{k}$. We only have to show that these scalars are as claimed. For any $j$ between $1$ and $k$,
@@ -135,7 +137,8 @@ For any $\vect{v}$ in $V$, $\norm{\vect{u}-\vect{u}_{V}}\leq \norm{\vect{u}-\vec
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:Orthogonality:BestApprox`
+:class: myproof
Recall that the inner product of any vector with itself is non-negative and that $\vect{u}_{V}\ip\vect{u}_{V^{\bot}}=0$.
@@ -201,7 +204,8 @@ Suppose $V$ is a subspace of $\R^{n}$ with orthogonal basis $\vect{v}_{1},...,\v
::::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:OrthoBase:OrthoDecomp`
+:class: myproof
Put
@@ -227,6 +231,7 @@ It is worthwhile to compare this result to the formula for the projection of one
:fig: Images/Fig-OrthoBase-DecompAs2Proj.svg
:name: Fig:OrthoBase:DecompAs2Proj
:position: -5.49,7.46,20.08
+:class: dark-light
A vector and its orthogonal projection on the subspace $V$. Note that this projection is the sum of the projections of $\vect{u}$ on the orthogonal basis $\vect{v}_{1},\vect{v}_{2}$.
```
@@ -446,7 +451,8 @@ An $n\times n$-matrix $A$ is orthogonal if and only if $A^{T}A=I_{n}$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoBase:OrthoMat`
+:class: myproof
Let $\vect{v}_{1},\vect{v}_{2}...,\vect{v}_{n}$ be the columns of $A$, so $\vect{v}_{1}^{T},\vect{v}_{2}^{T},...,\vect{v}_{n}^{T}$ are the rows of $A^{T}$. Consequently,
@@ -517,7 +523,8 @@ $\angle(A\vect{v}_{1},A\vect{v}_{2})=\angle(\vect{v}_{1},\vect{v}_{2})$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoBase:OrthoMatandInnerProd`
+:class: myproof
Using $A^{T}A=I_{n}$, we find:
@@ -536,12 +543,14 @@ Many statements about orthogonal matrices still hold for non-square matrices, as
You could of course also consider matrices for which the _rows_ are orthonormal. It turns out, however, that this yields the exact same concept.
:::{prf:proposition}
+:label: Prop:OrthoBase:Rows
An $n\times n$-matrix $A$ is orthogonal if and only if its rows are orthonormal.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoBase:Rows`
+:class: myproof
We know that $A$ is orthogonal if and only if $A^{T}A=I_{n}$. But this implies $A^{T}=A^{-1}$ and therefore also $AA^{T}=I_{n}$. Since $(A^{T})^{T}A^{T}=AA^{T}=I_{n}$, $A^{T}$ must be orthogonal by {prf:ref}`Prop:OrthoBase:OrthoMat`. Hence the columns of $A^{T}$, which are the rows of $A$, must be orthonormal.
@@ -550,6 +559,7 @@ We know that $A$ is orthogonal if and only if $A^{T}A=I_{n}$. But this implies $
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c443c628-427c-4ba6-a55b-d7fd0a562904?id=87842
:label: grasple_exercise_7_2_1
:dropdown:
@@ -558,6 +568,7 @@ We know that $A$ is orthogonal if and only if $A^{T}A=I_{n}$. But this implies $
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/825980e7-e25f-497e-a077-e009cedd55c4?id=87843
:label: grasple_exercise_7_2_2
:dropdown:
@@ -566,6 +577,7 @@ We know that $A$ is orthogonal if and only if $A^{T}A=I_{n}$. But this implies $
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c815026e-df41-461c-b0dc-0b06a387d0c9?id=91876
:label: grasple_exercise_7_2_3
:dropdown:
diff --git a/Chapter7/OrthoComp.md b/Chapter7/OrthoComp.md
index 5d3b39d..b8d8a77 100644
--- a/Chapter7/OrthoComp.md
+++ b/Chapter7/OrthoComp.md
@@ -33,7 +33,8 @@ Prove {prf:ref}`Prop:OrthoComp:OrthotoSpanningSet`.
::::
-:::{dropdown} Solution to {numref}`Exc:OrthoComp:OrthotoSpanningSet` (_click to show_)
+:::{admonition} Solution to {numref}`Exc:OrthoComp:OrthotoSpanningSet`
+:class: solution, dropdown
Assume the vector $\vect{u}$ is orthogonal to every $\vect{v}_{i}$. If $\vect{v}_{1},...,\vect{v}_{n}$ spans $V$, then any $\vect{v}$ in $V$ can be written as $c_{1}\vect{v}_{1}+\cdots c_{n}\vect{v}_{n}$ for certain $c_{1},...,c_{n}$ in $\R$. But then $\vect{u}\cdot\vect{v}=c_{1}\vect{u}\cdot\vect{v}_{1}+\cdots+c_{n}\vect{u}\cdot\vect{v}_{n}=0$, so $\vect{u}$ is orthogonl to $\vect{v}$.
@@ -112,6 +113,7 @@ Both examples are illustrated in {numref}`Figure %s <Fig:OrthoComp:OrthoComp>`.
:url: ortho/orthocomp
:fig: Images/Fig-OrthoComp-OrthoComp.svg
:name: Fig:OrthoComp:OrthoComp
+:class: dark-light
The orthogonal complement of a 1-dimensional subspace of $\R^{2}$ (left) and of a 2-dimensional subspace of $\R^{3}$ (right).
```
@@ -125,7 +127,8 @@ For any subspace $V$ of $\R^{n}$, the orthogonal complement $V^{\bot}$ is a subs
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoComp:OrthoComplisSpace`
+:class: myproof
Since the zero vector is orthogonal to any vector, it is in $V^{\bot}$. Suppose now that $\vect{u}_{1}$ and $\vect{u}_{2}$ are in $V^{\bot}$. Then, for arbitrary $\vect{v}$ in $V$, $(\vect{u}_{1}+\vect{u}_{2})\ip \vect{v}=\vect{u}_{1}\ip\vect{v}+\vect{u}_{2}\ip\vect{v}=0$, so $\vect{u}_{1}+\vect{u}_{2}$ is in $V^{\bot}$.
@@ -144,7 +147,8 @@ For any matrix $A$ we have $\mathrm{Col}(A^{T})^{\bot}=\mathrm{Nul}(A)$ and $\ma
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoComp:OrthoComplementNulA`
+:class: myproof
Note that the second claim is easily derived from the first by substituting $A^{T}$ for $A$. Let $\vect{r}_{1},...,\vect{r}_{n}$ be the rows of $A$. Then $\vect{r}_{1}^{T},...,\vect{r}_{n}^{T}$ are the columns of $A^{T}$. For any vector $\vect{x}$ in $\R^{m}$, we have
@@ -223,12 +227,14 @@ $$
:::
:::{prf:proposition}
+:label: Prop:OrthoComp:OrthoComplementSum
If $V$ is a subspace of $\R^{n}$, then $\dim(V)+\dim(V^{\bot})=n$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoComp:OrthoComplementSum`
+:class: myproof
Let $A$ be a matrix for which the columns are a basis of $V$. Then $n$ is the number of rows of $A$ which in turn is the number of columns of $A^{T}$. By {prf:ref}`Thm:BasisDim:DimensionTheorem`, $\dim(\mathrm{Col}(A^{T}))+\dim(\mathrm{Nul}(A^{T}))$ is the number of columns of $A^{T}$, which is the number of rows of $A$. Using {prf:ref}`Prop:OrthoComp:OrthoComplementNulA`, this yields
@@ -251,7 +257,8 @@ Let $V$ be a subspace of $\R^{n}$. For an arbitrary vector $\vect{u}$ in $\R^{n}
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:OrthoComp:PrthoDecomp`
+:class: myproof
Let $\vect{v}_{1},...,\vect{v}_{k}$ be a basis for $V$ and let $\vect{v}_{k+1},...,\vect{v}_{n}$ be a basis for $V^{\bot}$. We claim that the vectors $\vect{v}_{1},...,\vect{v}_{k},\vect{v}_{k+1},...,\vect{v}_{n}$ are linearly independent. Indeed, if there were a linear combination
@@ -281,6 +288,7 @@ Putting $\vect{u}_{V}=c_{1}\vect{v}_{1}+\cdots+c_{k}\vect{v}_{k}$ and $\vect{u}_
:url: ortho/orthodecomp
:fig: Images/Fig-OrthoComp-OrthoDecomp.svg
:name: Fig:OrthoBase:OrthoDecomp
+:class: dark-light
A subspace $V$, a vector $\vect{u}$ and the orthogonal decomposition of $\vect{u}$ with respect to $V$.
```
@@ -329,25 +337,52 @@ It is easy to check that, as the notation suggests, $\vect{u}_{V}$ is in $V$ (si
## Grasple Exercises
::::{grasple}
-:url: https://embed.grasple.com/exercises/f216b122-e2f3-4cd4-9268-7a814e12cec3?id=91429
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/d69cfb9d-325b-433d-9210-c322fa272d14?id=108892
:label: grasple_exercise_7_1_1
:dropdown:
-:description: Find a basis for the orthogonal complement of the span.
+:description: Finding vectors orthogonal to two given vectors in $\R^4$.
::::
+
::::{grasple}
-:url: https://embed.grasple.com/exercises/3e9132d6-c280-4361-a653-002ad50b4784?id=91433
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/19e0688c-b00c-46c2-aad8-200a230687b9?id=91435
:label: grasple_exercise_7_1_2
:dropdown:
-:description: Find a basis for the orthogonal complement of the column space.
+:description: Find the orthogonal complement of a vector $\vect{u}$ in $\R^3$ w.r.t. span$\{\vect{v}_1,\vect{v}_2\}$.
::::
+
+
::::{grasple}
-:url: https://embed.grasple.com/exercises/3e9132d6-c280-4361-a653-002ad50b4784?id=91433
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/c0918add-0c17-481a-80e3-c6688e65480a?id=91434
:label: grasple_exercise_7_1_3
:dropdown:
:description: Find a geometric description of $V^{\bot}$.
::::
+
+
+::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/f216b122-e2f3-4cd4-9268-7a814e12cec3?id=91429
+:label: grasple_exercise_7_1_4
+:dropdown:
+:description: Find a basis for the orthogonal complement of span$\{\vect{v}_1,\vect{v}_2\}$ in $\R^4$.
+
+::::
+
+::::{grasple}
+:iframeclass: dark-light
+:url: https://embed.grasple.com/exercises/3e9132d6-c280-4361-a653-002ad50b4784?id=91433
+:label: grasple_exercise_7_1_5
+:dropdown:
+:description: Find a basis for the orthogonal complement of the column space of a matrix.
+
+::::
+
+
diff --git a/Chapter7/Orthogonality.md b/Chapter7/Orthogonality.md
deleted file mode 100644
index ec1907d..0000000
--- a/Chapter7/Orthogonality.md
+++ /dev/null
@@ -1,332 +0,0 @@
-(Sec:Orthogonality)=
-
-# Orthogonality
-
-## Orthogonal complements
-
-:::{prf:definition}
-
-Suppose $V$ is a subspace of $\R^{n}$. Then the _orthogonal complement_ of $V$ is the set
-
-$$
-
-V^{\bot}=\left\{\vect{u}\in\R^{n}\mid \vect{u}\ip\vect{v}=0\text{ for all } \vect{v}\text{ in }V\right\}.
-
-
-$$
-
-in other words, it is the set of all vectors that are orthogonal to all of $V$.
-
-:::
-
-Note that, for a vector to be in $V^{\bot}$, it suffices that it is orthogonal to all elements in a basis of $V$ or, slightly more general, to all elements in a spanning set of $V$. Let us consider s simple example.
-
-:::{prf:Example}
-:label: Ex:Ortho:OrthoCompOfVect
-
-Let $V$ be the subspace spanned by a single vector $\vect{v}$, say, by
-
-$$\vect{v}=\begin{bmatrix}1\\2\\-1\end{bmatrix}\text{ and let }\vect{u}=\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}\text{ be any vector in $\R^{3}$}.$$
-
-Then $\vect{u}$ is in $V^{\bot}$ is and only if $\vect{u}\ip\vect{v}=a_{1}+2a_{2}-a_{3}=0$. So we find that $V^{\bot}$ is the plane described by the equation $a_{1}+2a_{2}-a_{3}=0$. The vector $\vect{v}$ is a normal vector to this plane.
-
-:::
-
-```{figure} Images/Fig-Ortho-OrthoComp.svg
-:name: Fig:Ortho:OrthoComp
-
-The orthogonal complement of a 1-dimensional subspace of $\R^{2}$ (left) and a of a 2-dimensional subspace of $\R^{3}$ (right).
-```
-
-The plane we found for $V^{\bot}$ in {prf:ref}`Ex:Ortho:OrthoCompOfVect` is a plane through the origin. It is therefore also a subspace. This is not a coincidence, as {prf:ref}`Prop:Ortho:OrthoComplisSpace` shows.
-
-:::{prf:proposition}
-:label: Prop:Ortho:OrthoComplisSpace
-
-For any subspace $V$ of $\R^{n}$, the orthogonal complement $V^{\bot}$ is a subspace, too. Moreover, the only vector that is both in $V$ and in $V^{\bot}$ is $\vect{0}$.
-
-:::
-
-:::{prf:proof}
-
-Since the zero vector is orthogonal to any vector, it is in $V^{\bot}$. Suppose now that $\vect{u}_{1}$ and $\vect{u}_{2}$ are in $V^{\bot}$. Then, for arbitrary $\vect{v}$ in $V$, $(\vect{u}_{1}+\vect{u}_{2})\ip \vect{v}=\vect{u}_{1}\ip\vect{v}+\vect{u}_{2}\ip\vect{v}=0$, so $\vect{u}_{1}+\vect{u}_{2}$ is in $V^{\bot}$.
-
-Assume now that $\vect{u}$ is in $V^{\bot}$ and that $c$ is any scalar. Then, again for every $\vect{v}$ in $V$, $(c\vect{u})\ip\vect{v}=c(\vect{u}\ip\vect{v})=0$ so $c\vect{u}$ is in $V^{\bot}$.
-
-If $\vect{v}$ is both in $V$ and $V^{\bot}$, then $\vect{v}\ip\vect{v}=0$ so $\vect{v}=\vect{0}$.
-
-:::
-
-```{margin} TODO
-
-Add reference to Section Sec:SubspacesRn
-
-```
-
-As we have seen in Section, both the column space and null space of any $n\times m$ matrix are subspaces of $\R^{n}$ and $\R^{m}$, respectively. It turns out that the transposition ${}^{T}$ and the orthogonal complement ${}^{\bot}$ relate these two spaces to each other.
-
-:::{prf:proposition}
-:label: Prop:Orthogonality:OrthoComplementNulA
-
-For any $n\times m$ matrix $M$ we have $\mathrm{Col}(A^{T})^{\bot}=\mathrm{Nul}(A)$ and $\mathrm{Col}(A)^{\bot}=\mathrm{Nul}(A^{T})$.
-
-:::
-
-:::{prf:proof}
-
-Note that the second claim is easily derived from the first by substituting $A^{T}$ for $A$. Let $\vect{r}_{1},...,\vect{r}_{n}$ be the rows of $A$. Then $\vect{r}_{1}^{T},...,\vect{r}_{n}^{T}$ are the columns of $A^{T}$. For any vector $\vect{x}$ in $\R^{m}$, we have
-
-$$A\vect{x}=\begin{bmatrix}\vect{r}_{1}\vect{x}\\\vdots\\\vect{r}_{n}\vect{x}\end{bmatrix}=\begin{bmatrix}\vect{r}_{1}^{T}\ip\vect{x}\\\vdots\\\vect{r}_{n}^{T}\ip\vect{x}\end{bmatrix}.$$
-
-Now, $\vect{x}$ is in $\mathrm{Nul}(A)$ precisely when $A\vect{x}=\vect{0}$ or, in other words, when $\vect{r}_{i}^{T}\ip\vect{x}=0$ for any $i$. Since the set $\left\{\vect{r}_{1}^{T},..,\vect{r}_{n}^{T}\right\}$ spans $\mathrm{Col}(A^{T})$, this is equivalent to $\vect{x}$ being in $\mathrm{Col}(A^{T})^{\bot}$.
-
-:::
-
-## Orthogonal and orthonormal bases
-
-:::{prf:definition}
-
-A subset $S$ of $\R^{n}$ is called _orthogonal_ if any two distinct vectors $\vect{v}_{1}$ and $\vect{v}_{2}$ in $S$ are orthogonal to each other. If $S$ is a basis for a subspace $V$ and $S$ is orthogonal, we say it is an _orthogonal basis_ for $V$.
-
-:::
-
-:::{prf:Example}
-:label: Ex:Ortho:ExOfOrthoBase
-
-Consider the plane
-
-$$\mathcal{P}=\left\{\begin{bmatrix}x\\y\\z\end{bmatrix}\mid x+y+z=0\right\}\text{ and the vectors }\vect{v}_{1}=\begin{bmatrix}1\\-1\\0\end{bmatrix},\quad \vect{v}_{2}=\begin{bmatrix}1\\1\\-2\end{bmatrix}.$$
-
-Both $\vect{v}_{1}$ and $\vect{v}_{2}$ lie in $\mathcal{P}$. The set $\mathcal{B}=\left\{\vect{v}_{1},\vect{v}_{2}\right\}$ is a linearly independent set of two vectors in $\mathcal{P}$. Since $\dim(\mathcal{P})=2$, it must therefore be a basis. Furthermore, $\vect{v}_{1}\ip\vect{v}_{2}=1-1-0=0$ so $\vect{v}_{1}$ is orthogonal to $\vect{v}_{2}$. Hence $\mathcal{B}$ is an orthogonal basis for $\mathcal{P}$.
-
-:::
-
-Since $\vect{0}$ is orthogonal to every vector, adding it to a set or removing it from a set does not change whether the set is orthogonal or not.
-
-:::{prf:proposition}
-:label: Prop:Ortho:OrthoSetLinInd
-
-An orthogonal set $S$ which does not contain $\vect{0}$ is linearly independent.
-
-:::
-
-:::{prf:proof}
-
-Assume $S$ is linearly dependent. Then there are vectors $\vect{v}_{1},...,\vect{v}_{n}$ in $S$ and scalars $c_{1},...,c_{n}$, not all zero, such that $\vect{0}=c_{1}\vect{v}_{1}+\cdots +c_{n}\vect{v}_{n}.$ But then, for any $i$:
-
-$$0=\vect{0}\ip \vect{v}_{i}=(c_{1}\vect{v}_{1}+\cdots +c_{n}\vect{v}_{n})\ip\vect{v}_{i}=c_{i}(\vect{v}_{i}\ip\vect{v}_{i}).$$
-
-Since no $\vect{v}_{i}$ is $\vect{0}$, all $\vect{v}_{i}\ip\vect{v}_{i}$ are non-zero, hence all $c_{i}$ must be zero, which contradicts our assumption.
-
-:::
-
-As a consequence of {prf:ref}`Prop:Ortho:OrthoSetLinInd`, any orthogonal set that does not contain $\vect{0}$ is an orthogonal basis for its span.
-
-:::{prf:definition}
-
-An orthogonal basis is called _orthonormal_ if all elements in the basis have norm $1$.
-
-:::
-
-:::{prf:remark}
-
-If $\vect{v}_{1},...,\vect{v}_{n}$ is an orthogonal basis for a subspace $V$, then an orthonormal basis for $V$ can be obtained by dividing each $\vect{v}_{i}$ by its norm.
-
-:::
-
-:::{prf:example}
-:label: Ex:Ortho:OrthonormalBase
-
-Consider the plane $\mathcal{P}$, the vectors $\vect{v}_{1},\vect{v}_{2}$ and the basis $\mathcal{B}$ from {prf:ref}`Ex:Ortho:ExOfOrthoBase`. This $\mathcal{B}$ is an orthogonal basis, but $\norm{\vect{v}_{1}}=\sqrt{2}$ and $\norm{\vect{v}_{2}}=\sqrt{6}$ so it is not orthonormal.
-
-We can remedy this by considering the basis $\mathcal{B}_{2}=\left\{\vect{u}_{1},\vect{u}_{2}\right\}$ where
-
-$$\vect{u}_{1}=\frac{\vect{v}_{1}}{\norm{\vect{v}_{1}}}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\\0\end{bmatrix}\quad\text{and}\quad \vect{u}_{2}=\frac{\vect{v}_{2}}{\norm{\vect{v}_{2}}}=\frac{1}{\sqrt{6}}\begin{bmatrix}1\\1\\-2\end{bmatrix}.$$
-
-This new basis $\mathcal{B}_{2}$ is an orthonormal basis. We have kept the directions of $\vect{v}_{1}$ and $\vect{v}_{2}$, but we have made sure that our step length is now 1.
-
-:::
-
-The essence of {prf:ref}`Thm:Ortho:WeightsOrthoBase` is that it is easy to find the coordinates of any vector in a subspace $V$ with respect to a given orthogonal basis of $V$. In fact, this is largely why we are interested in such bases.
-
-:::{prf:theorem}
-:label: Thm:Ortho:WeightsOrthoBase
-
-Let $V$ be a subspace of $\R^{n}$ and assume $\vect{v}_{1},...,\vect{v}_{k}$ is an orthogonal basis for $V$. Then any vector $\vect{v}$ in $V$ can be written as:
-
-$$\vect{v}=\frac{\vect{v}\ip\vect{v}_{1}}{\vect{v}_{1}\ip\vect{v}_{1}}\vect{v}_{1}+\cdots +\frac{\vect{v}\ip\vect{v}_{k}}{\vect{v}_{k}\ip\vect{v}_{k}}\vect{v}_{k}.$$
-
-In particular, if $\vect{v}_{1},..,\vect{v}_{k}$ is an orthonormal basis, then any $\vect{v}$ in $V$ can be written as:
-
-$$\vect{v}=(\vect{v}\ip\vect{v}_{1})\vect{v}_{1}+\cdots +(\vect{v}\ip\vect{v}_{k})\vect{v}_{k}.$$
-
-:::
-
-:::{prf:proof}
-
-Since $\vect{v}_{1},...,\vect{v}_{k}$ is a basis for $V$ and $\vect{v}$ is in $V$, there are scalars $c_{1},...,c_{k}$ such that $\vect{v}=c_{1}\vect{v}_{1}+\cdots +c_{k}\vect{v}_{k}$. We only have to show that these scalars are as claimed. For any $j$ between $1$ and $k$,
-
-$$\vect{v}\ip\vect{v}_{j}=(c_{1}\vect{v}_{1}+\cdots +c_{k}\vect{v}_{k})\ip\vect{v}_{j}=c_{j}(\vect{v}_{j}\ip\vect{v}_{j}$$
-
-by the orthogonality of $\left\{\vect{v}_{1},...,\vect{v}_{k}\right\}$. This implies $c_{j}=\frac{\vect{v}\ip\vect{v}_{j}}{\vect{v}_{j}\ip\vect{v}_{j}}$ as claimed.
-
-If $\vect{v}_{1},...,\vect{v}_{k}$ is orthonormal, then $\vect{v}_{j}\ip\vect{v}_{j}=1$ for every $j$, so this reduces to $c_{j}=\vect{v}\ip\vect{v}_{j}$.
-
-:::
-
-## Orthogonal projections revisited
-
-```{margin} TODO
-
-Add reference to Section Sec:GeomLinTrans
-
-```
-
-In Section we have already briefly toched upon orthogonal projections in higher dimension. Now that we know about orthogonal bases, we can make this more concrete.
-
-:::{prf:definition}
-:label: Dfn:Orthogonality:OrthoProjection
-
-Let $V$ be a subspace of $\R{n}$ and let $\vect{v}_{1},...,\vect{v}_{k}$ be an orthogonal basis for $V$. For any $\vect{w}$ in $\R^{n}$, we define the _orthogonal projection_ of $\vect{w}$ on $V$ as
-
-$$\proj_{V}(w)=\frac{\vect{w}\ip\vect{v}_{1}}{\vect{v}_{1}\ip\vect{v}_{1}}\vect{v}_{1}+\cdots +\frac{\vect{w}\ip\vect{v}_{k}}{\vect{v}_{k}\ip\vect{v}_{k}}\vect{v}_{k}.$$
-
-:::
-
-```{figure} Images/Fig-Ortho-DecompAs2Proj.svg
-:name: Fig:Ortho:DecompAs2Proj
-
-A vector and its orthogonal projection on the subspace $V$.
-```
-
-:::{prf:example}
-
-Let us revisit the plane $\mathcal{P}$ with basis $\mathcal{B}=\left\{\vect{v}_{1},\vect{v}_{2}\right\}$ from {prf:ref}`Ex:Ortho:ExOfOrthoBase`. Take a vector, say, $\vect{w}=\begin{bmatrix}-1\\1\\2\end{bmatrix}$. We find $\vect{w}\ip\vect{v}_{1}=-2,\vect{w}\ip\vect{v}_{2}=-4,$ and $\vect{v}_{1}\ip\vect{v}_{1}=2,\vect{v}_{2}\ip\vect{v}_{2}=6$. Consequently,
-
-$$
-\proj_{\mathcal{P}}(\vect{w})=\frac{\vect{w}\ip\vect{v}_{1}}{\vect{v}_{1}\ip\vect{v}_{1}}\vect{v}_{1}+\frac{\vect{w}\ip\vect{v}_{2}}{\vect{v}_{2}\ip\vect{v}_{2}}\vect{v}_{2}=-\frac{2}{2}\vect{v}_{1}-\frac{4}{6}\vect{v}_{2}=
-\begin{bmatrix}
- -\frac{5}{3} \\
- {-\frac{1}{3}} \\
- {\frac{4}{3}}
-\end{bmatrix}
-$$
-
-is the orthogonal projection of $\vect{w}$ on $\mathcal{P}$.
-
-:::
-
-This orthogonal projection can be thought of as the best approximation of $\vect{w}$ using vectors from $V$. Perhaps surprisingly, it is independent of the choice of the orthogonal basis $\vect{v}_{1},...,\vect{v}_{k}$ by {prf:ref}`Thm:Ortho:OrthoDecomp`. This justifies the fact that the chosen orthogonal base is left out of the notation.
-
-:::{prf:Theorem}
-:label: Thm:Ortho:OrthoDecomp
-
-Suppose $V$ is a subpsace of $\R^{n}$ and $\vect{w}$ is a vector in $\R^{n}$. Then there is a unique decomposition
-
-$$\vect{w}=\hat{\vect{w}}+\vect{x}\quad\text{where $\hat{\vect{w}}$ is in $V$ and $\vect{x}$ is in $V^{\bot}$.}$$
-
-In fact, $\hat{\vect{w}}=\proj_{V}(\vect{w})$. This is called the _orthogonal decomposition_ of $\vect{w}$ with respect to $V$.
-
-:::
-
-:::{prf:proof}
-
-Fix an orthogonal basis $\vect{v}_{1},..,\vect{v}_{k}$ for $V$ and put $\vect{x}=\vect{w}-\proj_{V}(\vect{w})$. Clearly, $\proj_{V}(\vect{w})$ is in $V$. If we can show $\vect{x}\bot \vect{v}_{i}$ for any $i$, it will follow that $\vect{x}$ is in $V^{\bot}$. This follows readily:
-
-\begin{align*}
-\vect{x}\ip \vect{v}*{i}&=\left(\vect{w}-\frac{\vect{w}\ip\vect{v}_{1}}{\vect{v}_{1}\ip\vect{v}_{1}}\vect{v}_{1}+\cdots +\frac{\vect{w}\ip\vect{v}_{k}}{\vect{v}_{k}\ip\vect{v}_{k}}\vect{v}_{k}\right)\ip\vect{v}_{i}\\
-&=\vect{w}\ip\vect{v}_{i}-\frac{\vect{w}\ip\vect{v}_{i}}{\vect{v}_{i}\ip\vect{v}_{i}}(\vect{v}_{i}\ip\vect{v}_{i})=0.
-\end{align_}
-
-The only thing left to show is that this decomposition is unique. Suppose $\vect{w}=\hat{\vect{w}}_{1}+\vect{x}_{1}$ and $\vect{w}=\hat{\vect{w}}_{2}+\vect{x}_{2}$ where $\hat{\vect{w}}_{1},\hat{\vect{w}}_{2}$ are in $V$ and $\vect{x}_{1},\vect{x}_{2}$ are in $V^{\bot}$. Then
-
-$$\hat{\vect{w}}_{1}-\hat{\vect{w}}_{2}=\vect{x}_{2}-\vect{x}_{1}.$$
-
-Since $V$ is a subspace, the left hand side is in $V$ and since $V^{\bot}$ is a subspace the right hand side is in $V^{\bot}$. But the only element in both $V$ and $V^{\bot}$ is $\vect{0}$. So $\vect{x}_{1}=\vect{x}_{2}$ and $\hat{\vect{w}}_{1}=\hat{\vect{w}}_{2}$.
-
-:::
-
-```{figure} Images/Fig-Ortho-OrthoDecomp.svg
-:name: Fig:Ortho:OrthoDecomp
-
-A vector, its orthogonal projection on a subspace, and the difference of the two. Note that the difference between the vector and its projection on $V$ if orthogonal to $V$, i.e. it is in $V^{\bot}$.
-```
-
-{prf:ref}`Thm:Ortho:OrthoDecomp` is quite a powerful tool. For example, it allows us to establish the following useful facts. Of particular interest is [iii.](#It:Ortho:ProjIsClosest), which states in essence that $\proj_{V}(\vect{w})$ is the best approximation of $\vect{w}$ with a vector from $V$ or, in other words, that the projection of $\vect{w}$ onto $V$ is the point in $V$ which is closest to $\vect{w}$.
-
-:::{prf:Proposition}
-:label: Prop:Orthogonality:BestApprox
-
-Let $V$ be a subspace of $\R^{n}$ and let $\vect{w}$ be an arbitrary vector in $\R^{n}$. Then:
-
-<ol type="i">
-<li>
-
-$\norm{\vect{w}}\geq \norm{\proj_{V}(\vect{w})}$.
-
-</li>
-<li>
-
-$\vect{w}\ip\proj_{V}(\vect{w})\geq 0$ and $\vect{w}\ip\proj_{V}(\vect{w})=0$ precisely when $\vect{w}$ is in $V^{\bot}$.
-
-</li>
-<li id="It:Ortho:ProjIsClosest">
-
-For any $\vect{v}$ in $V$, $\norm{\vect{w}-\vect{v}}\leq\norm{\vect{w}-\proj_{V}(\vect{w})}$.
-
-</li>
-
-</ol>
-
-:::
-
-:::{prf:proof}
-
-Let $\vect{w}=\proj_{V}(\vect{w})+\vect{x}$ where $\vect{x}$ is in $V^{\bot}$.
-
-<ol type="i">
-<li>
-
-Since the inproduct of any vector with itself is non-negative, we find:
-
-$$
-\begin{align*}
-\norm{\vect{w}}&=\sqrt{\vect{w}\ip\vect{w}}=\sqrt{(\proj_{V}(\vect{w})+\vect{x})\ip(\proj_{V}(\vect{w})+\vect{x})}\\
-&=\sqrt{\proj_{V}(\vect{w})\ip\proj_{V}(\vect{w})+\vect{x}\ip\vect{x}}\\
-&\geq\sqrt{\proj_{V}(\vect{w})\ip\proj_{V}(\vect{w})}=\norm{\proj_{V}(\vect{w})}
-\end{align*}
-$$
-
-</li>
-<li>
-
-We have:
-
-$$
-\begin{align*}
-\vect{w}\ip\proj_{V}(\vect{w})&=(\proj_{V}(\vect{w})+\vect{x})\ip\proj_{V}(\vect{w})\\
-&=\proj_{V}(\vect{w})\ip\proj_{V}(\vect{w})\geq 0.
-\end{align*}
-$$
-
-Furthermore, $\proj_{V}(\vect{w})\ip\proj_{V}(\vect{w})=0$ implies $\proj_{V}(\vect{w})=\vect{0}$, so $\vect{w}=\vect{x}$ which is in $V^{\bot}$.
-
-</li>
-<li>
-
-</li>
-
-We find, for arbitrary $\vect{v}$ in $V$:
-
-$$
-\begin{align*}
-\norm{\vect{w}-\vect{v}}&=\sqrt{(\proj_{V}(\vect{w})+\vect{x}-\vect{v})\ip(\proj_{V}(\vect{w})+\vect{x}-\vect{v})}\\
-&=\sqrt{(\proj_{V}(\vect{w})-\vect{v})\ip(\proj_{V}(\vect{w})-\vect{v})+\vect{x}\ip\vect{x}}
-\end{align*}
-$$
-
-Again using the fact that the inproduct of any vector with itself is non-negative, we conclude that this is minimal when $\proj_{V}(\vect{w})-\vect{v}=\vect{0}$, i.e. when $\proj_{V}(\vect{w})=\vect{v}$.
-
-</ol>
-
-:::
diff --git a/Chapter8/Images/Fig-SVD-Decomposition.svg b/Chapter8/Images/Fig-SVD-Decomposition.svg
index d5a9ed3..e7c2664 100644
--- a/Chapter8/Images/Fig-SVD-Decomposition.svg
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diff --git a/Chapter8/QuadraticForms.md b/Chapter8/QuadraticForms.md
index 9d0d1f4..47169e7 100644
--- a/Chapter8/QuadraticForms.md
+++ b/Chapter8/QuadraticForms.md
@@ -130,8 +130,9 @@ the graph of a linear function $z = a_1x_1 + a_2x_2 + b$ is a plane.
:::{figure} Images/Fig-QuadForms-Plane1.svg
:name: Fig:QuadForms:Plane1
+:class: dark-light
-The plane $z = \frac13x_1 - x_2 +2$
+The plane $z = \frac13x_1 - x_2 +2$.
:::
The graph of a quadratic function is a curved surface.
@@ -140,14 +141,16 @@ The graph of a quadratic function is a curved surface.
:::{figure} Images/Fig-QuadForms-QuadSurface1.png
:name: Fig:QuadForms:QuadSurface1
+:class: dark-light
-The surface $z = -\frac13x_1^2 + \frac13x_2^2 + 2 $
+The surface $z = -\frac13x_1^2 + \frac13x_2^2 + 2 $.
:::
:::{figure} Images/Fig-QuadForms-QuadSurface2.png
:name: Fig:QuadForms:QuadSurface2
+:class: dark-light
-The surface $z = -\frac12x_1^2 - \frac14x_2^2 + x_1 - x_2 + 2$
+The surface $z = -\frac12x_1^2 - \frac14x_2^2 + x_1 - x_2 + 2$.
:::
The shape of the surfaces is in most cases determined by the quadratic part $\vect{x}^TA\vect{x}$. The linear part is then only relevant for the position.
@@ -309,7 +312,8 @@ $$
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:QuadForms:Substitution`
+:class: myproof
If we put $\vect{x} = P\vect{y}$ we get
@@ -412,7 +416,8 @@ $$
where $\lambda_1, \ldots, \lambda_n$ are the _eigenvalues_ of the matrix $A$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:QuadForms:Diagonalize`
+:class: myproof
If we make the substitution $\vect{x} = Q\vect{y}$ we find that
@@ -501,6 +506,7 @@ The property is known as _Sylvester's Law of Inertia_.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5a3d937b-6ecb-4fe8-b805-424af7e7ac55?id=90077
:label: grasple_exercise_8_2_T1
:dropdown:
@@ -607,7 +613,8 @@ $q_A$ is **indefinite** if at least one eigenvalue is positive, and at least one
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Thm:QuadForms:Classification`
+:class: myproof
This immediately follows from {prf:ref}`Prop:QuadForms:Diagonalize`. If we make the substitution $\vect{x} = Q\vect{y}$ with the matrix $Q$ of the orthogonal diagonalization, i.e.,
@@ -736,6 +743,7 @@ A _conic section_ or _conic_ is a curve that results when a circular cone is int
:::{figure} Images/Fig-QuadForms-ConicSections.png
:name: Fig:QuadForms:ConeWithPlanes
+:class: dark-light
Intersections of a cone with several planes (not going through the apex).
@@ -809,8 +817,9 @@ Both curves have the coordinates axes as axes of symmetry. In this context they
:::{figure} Images/Fig-QuadForms-EllipseHyperbola.svg
:name: Fig:QuadForms:EllipseHyperbola
+:class: dark-light
-(Standard) Hyperbola and Ellipse
+(Standard) Hyperbola and Ellipse.
:::
::::{exercise}
@@ -1045,8 +1054,9 @@ See Figure {numref}`Fig:QuadForms:Ellipses`
:::{figure} Images/Fig-QuadForms-Ellipses(2).svg
:name: Fig:QuadForms:Ellipses
+:class: dark-light
-The two ellipses
+The two ellipses.
:::
::::
@@ -1055,6 +1065,7 @@ The two ellipses
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b0668dff-a174-447b-8008-09b242a804fb?id=87448
:label: grasple_exercise_8_2_1
:dropdown:
@@ -1064,6 +1075,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/eaec14bd-fe7a-4c7a-a269-68e1d369bc2b?id=90207
:label: grasple_exercise_8_2_2
:dropdown:
@@ -1073,6 +1085,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7044809f-28ca-4caf-b3fc-139010112ca1?id=90052
:label: grasple_exercise_8_2_3
:dropdown:
@@ -1082,6 +1095,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/b71d8b9f-a3e8-48f6-b236-58f85a4818a6?id=90997
:label: grasple_exercise_8_2_4
:dropdown:
@@ -1091,6 +1105,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f0f9e677-eb21-4f84-9850-039b24ee0999?id=93112
:label: grasple_exercise_8_2_5
:dropdown:
@@ -1101,6 +1116,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f4657e73-219d-4ac9-bf17-81b240ddac96?id=93113
:label: grasple_exercise_8_2_6
:dropdown:
@@ -1110,6 +1126,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/21ad829b-77f5-4e14-9808-4fbbb901c9b4?id=93119
:label: grasple_exercise_8_2_7
:dropdown:
@@ -1119,6 +1136,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/03704333-e9db-46f0-b292-eb235aee6b22?id=91025
:label: grasple_exercise_8_2_8
:dropdown:
@@ -1128,6 +1146,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8b851997-932f-4ee6-8dfd-785bd7908e1c?id=91091
:label: grasple_exercise_8_2_9
:dropdown:
@@ -1136,6 +1155,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7c47dc31-b7d8-409e-9334-8a5de188c928?id=91912
:label: grasple_exercise_8_2_10
:dropdown:
@@ -1146,6 +1166,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/ee9c377f-5150-4264-8d3e-1150c482fd7f?id=93116
:label: grasple_exercise_8_2_11
:dropdown:
@@ -1155,6 +1176,7 @@ The two ellipses
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/51f56e96-3761-44c5-8d20-4cf0047a1ea4?id=93115
:label: grasple_exercise_8_2_12
:dropdown:
@@ -1165,6 +1187,7 @@ The two ellipses
The following exercises are a more theoretical.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/01a3d009-b2e0-4f2b-9d49-fcca955d6c5d?id=91048
:label: grasple_exercise_8_2_13
:dropdown:
@@ -1174,6 +1197,7 @@ The following exercises are a more theoretical.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7fea2ed7-c54f-4665-8ac8-611a8b0f6c5e?id=93114
:label: grasple_exercise_8_2_14
:dropdown:
@@ -1183,6 +1207,7 @@ The following exercises are a more theoretical.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/78b55ab4-4f27-4d32-90ec-9bb28bb8f7b8?id=91021
:label: grasple_exercise_8_2_15
:dropdown:
diff --git a/Chapter8/SingularValueDecomp.md b/Chapter8/SingularValueDecomp.md
index e00ff01..b6c00b0 100644
--- a/Chapter8/SingularValueDecomp.md
+++ b/Chapter8/SingularValueDecomp.md
@@ -2,7 +2,7 @@
# Singular Value Decomposition (SVD)
-We have seen already several ways to factorise matrices. In {numref}`Sec:MatFactor`, we studied the $LU$ and the $PLU$ factorisations, and in {numref}`Sec:Gram-Schmidt:QRdecomp`
+We have seen already several ways to factorise matrices. In {numref}`Sec:LU-decomp`, we studied the $LU$ and the $PLU$ factorisations, and in {numref}`Sec:Gram-Schmidt:QRdecomp`
we laid the QR Decomposition on the table. In {numref}`Sec:SymmetricMat` we showed that every symmetric (square) matrix $A$ can be written as $A = QDQ^{-1} = QDQ^T$. In this section it is in a sense this last decomposition we will generalize to non-symmetric matrices, and even to non-square matrices.
We will introduce and study the so-called **singular value decomposition** (SVD) of a matrix.
In the first subsection ({numref}`Subsec:SVD:Definition`) we will give the definition of the SVD, and illustrate it with a few examples. In the second subsection ({numref}`Subsec:SVD:Existence`) an algorithm to compute the SVD is presented and illustrated. And it will be shown that this algorithm always yields a proper SVD.
@@ -162,7 +162,8 @@ Suppose $A = U\Sigma V^T$, with $U, \Sigma, V$ as in the definition.
::::
-::::{dropdown} Proof of {prf:ref}`Prop:SVD:BasicProp`
+::::{admonition} Proof of {prf:ref}`Prop:SVD:BasicProp`
+:class: myproof, dropdown
Suppose $\sigma_1>0, \ldots, \sigma_r>0$ and $\sigma_{r+1}=0 , \ldots, \sigma_p=0$, where $p=$min$\{m,n\}$.
@@ -236,8 +237,8 @@ Moreover, the columns of $V$ are corresponding eigenvectors (of $A^TA$).
::::
-
-::::{dropdown} Proof of {prf:ref}`Prop:SVD:singularvalues`
+::::{admonition} Proof of {prf:ref}`Prop:SVD:singularvalues`
+:class: myproof, dropdown
First of all, because of the properties of the matrices $U$, $\Sigma$ and $V$ we have that
@@ -377,7 +378,8 @@ Let $A$ be an $m\times n$ matrix with real entries. Then the following propertie
:::::
-::::{dropdown} Proof of {prf:ref}`Prop:SVD:propertiesATA`
+::::{admonition} Proof of {prf:ref}`Prop:SVD:propertiesATA`
+:class: myproof, dropdown
:::{latexlist}
:enumerated: true
@@ -480,7 +482,8 @@ For every $m \times n$ matrix $A$ a singular value decomposition exists
The proof consists in showing that all steps in the algorithm do what they are supposed to do, and that the final result consists of three matrices $U, \Sigma, V$ that can act as a
singular value decomposition of $A$.
-::::{dropdown} Proof of {prf:ref}`Thm:SVD:Existence`
+::::{admonition} Proof of {prf:ref}`Thm:SVD:Existence`
+:class: myproof, dropdown
Let us first consider the six steps of the algorithm.
@@ -666,13 +669,13 @@ In this section we will have a deeper look at the decomposition and its meaning.
By definition, the matrices $U$ and $V$ in the SVD of an $m\times n$ matrix $A$
are orthogonal matrices. Thus the columns of $U$ give an orthonormal basis of $\R^m$, the columns of $V$ an orthonormal basis of $\R^n$. The decomposition $U\Sigma V^T$ then becomes a composition of transformations. We can visualise this using the graph in {numref}`Figure %s <Fig:SVD:decomposition>`:
-:::{figure} Images/Fig-SVD-Decomposition.svg
-:width: 300px
+:::::{figure} Images/Fig-SVD-Decomposition.svg
:name: Fig:SVD:decomposition
+:class: dark-light
Diagram showing the SVD as a composition of linear transformations.
-::::
+:::::
Let us first consider the case where $A$ is a $2 \times 2$ matrix, as in that case everything takes place in the plane, and we can make an exact drawing of what is going on.
@@ -774,8 +777,9 @@ $$
:::{figure} Images/Fig-SVD-GeometricView.svg
:name: Fig:SVD:GeometricView
+:class: dark-light
-Geometric decomposition of $\tilde{A} = \dfrac{1}{\sqrt{5}}\begin{bmatrix} 5 & 2 \\ 0 & 6 \end{bmatrix} = U\tilde{\Sigma}V^T$
+Geometric decomposition of $\tilde{A} = \dfrac{1}{\sqrt{5}}\begin{bmatrix} 5 & 2 \\ 0 & 6 \end{bmatrix} = U\tilde{\Sigma}V^T$.
:::
::::
@@ -963,7 +967,7 @@ We expect $A_3$ to be a good approximation of $A$.
$$
- As you can see, $A_3$ closely resembles $A$. (You have to trust us with regard to the hidden columns. ;-) <BR>
+ As you can see, $A_3$ closely resembles $A$. (You have to trust us with regard to the hidden columns. <html>😉</html> <BR>
The gain: to store the $12\times10$ matrix $A$, we have to store $120$ reals. <BR>
To store $U_3, \Sigma_{33}$ and $V_3$ , we only have to store $12\times3 + 3 + 10\times 3 = 69$ numbers. The middle 3 comes from first three elements ($=$ singular values) on the diagonal of $\Sigma$.
@@ -977,6 +981,7 @@ We expect $A_3$ to be a good approximation of $A$.
## Grasple Exercises
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/27adae2a-db2a-46fa-800f-49e4c0dfe4fa?id=93487
:label: grasple_exercise_8_3_9
:dropdown:
@@ -985,6 +990,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/caac29d1-9700-4a30-8c7c-19ea6148258f?id=93490
:label: grasple_exercise_8_3_10
:dropdown:
@@ -994,6 +1000,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/e4c651aa-a998-4e19-957b-20ddf41509bf?id=93468
:label: grasple_exercise_8_3_2
:dropdown:
@@ -1001,6 +1008,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/47ebaa77-9f3c-4363-a57e-d37242c6e598?id=93471
:label: grasple_exercise_8_3_3
:dropdown:
@@ -1008,6 +1016,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/79d22478-56e3-49ee-9b19-77ab1ad06eaf?id=93470
:label: grasple_exercise_8_3_4
:dropdown:
@@ -1017,6 +1026,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/10affbae-4221-40f8-bf0e-df626a0e64ae?id=93479
:label: grasple_exercise_8_3_5
:dropdown:
@@ -1025,6 +1035,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/37ea17f1-1bfb-4a19-b9e2-1292a593dfa3?id=93480
:label: grasple_exercise_8_3_6
:dropdown:
@@ -1033,6 +1044,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/20dd219a-35f3-48d7-ad9d-35038047336b?id=92586
:label: grasple_exercise_8_3_7
:dropdown:
@@ -1041,6 +1053,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9848d7be-1530-46b0-941f-9ae76e95abfa?id=93481
:label: grasple_exercise_8_3_8
:dropdown:
@@ -1049,6 +1062,7 @@ We expect $A_3$ to be a good approximation of $A$.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3fdad317-fc18-4f88-b416-87cbd1d5e708?id=93495
:label: grasple_exercise_8_3_1
:dropdown:
diff --git a/Chapter8/SymmetricMatrices.md b/Chapter8/SymmetricMatrices.md
index 6933ccc..eb59118 100644
--- a/Chapter8/SymmetricMatrices.md
+++ b/Chapter8/SymmetricMatrices.md
@@ -23,7 +23,7 @@ Note that this definition implies that a symmetric matrix must be a square matri
The matrices
$$
- A_1 = \begin{bmatrix} 2&\color{blue}3&\color{red}4\\\color{blue}3&1&\color{green}5 \\\color{red}4&\color{green}5&7 \end{bmatrix} \quad \text{and} \quad
+ A_1 = \begin{bmatrix} 2&\class{blue}3&\class{red}4\\\class{blue}3&1&\class{green}5 \\\class{red}4&\class{green}5&7 \end{bmatrix} \quad \text{and} \quad
A_2 = \begin{bmatrix} 0&2&3&4\\
2&0&1&5 \\
3&1&0&6 \\
@@ -68,7 +68,8 @@ In other contexts the word _spectrum_ of a transformation is used for the set of
So, for a symmetric matrix an orthonormal basis of eigenvectors always exists. For the inertia tensor of a 3D body such a basis corresponds to the (perpendicular) principal axes.
-::::{prf:proof} Of the converse of {prf:ref}`Thm:SymmetricMat:OrthogDiag`.
+::::{admonition} Of the converse of {prf:ref}`Thm:SymmetricMat:OrthogDiag`
+:class: myproof
Recall that an orthogonal matrix is a matrix $Q$ for which $Q^{-1} = Q^T$.
@@ -118,6 +119,7 @@ the image of the unit circle under the transformation $\vect{x} \mapsto A\vect{x
:::{figure} Images/Fig-SymmetricMat-Evectors.svg
:name: Fig:SymmetricMat:Evectors
+:class: dark-light
The transformation $T(\vect{x}) = \begin{bmatrix} 1&2\\2&-2 \end{bmatrix}\vect{x}$.
:::
@@ -144,7 +146,8 @@ Suppose $A$ is a symmetric matrix.
If $\mathbf{v}_1$ and $\mathbf{v}_2$ are eigenvectors of $A$ for _different_ eigenvalues, then $\mathbf{v}_1\perp \mathbf{v}_2$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:SymmetricMat:OrthogonalEigenvectors`
+:class: myproof
Suppose $\mathbf{v}_1$ and $\mathbf{v}_2$ are eigenvectors of the symmetric matrix $A$ for the different eigenvalues $\lambda_1,\lambda_2$.
We want to show that $\mathbf{v}_1 \ip \mathbf{v}_2 = 0$.
@@ -214,7 +217,8 @@ If $\vect{u}$ is an eigenvector of $A$ for the eigenvalue $\lambda$, and $\vect{
::::
-::::{dropdown} Solution to {numref}`Exc:SymmetricMat:uTAv` (_click to show_)
+::::{admonition} Solution to {numref}`Exc:SymmetricMat:uTAv`
+:class: solution, dropdown
The proof is completely analogous to the proof of {prf:ref}`Prop:SymmetricMat:OrthogonalEigenvectors`.
Suppose
@@ -257,7 +261,8 @@ All eigenvalues of symmetric matrices are real.
The easiest proof is via complex numbers. Feel free to skip it, in particular when you don't feel comfortable with complex numbers.
-::::{dropdown} Proof of {prf:ref}`Prop:SymmetricMat:RealEigenvalues`
+::::{admonition} Proof of {prf:ref}`Prop:SymmetricMat:RealEigenvalues`
+:class: myproof, dropdown
For two vectors $\mathbf{u},\mathbf{v}$ in $\C^n$ we consider the expression
@@ -533,8 +538,8 @@ $$
And now it's time for the proof of the main theorem. The proof is of the type technical and intricate. Skip it if you like.
-::::{dropdown} Proof of {prf:ref}`Thm:SymmetricMat:OrthogDiag`
-%::::{prf:proof} (of {prf:ref}`Thm:SymmetricMat:OrthogDiag`)
+::::{admonition} Proof of {prf:ref}`Thm:SymmetricMat:OrthogDiag`
+:class: myproof, dropdown
Suppose that $A$ is a symmetric $n \times n$ matrix. We know there are $n$ real, possibly multiple, eigenvalues
$\lambda_1, \lambda_2, \ldots, \lambda_n$.
@@ -870,7 +875,9 @@ of $n$ matrices $P_i$ that represent orthogonal projections onto one-dimensional
Formula {eq}`Eq:SymmetricMat:SpectralDecomp` is referred to as being a **spectral decomposition** of the matrix $A$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Thm:SymmetricMat:SpectralDecomp`
+:class: myproof
+
For a general $n\times n$ symmetric matrix $A$, there exists an orthogonal diagonalization
$$
@@ -911,7 +918,8 @@ $$
where $P_i$ denotes the orthogonal projection onto the eigenspace $E_{\lambda_i}$.
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Cor:SymmetricMat:SpectralThm-2`
+:class: myproof
We know that
@@ -983,6 +991,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/f76823e6-8936-4edf-bd0b-fa3a2aa7246f?id=88040
:label: grasple_exercise_8_1_1
:dropdown:
@@ -991,6 +1000,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9828a4b4-98f7-46c3-8dab-74ac04fc1955?id=88032
:label: grasple_exercise_8_1_2
:dropdown:
@@ -999,6 +1009,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/8af926a0-80d8-459f-af55-c37a492a18c6?id=88045
:label: grasple_exercise_8_1_3
:dropdown:
@@ -1007,6 +1018,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/03d75a31-7e1b-4dd2-be0a-5e9a93a0ef09?id=94940
:label: grasple_exercise_8_1_4
:dropdown:
@@ -1015,6 +1027,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/926933aa-a33e-40f5-8e70-84bb9ed63fc8?id=87465
:label: grasple_exercise_8_1_5
:dropdown:
@@ -1023,6 +1036,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/9aac9c37-aa3b-4d5a-bb92-f00c09e5f052?id=94943
:label: grasple_exercise_8_1_6
:dropdown:
@@ -1031,6 +1045,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/a6a95823-15e4-4354-b89d-559306a5a7fa?id=94941
:label: grasple_exercise_8_1_7
:dropdown:
@@ -1039,6 +1054,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/0403af25-edba-4bc6-b077-3de227253419?id=56931
:label: grasple_exercise_8_1_8
:dropdown:
@@ -1047,6 +1063,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/3a45e358-4898-4d1d-b6f4-ba9679dd13e0?id=87765
:label: grasple_exercise_8_1_9
:dropdown:
@@ -1055,6 +1072,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/eb8b0e2f-d909-47ce-8ef1-50ad67e2b0f6?id=87905
:label: grasple_exercise_8_1_10
:dropdown:
@@ -1064,6 +1082,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5ce15529-61a7-43d0-9fd3-5ad5469618e8?id=89131
:label: grasple_exercise_8_1_11
:dropdown:
@@ -1073,6 +1092,7 @@ $$
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+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/5511e064-f22d-4601-9156-f00545d59f80?id=88649
:label: grasple_exercise_8_1_12
:dropdown:
@@ -1081,6 +1101,7 @@ $$
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/c994fa76-f723-4700-922b-2f05ff0ef822?id=87760
:label: grasple_exercise_8_1_13
:dropdown:
@@ -1089,13 +1110,16 @@ $$
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+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/4fd8d027-0e63-46ec-aaf5-f2d10d8707c9?id=87038
:label: grasple_exercise_8_1_14
:dropdown:
:description: To give an example of a 3x3 symm matrix with given eigenvalues and eigenspace.
+::::
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/77b08679-8974-453a-8f68-7e08e8ecfaf5?id=94944
:label: grasple_exercise_8_1_15
:dropdown:
@@ -1105,6 +1129,7 @@ $$
The following exercise have a more theoretical flavour.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/6e0ebf73-fba2-46d0-aaa8-44e53ea07e53?id=88034
:label: grasple_exercise_8_1_16
:dropdown:
@@ -1113,6 +1138,7 @@ The following exercise have a more theoretical flavour.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/73c272d7-dbb0-47c9-8bee-074b1f8cc154?id=82845
:label: grasple_exercise_8_1_17
:dropdown:
@@ -1121,6 +1147,7 @@ The following exercise have a more theoretical flavour.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/7959665f-09d0-4362-a0e8-c0a3e613399f?id=82848
:label: grasple_exercise_8_1_18
:dropdown:
@@ -1129,6 +1156,7 @@ The following exercise have a more theoretical flavour.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/33f5be5a-1cfa-4056-ac91-c2282de234b1?id=87864
:label: grasple_exercise_8_1_19
:dropdown:
@@ -1137,6 +1165,7 @@ The following exercise have a more theoretical flavour.
::::{grasple}
+:iframeclass: dark-light
:url: https://embed.grasple.com/exercises/59c4c327-1603-4cc1-8b92-7415c691098b?id=87873
:label: grasple_exercise_8_1_20
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diff --git a/Chapter9/DynSystContinuous.md b/Chapter9/DynSystContinuous.md
index 727a2c9..7d0c92f 100644
--- a/Chapter9/DynSystContinuous.md
+++ b/Chapter9/DynSystContinuous.md
@@ -81,19 +81,21 @@ $$
:::{prf:definition}
-In this context, we call $\vect{x}'=A\vect{x}$ a **system of (linear) differential equations** or a **synamical system**, $\vect{x}$ a **vector-valued function**, $\vect{x}'$ the **derivative** of $\vect{x}$, and the $x_{i}$'s the **component functions** of $\vect{x}$. Any $\vect{y}$ for which $\vect{y}'=A\vect{y}$ holds is called a **solution** to the system of differential equations.
+In this context, we call $\vect{x}'=A\vect{x}$ a **system of (linear) differential equations** or a **dynamical system**, $\vect{x}$ a **vector-valued function**, $\vect{x}'$ the **derivative** of $\vect{x}$, and the $x_{i}$'s the **component functions** of $\vect{x}$. Any $\vect{y}$ for which $\vect{y}'=A\vect{y}$ holds is called a **solution** to the system of differential equations.
:::
The following proposition will be quite useful to us. It tells us that, in order to find the full (infinite) solution set, it suffices to find a (finite) basis of solutions.
:::{prf:proposition}
+:label: Prop:DynSystContinuous:LinComb
If $\vect{y}$ and $\vect{z}$ are solutions of $\vect{x}'=A\vect{x}$ and $c$ and $d$ are arbitrary real numbers, then $c\vect{y}+d\vect{z}$ is also a solution of $\vect{x}=A\vect{x}'$.
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:DynSystContinuous:LinComb`
+:class: myproof
Exercise.
@@ -184,13 +186,13 @@ A=\begin{bmatrix}
\end{bmatrix}.
$$
-Since $A$ is an upper diagonal matrix, we can conclude that its eigenvalues are $-\beta$ and $-\alpha$, which for simplicity's sake we will assume to be different. Therefore, a solution to the system of linear differential equations $\vect{y}'=A\vect{y}$ is given by
+Since $A$ is an upper diagonal matrix, we can conclude that its eigenvalues are $-\beta$ and $-\alpha$, which, for simplicity's sake, we will assume to be different. Therefore, a solution to the system of linear differential equations $\vect{y}'=A\vect{y}$ is given by
$$
\vect{y}=c_{1}\vect{v}_{-\beta}e^{-\beta t}+c_{2}\vect{v}_{-\alpha}e^{-\alpha t}
$$
-where $c_{1}$ and $c_{2}$ are some constants while $\vect{v}_{-\beta}$ and $\vect{v}_{-\alpha}$ are the eigenvectors of $A$ corresponding $-\beta $ and $-\alpha$, respectively. In particular, if $t$ gets very large, we find very large but negative exponents on the right hand side. That is, both $\lim_{t\to\infty}S(t)$ and $\lim_{t\to\infty} I(t)$ are $0$. This makes perfect sense intuitively, as we expect all members of the population to get infected and recover. After that, they are neither susceptible nor infected anymore.
+where $c_{1}$ and $c_{2}$ are some constants while $\vect{v}_{-\beta}$ and $\vect{v}_{-\alpha}$ are the eigenvectors of $A$ corresponding to $-\beta $ and $-\alpha$, respectively. In particular, if $t$ gets very large, we find very large but negative exponents on the right hand side. That is, both $\lim_{t\to\infty}S(t)$ and $\lim_{t\to\infty} I(t)$ are $0$. This makes perfect sense intuitively, as we expect all members of the population to get infected and recover. After that, they are neither susceptible nor infected anymore.
Note that, in the long run, we will end up arbitrarily close to $\vect{0}$ regardless of the starting values of $S$ and $I$. That is, if we start in any $\vect{v}$ and follow the solution $\vect{y}(t)$ of the system of linear differential equations satisfying the initial condition $\vect{y}(0)=\vect{v}$, then we will always end up in $\vect{0}$. In other words, $\vect{v}$ *attracts* all points.
@@ -222,8 +224,11 @@ a **saddle point** if $\lambda_{1}\lambda_{2}<0$, i.e. if $\lambda_{1}$ and $\la
</ul>
+The three different behaviours are illustrated in {numref}`Figure %s <Fig:DynSystContinuous:Trajectories>`.
+
:::
+
Let us once again consider the system $\vect{y}'=A\vect{y}$. By {prf:ref}`Prop:DynSystContinuous:EVsgiveSols`, we can find solutions $\vect{y}=\vect{v}e^{\lambda t}$ where $\lambda$ is an eigenvalue of $A$ and $\vect{v}$ is a corresponding eigenvector. But if $\lambda$ is not a real number, this does not give a real-valued function. In some applications that's perfectly fine, but often we're interested in real solutions to systems of linear differential equations. Can we stil find any of those if some eigenvalues are complex?
Yes, we can! First, we can use the following well-known fact from calculus:
@@ -273,14 +278,22 @@ $$
$$
-are linearly independent solutions to the linear system of differential equations $\vect{y}'=A\vect{y}$. In this case, the origin is called a **spiral point**.
+are linearly independent solutions to the linear system of differential equations $\vect{y}'=A\vect{y}$. In this case, the origin is called a **spiral point**. An example of a spiral point can be seen in {numref}`Figure %s <Fig:DynSystContinuous:Trajectories>`.
:::
If $a<0$ in this proposition, then $e^{at}$ will become arbitrarily small, so as $t$ increases, $\vect{y}(t)$ will approach $0$. In this case, the trajectory will spiral towards the origin. If $a>0$, then $e^{at}$ becomes arbitrarily large and the trajectory will spiral away from the origin.
+
+::::{figure} Images/Fig-DynSystContinuous-Trajectories.svg
+:name: Fig:DynSystContinuous:Trajectories
+:class: dark-light
+
+The possible behaviours of the origin illustrated. On the top left, it's an attractor, on the top right a repeller, on the bottom left a saddle point, and on the bottom right a spiral point. For the spiral point, do you expect the real part of the eigenvalues to be positive or negative, given the figure?
+::::
+
## Decoupling a dynamical system
-In the previous section, we say that the eigenvalues and eigenvectors determine the long-term behaviour of a dynamical system. This leads naturally to the suspicions that, perhaps, diagonalizing a matrix can help us solve a system of linear differential equations. This is indeed the case.
+In the previous section, we saw that the eigenvalues and eigenvectors determine the long-term behaviour of a dynamical system. This leads naturally to the suspicion that, perhaps, diagonalizing a matrix can help us solve a system of linear differential equations. This is indeed the case.
-Let us assume $A$ is an $n\times n$-matrix with eigenfunctions $\vect{y}_{1},...,\vect{v}_{n}$, that is $\vect{y}_{i}=\vect{v}_{i}e^{\lambda_{i}t}$ where $\lambda_{i}$ is an eigenvalue of $A$ with associated eigenvector $\vect{v}_{i}$.
\ No newline at end of file
+Let us assume $A$ is an $n\times n$-matrix with eigenfunctions $\vect{y}_{1},...,\vect{y}_{n}$, that is $\vect{y}_{i}=\vect{v}_{i}e^{\lambda_{i}t}$ where $\lambda_{i}$ is an eigenvalue of $A$ with associated eigenvector $\vect{v}_{i}$.
\ No newline at end of file
diff --git a/Chapter9/DynSystDiscrete.md b/Chapter9/DynSystDiscrete.md
index 5829e90..67deffa 100644
--- a/Chapter9/DynSystDiscrete.md
+++ b/Chapter9/DynSystDiscrete.md
@@ -45,6 +45,7 @@ individuals of the age group 'full grown' reproduce 4 offspring and with probabi
:::{figure} Images/Fig-DynSystDiscrete-LeslieGraph.svg
:name: Fig:DynSystDiscrete:Leslie1
+:class: dark-light
The graph of the population model.
:::
@@ -179,7 +180,8 @@ The expression for $\vect{x}_k$ with unspecified parameters $c_1,\ldots,c_n$ is
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:DynSystDiscrete:DiagCase`
+:class: myproof
There is nothing much to prove.
@@ -262,7 +264,8 @@ If $|\lambda_1| = 1$ the origin is stable but not asymptotically stable.
</ul>
::::
-::::{prf:proof}
+::::{admonition} Proof of {prf:ref}`Prop:DynSystDiscrete:DiagCase2`
+:class: myproof
From Equation {eq}`Eq:DynSystDiscrete:GenSolDiagble` in {prf:ref}`Prop:DynSystDiscrete:DiagCase` it follows immediately that if all
@@ -384,8 +387,9 @@ The trajectories in $\R^4$ are hard to plot. Instead we can plot the progression
:::{figure} Images/Fig-DynSystDiscrete-Leslie-2.svg
:name: Fig:DynSystDiscrete:Leslie2
+:class: dark-light
-The evolvement in time of the population model
+The evolvement in time of the population model.
:::
What is not so clear in the picture is that for large $k$ the state vectors $\vect{x}_k$ are approximately eigenvectors for the matrix $M$. However, numerical values shed light on this phenomenon. For instance, the last two 'states' are given by
@@ -467,8 +471,9 @@ In {numref}`Fig:DynSystDiscrete:SimplestSystem` the paths are shown for the star
:::{figure} Images/Fig-DynSystDiscrete-SimplestSystem.svg
:name: Fig:DynSystDiscrete:SimplestSystem
+:class: dark-light
-A very simple dynamical system
+A very simple dynamical system.
:::
@@ -499,8 +504,9 @@ matrix $A = \left[\begin{array}{cc} 0.5 & 0.2 \\ -0.2 & 1.0 \end{array}\right]$.
:::{figure} Images/Fig-DynSystDiscrete-NiceNode.svg
:name: Fig:DynSystDiscrete:NiceNode
+:class: dark-light
-A dynamical system with a stable node
+A dynamical system with a stable node.
:::
@@ -549,8 +555,9 @@ $(1,-1)$ and $(-1,1)$. On each of them the direction of the points $\vect{x}_k$
:::{figure} Images/Fig-DynSystDiscrete-Spiral1A.svg
:name: Fig:DynSystDiscrete:Spiral1A
+:class: dark-light
-A dynamical system with a spiral point
+A dynamical system with a spiral point.
:::
@@ -599,8 +606,9 @@ $$
:::{figure} Images/Fig-DynSystDiscrete-Spiral1B.svg
:name: Fig:DynSystDiscrete:Spiral1B
+:class: dark-light
-One trajectory $\vect{y}_0$, $\vect{y}_1$, $\vect{y}_2$, .....
+One trajectory $\vect{y}_0$, $\vect{y}_1$, $\vect{y}_2$, $\ldots$.
:::
diff --git a/Chapter9/Images/Fig-DynSystContinuous-Trajectories.svg b/Chapter9/Images/Fig-DynSystContinuous-Trajectories.svg
new file mode 100644
index 0000000..e97f0d2
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81.390023 80.259549 C 81.34696 80.232145 81.268664 80.185168 81.221687 80.157764 C 81.174709 80.130361 81.100328 80.087298 81.053351 80.063809 C 81.006374 80.04032 80.931993 80.001173 80.88893 79.981599 C 80.841952 79.95811 80.767571 79.922877 80.724509 79.907218 C 80.681446 79.887644 80.607065 79.856325 80.564002 79.840666 C 80.52094 79.825007 80.454388 79.801518 80.41524 79.789774 C 80.372178 79.774115 80.305626 79.754541 80.270393 79.746711 C 80.231245 79.734967 80.168609 79.719308 80.133376 79.715393 C 80.098142 79.707563 80.039421 79.695819 80.008102 79.691904 C 79.972869 79.687989 79.921977 79.684075 79.890659 79.684075 C 79.863255 79.68016 79.816278 79.68016 79.788874 79.684075 C 79.761471 79.684075 79.722323 79.687989 79.698834 79.695819 C 79.675345 79.699734 79.640112 79.707563 79.620538 79.715393 C 79.600964 79.723222 79.569646 79.734967 79.553987 79.742796 C 79.538328 79.754541 79.514839 79.7702 79.503094 79.781944 C 79.49135 79.793689 79.475691 79.817177 79.467861 79.828922 C 79.456117 79.844581 79.448287 79.86807 79.444373 79.883729 C 79.436543 79.899388 79.432628 79.926792 79.432628 79.942451 C 79.432628 79.962025 79.432628 79.989428 79.436543 80.009002 C 79.436543 80.028576 79.444373 80.059894 79.448287 80.079468 C 79.456117 80.099042 79.467861 80.134275 79.475691 80.153849 C 79.487435 80.173423 79.503094 80.208656 79.514839 80.232145 C 79.526583 80.251719 79.550072 80.290867 79.565731 80.310441 C 79.58139 80.33393 79.604879 80.369163 79.624453 80.392651 C 79.640112 80.41614 79.67143 80.451373 79.691004 80.474862 C 79.710578 80.498351 79.745811 80.533584 79.7693 80.557073 C 79.788874 80.580561 79.828022 80.615794 79.851511 80.635368 C 79.874999 80.658857 79.914147 80.69409 79.941551 80.713664 C 79.96504 80.737153 80.008102 80.768471 80.035506 80.79196 C 80.058995 80.811534 80.105972 80.842852 80.133376 80.862426 C 80.160779 80.882 80.203842 80.913318 80.231245 80.932892 C 80.258649 80.948552 80.305626 80.97987 80.33303 80.995529 C 80.360433 81.011188 80.407411 81.038592 80.434814 81.054251 C 80.462218 81.06991 80.509195 81.093399 80.536599 81.109058 C 80.564002 81.120802 80.61098 81.144291 80.638383 81.156035 C 80.665787 81.16778 80.70885 81.187354 80.732338 81.195183 C 80.759742 81.206928 80.802805 81.222587 80.826293 81.230416 C 80.853697 81.238246 80.892845 81.253905 80.916333 81.25782 C 80.939822 81.26565 80.97897 81.277394 81.002459 81.281309 C 81.025947 81.285223 81.061181 81.293053 81.080755 81.296968 C 81.100328 81.300883 81.135562 81.304797 81.151221 81.304797 C 81.170795 81.308712 81.202113 81.308712 81.217772 81.308712 C 81.237346 81.308712 81.260835 81.304797 81.276494 81.304797 C 81.292153 81.300883 81.315642 81.296968 81.327386 81.293053 C 81.339131 81.289138 81.358705 81.285223 81.370449 81.277394 C 81.378279 81.273479 81.393938 81.26565 81.405682 81.25782 C 81.413512 81.24999 81.425256 81.238246 81.429171 81.230416 C 81.437 81.222587 81.44483 81.210842 81.448745 81.199098 C 81.45266 81.191269 81.456574 81.175609 81.456574 81.163865 C 81.460489 81.156035 81.460489 81.140376 81.460489 81.128632 C 81.460489 81.116888 81.456574 81.097314 81.45266 81.085569 C 81.448745 81.073825 81.44483 81.054251 81.440915 81.042507 C 81.433086 81.030762 81.425256 81.007273 81.417426 80.995529 C 81.413512 80.983785 81.401767 80.960296 81.390023 80.948552 C 81.382193 80.932892 81.370449 80.913318 81.358705 80.897659 C 81.34696 80.885915 81.331301 80.862426 81.319557 80.850682 C 81.307812 80.835023 81.288238 80.811534 81.276494 80.79979 C 81.260835 80.78413 81.241261 80.764556 81.225602 80.748897 C 81.213857 80.737153 81.190369 80.717579 81.174709 80.70192 C 81.15905 80.690175 81.135562 80.670602 81.119902 80.654942 C 81.104243 80.643198 81.07684 80.623624 81.061181 80.61188 C 81.045521 80.600135 81.018118 80.580561 81.002459 80.568817 C 80.982885 80.557073 80.955481 80.541413 80.939822 80.529669 C 80.924163 80.517925 80.896759 80.502266 80.877185 80.490521 C 80.861526 80.482692 80.834123 80.467032 80.814549 80.459203 C 80.79889 80.451373 80.771486 80.435714 80.755827 80.427885 C 80.740168 80.420055 80.712764 80.408311 80.697105 80.400481 C 80.681446 80.396566 80.654043 80.384822 80.638383 80.380907 C 80.622724 80.373078 80.599235 80.365248 80.583576 80.361333 C 80.567917 80.357418 80.544428 80.349589 80.528769 80.345674 C 80.517025 80.341759 80.493536 80.337844 80.481792 80.33393 C 80.470047 80.33393 80.446559 80.330015 80.434814 80.3261 C 80.42307 80.3261 80.403496 80.3261 80.395666 80.3261 C 80.383922 80.3261 80.368263 80.3261 80.356519 80.3261 C 80.348689 80.3261 80.33303 80.330015 80.3252 80.330015 C 80.317371 80.330015 80.305626 80.33393 80.297797 80.337844 C 80.289967 80.341759 80.278223 80.345674 80.274308 80.349589 C 80.270393 80.353504 80.262564 80.361333 80.254734 80.365248 C 80.250819 80.369163 80.246904 80.376992 80.24299 80.380907 C 80.239075 80.384822 80.239075 80.396566 80.23516 80.400481 C 80.23516 80.408311 80.231245 80.41614 80.231245 80.42397 C 80.231245 80.431799 80.23516 80.439629 80.23516 80.447459 C 80.23516 80.455288 80.239075 80.467032 80.239075 80.474862 C 80.24299 80.482692 80.246904 80.494436 80.250819 80.502266 C 80.254734 80.510095 80.262564 80.52184 80.266478 80.529669 C 80.270393 80.537499 80.278223 80.553158 80.286052 80.560987 C 80.289967 80.568817 80.301711 80.580561 80.305626 80.588391 C 80.313456 80.596221 80.3252 80.61188 80.33303 80.619709 C 80.340859 80.627539 80.352604 80.639283 80.360433 80.651028 C 80.368263 80.658857 80.383922 80.670602 80.391752 80.678431 C 80.399581 80.686261 80.41524 80.698005 80.42307 80.705835 C 80.434814 80.713664 80.450473 80.725409 80.458303 80.733238 C 80.470047 80.741068 80.485707 80.752812 80.493536 80.760642 C 80.505281 80.768471 80.52094 80.780216 80.532684 80.78413 C 80.540514 80.79196 80.560088 80.803704 80.567917 80.807619 C 80.579662 80.815449 80.595321 80.823278 80.607065 80.831108 C 80.614895 80.835023 80.634469 80.842852 80.642298 80.850682 C 80.654043 80.854597 80.669702 80.862426 80.681446 80.866341 C 80.689276 80.870256 80.704935 80.878085 80.716679 80.882 C 80.724509 80.885915 80.740168 80.88983 80.747997 80.893745 C 80.759742 80.897659 80.771486 80.901574 80.783231 80.901574 C 80.79106 80.905489 80.802805 80.909404 80.810634 80.909404 C 80.822378 80.913318 80.834123 80.913318 80.841952 80.917233 C 80.849782 80.917233 80.861526 80.917233 80.865441 80.917233 C 80.873271 80.921148 80.885015 80.921148 80.88893 80.921148 C 80.896759 80.917233 80.904589 80.917233 80.912419 80.917233 C 80.916333 80.917233 80.924163 80.913318 80.928078 80.913318 C 80.931993 80.913318 80.939822 80.909404 80.943737 80.905489 C 80.947652 80.905489 80.951566 80.901574 80.955481 80.897659 C 80.959396 80.897659 80.963311 80.893745 80.967226 80.88983 C 80.967226 80.885915 80.97114 80.882 80.97114 80.878085 C 80.97114 80.874171 80.975055 80.870256 80.975055 80.866341 C 80.975055 80.862426 80.975055 80.854597 80.975055 80.850682 C 80.975055 80.846767 80.97114 80.838937 80.97114 80.835023 C 80.97114 80.831108 80.967226 80.823278 80.967226 80.819364 C 80.963311 80.815449 80.959396 80.807619 80.959396 80.803704 C 80.955481 80.795875 80.951566 80.788045 80.947652 80.78413 C 80.943737 80.780216 80.939822 80.772386 80.935907 80.764556 C 80.931993 80.760642 80.924163 80.752812 80.920248 80.748897 C 80.916333 80.744983 80.908504 80.737153 80.904589 80.729323 C 80.900674 80.725409 80.892845 80.717579 80.885015 80.713664 C 80.8811 80.705835 80.873271 80.698005 80.865441 80.69409 C 80.861526 80.690175 80.853697 80.682346 80.845867 80.678431 C 80.841952 80.674516 80.830208 80.666687 80.826293 80.662772 C 80.818464 80.658857 80.810634 80.651028 80.802805 80.647113 C 80.79889 80.643198 80.787145 80.635368 80.779316 80.631454 C 80.775401 80.627539 80.763657 80.623624 80.759742 80.619709 C 80.751912 80.615794 80.740168 80.607965 80.736253 80.607965 C 80.728424 80.60405 80.720594 80.600135 80.712764 80.596221 C 80.70885 80.592306 80.697105 80.588391 80.69319 80.584476 C 80.685361 80.584476 80.677531 80.580561 80.669702 80.576647 C 80.665787 80.576647 80.657957 80.572732 80.650128 80.572732 C 80.646213 80.568817 80.638383 80.568817 80.630554 80.564902 C 80.626639 80.564902 80.618809 80.564902 80.614895 80.560987 C 80.61098 80.560987 80.60315 80.560987 80.599235 80.560987 C 80.591406 80.560987 80.587491 80.560987 80.583576 80.560987 C 80.579662 80.560987 80.571832 80.560987 80.567917 80.560987 C 80.567917 80.560987 80.560088 80.560987 80.556173 80.560987 C 80.556173 80.560987 80.552258 80.564902 80.548343 80.564902 C 80.544428 80.564902 80.540514 80.568817 80.540514 80.568817 C 80.536599 80.572732 80.536599 80.572732 80.532684 80.576647 C 80.532684 80.576647 80.528769 80.580561 80.528769 80.580561 C 80.528769 80.584476 80.528769 80.588391 80.524854 80.588391 C 80.524854 80.592306 80.524854 80.596221 80.524854 80.596221 C 80.524854 80.600135 80.524854 80.60405 80.528769 80.607965 C 80.528769 80.607965 80.528769 80.61188 80.528769 80.615794 C 80.528769 80.619709 80.532684 80.623624 80.532684 80.627539 C 80.536599 80.627539 80.536599 80.635368 80.540514 80.635368 C 80.540514 80.639283 80.544428 80.643198 80.544428 80.647113 C 80.548343 80.651028 80.552258 80.654942 80.556173 80.658857 C 80.556173 80.662772 80.560088 80.666687 80.564002 80.670602 C 80.567917 80.670602 80.571832 80.678431 80.575747 80.678431 C 80.579662 80.682346 80.583576 80.686261 80.587491 80.690175 C 80.587491 80.69409 80.595321 80.698005 80.599235 80.70192 C 80.60315 80.705835 80.607065 80.709749 80.61098 80.709749 C 80.614895 80.713664 80.618809 80.717579 80.622724 80.721494 C 80.626639 80.721494 80.634469 80.725409 80.638383 80.729323 C 80.642298 80.733238 80.646213 80.737153 80.650128 80.737153 C 80.654043 80.741068 80.661872 80.744983 80.665787 80.744983 C 80.669702 80.748897 80.673616 80.748897 80.677531 80.752812 C 80.681446 80.752812 80.689276 80.756727 80.69319 80.756727 C 80.697105 80.760642 80.70102 80.760642 80.704935 80.764556 C 80.70885 80.764556 80.712764 80.768471 80.716679 80.768471 C 80.720594 80.768471 80.724509 80.772386 80.728424 80.772386 C 80.732338 80.772386 80.736253 80.772386 80.740168 80.776301 C 80.744083 80.776301 80.747997 80.776301 80.751912 80.776301 C 80.751912 80.776301 80.755827 80.776301 80.759742 80.776301 C 80.763657 80.776301 80.767571 80.776301 80.767571 80.776301 C 80.771486 80.776301 80.775401 80.776301 80.775401 80.776301 C 80.775401 80.776301 80.779316 80.776301 80.783231 80.776301 C 80.783231 80.772386 80.787145 80.772386 80.787145 80.772386 C 80.787145 80.772386 80.79106 80.768471 80.79106 80.768471 C 80.79106 80.768471 80.794975 80.768471 80.794975 80.764556 C 80.794975 80.764556 80.794975 80.760642 80.794975 80.760642 C 80.794975 80.760642 80.79889 80.756727 80.79889 80.756727 C 80.79889 80.756727 80.79889 80.752812 80.79889 80.752812 C 80.79889 80.748897 80.794975 80.748897 80.794975 80.744983 C 80.794975 80.744983 80.794975 80.741068 80.794975 80.741068 C 80.794975 80.737153 80.79106 80.733238 80.79106 80.733238 C 80.79106 80.733238 80.787145 80.729323 80.787145 80.725409 C 80.787145 80.725409 80.783231 80.721494 80.783231 80.721494 C 80.779316 80.717579 80.779316 80.713664 80.775401 80.713664 C 80.775401 80.709749 80.771486 80.709749 80.771486 80.705835 C 80.767571 80.705835 80.767571 80.70192 80.763657 80.70192 C 80.763657 80.698005 80.759742 80.69409 80.755827 80.69409 C 80.755827 80.69409 80.751912 80.690175 80.747997 80.686261 C 80.747997 80.686261 80.744083 80.682346 80.740168 80.682346 C 80.740168 80.682346 80.736253 80.678431 80.732338 80.674516 C 80.732338 80.674516 80.728424 80.674516 80.724509 80.670602 C 80.724509 80.670602 80.720594 80.666687 80.716679 80.666687 C 80.712764 80.666687 80.712764 80.662772 80.70885 80.662772 C 80.704935 80.662772 80.70102 80.658857 80.70102 80.658857 C 80.697105 80.658857 80.69319 80.654942 80.69319 80.654942 C 80.689276 80.654942 80.685361 80.651028 80.685361 80.651028 C 80.681446 80.651028 80.681446 80.651028 80.677531 80.651028 C 80.677531 80.647113 80.673616 80.647113 80.669702 80.647113 C 80.669702 80.647113 80.665787 80.647113 80.665787 80.647113 C 80.661872 80.647113 80.661872 80.647113 80.657957 80.647113 C 80.657957 80.647113 80.654043 80.647113 80.654043 80.647113 C 80.650128 80.647113 80.650128 80.647113 80.650128 80.647113 C 80.646213 80.647113 80.646213 80.647113 80.646213 80.647113 C 80.642298 80.647113 80.642298 80.647113 80.642298 80.647113 C 80.638383 80.647113 80.638383 80.647113 80.638383 80.651028 C 80.638383 80.651028 80.634469 80.651028 80.634469 80.651028 C 80.634469 80.651028 80.634469 80.654942 80.634469 80.654942 C 80.634469 80.658857 80.634469 80.658857 80.634469 80.658857 C 80.634469 80.658857 80.634469 80.662772 80.634469 80.662772 C 80.634469 80.662772 80.634469 80.666687 80.634469 80.666687 C 80.634469 80.666687 80.634469 80.670602 80.634469 80.670602 C 80.638383 80.670602 80.638383 80.674516 80.638383 80.674516 C 80.638383 80.674516 80.642298 80.678431 80.642298 80.678431 C 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76.168594 76.097228 76.192082 C 75.995443 76.215571 75.842766 76.262549 75.760556 76.293867 C 75.67443 76.3291 75.549157 76.395651 75.482606 76.438714 C 75.416054 76.485692 75.318184 76.567902 75.267292 76.622709 C 75.2164 76.677516 75.145934 76.775386 75.110701 76.838023 C 75.079382 76.904574 75.03632 77.018103 75.02066 77.088569 C 75.001087 77.16295 74.989342 77.288223 74.985427 77.366519 C 74.985427 77.44873 74.997172 77.585747 75.012831 77.671873 C 75.02849 77.757998 75.063723 77.902845 75.095041 77.9968 C 75.12636 78.08684 75.185082 78.239517 75.232059 78.337387 C 75.275122 78.431342 75.361247 78.591848 75.416054 78.689718 C 75.474776 78.791503 75.580475 78.952009 75.650942 79.053793 C 75.721408 79.155578 75.846681 79.319999 75.928892 79.421784 C 76.011102 79.523568 76.155949 79.687989 76.24599 79.789774 C 76.339945 79.895473 76.496536 80.05598 76.598321 80.157764 C 76.700105 80.259549 76.872356 80.420055 76.98197 80.517925 C 77.091584 80.619709 77.271664 80.776301 77.389108 80.870256 C 77.506552 80.968126 77.694462 81.116888 77.81582 81.210842 C 77.937179 81.300883 78.136833 81.44573 78.262106 81.531855 C 78.383465 81.617981 78.587034 81.754998 78.716222 81.837209 C 78.841495 81.915505 79.048979 82.044693 79.174252 82.119074 C 79.30344 82.193455 79.507009 82.310898 79.636197 82.37745 C 79.761471 82.447916 79.96504 82.553615 80.094228 82.612337 C 80.219501 82.674974 80.419155 82.768929 80.540514 82.819821 C 80.661872 82.874628 80.857612 82.952924 80.97897 82.999901 C 81.096414 83.046879 81.284324 83.11343 81.397852 83.152578 C 81.511381 83.187811 81.687547 83.242618 81.797161 83.273937 C 81.90286 83.30134 82.071196 83.344403 82.172981 83.363977 C 82.270851 83.387465 82.427442 83.414869 82.521397 83.426613 C 82.611437 83.438358 82.756284 83.454017 82.838495 83.457932 C 82.92462 83.461846 83.053808 83.465761 83.128189 83.461846 C 83.20257 83.457932 83.316099 83.446187 83.382651 83.438358 C 83.445287 83.426613 83.543157 83.403125 83.601879 83.387465 C 83.656686 83.367891 83.738897 83.332658 83.781959 83.30917 C 83.828937 83.285681 83.895488 83.238703 83.930721 83.207385 C 83.965954 83.176067 84.012932 83.12126 84.036421 83.086027 C 84.063824 83.046879 84.095142 82.984242 84.110802 82.941179 C 84.122546 82.898117 84.138205 82.827651 84.146035 82.780673 C 84.149949 82.733696 84.149949 82.651485 84.14212 82.600593 C 84.138205 82.5497 84.122546 82.46749 84.106887 82.412683 C 84.091228 82.357876 84.063824 82.267836 84.040335 82.209114 C 84.016847 82.150392 83.969869 82.056437 83.938551 81.997715 C 83.907233 81.938993 83.848511 81.841124 83.809363 81.778487 C 83.7663 81.719765 83.695834 81.617981 83.648856 81.559259 C 83.601879 81.496622 83.523583 81.394838 83.468776 81.336116 C 83.413969 81.273479 83.323929 81.171695 83.265207 81.112973 C 83.20257 81.050336 83.104701 80.952466 83.038149 80.88983 C 82.975513 80.831108 82.869813 80.733238 82.799347 80.674516 C 82.728881 80.615794 82.615352 80.525754 82.544886 80.467032 C 82.47442 80.412225 82.353061 80.322185 82.27868 80.267378 C 82.204299 80.216486 82.082941 80.130361 82.00856 80.079468 C 81.930264 80.028576 81.808905 79.95028 81.73061 79.903303 C 81.652314 79.856325 81.530955 79.781944 81.45266 79.738882 C 81.374364 79.695819 81.24909 79.629268 81.174709 79.59012 C 81.096414 79.554887 80.975055 79.49225 80.900674 79.460932 C 80.826293 79.425698 80.704935 79.374806 80.634469 79.343488 C 80.560088 79.31217 80.446559 79.269107 80.376092 79.245618 C 80.305626 79.218215 80.196012 79.182982 80.129461 79.163408 C 80.062909 79.143834 79.95721 79.11643 79.894573 79.100771 C 79.831937 79.085112 79.734067 79.065538 79.675345 79.057708 C 79.620538 79.045964 79.530498 79.03422 79.475691 79.030305 C 79.424799 79.02639 79.342588 79.01856 79.295611 79.01856 C 79.244718 79.01856 79.174252 79.022475 79.131189 79.02639 C 79.088127 79.03422 79.02549 79.042049 78.990257 79.053793 C 78.955024 79.061623 78.900217 79.081197 78.868899 79.092941 C 78.841495 79.104686 78.798432 79.132089 78.771029 79.147748 C 78.74754 79.167322 78.716222 79.194726 78.696648 79.218215 C 78.680989 79.237789 78.6575 79.273022 78.645756 79.300425 C 78.634011 79.323914 78.618352 79.363062 78.614437 79.390465 C 78.610522 79.417869 78.606608 79.464846 78.606608 79.496165 C 78.606608 79.527483 78.614437 79.57446 78.618352 79.609694 C 78.626182 79.641012 78.641841 79.691904 78.653585 79.727137 C 78.665329 79.76237 78.688818 79.817177 78.704477 79.852411 C 78.724051 79.887644 78.75537 79.946365 78.778858 79.985513 C 78.798432 80.020746 78.83758 80.079468 78.864984 80.118616 C 78.892387 80.153849 78.939365 80.216486 78.970683 80.251719 C 79.002001 80.290867 79.052894 80.349589 79.088127 80.388737 C 79.12336 80.42397 79.182082 80.482692 79.217315 80.52184 C 79.256463 80.557073 79.319099 80.615794 79.362162 80.651028 C 79.40131 80.690175 79.467861 80.744983 79.510924 80.780216 C 79.553987 80.815449 79.624453 80.870256 79.667516 80.905489 C 79.714493 80.936807 79.784959 80.987699 79.831937 81.019018 C 79.878914 81.050336 79.953295 81.101228 80.000273 81.128632 C 80.043335 81.15995 80.121631 81.206928 80.168609 81.234331 C 80.215586 81.261735 80.289967 81.300883 80.336945 81.324371 C 80.383922 81.351775 80.458303 81.387008 80.501366 81.410497 C 80.548343 81.433985 80.622724 81.465304 80.665787 81.484878 C 80.70885 81.504452 80.783231 81.531855 80.826293 81.547514 C 80.869356 81.563174 80.935907 81.590577 80.975055 81.602321 C 81.018118 81.614066 81.080755 81.63364 81.119902 81.645384 C 81.15905 81.653214 81.221687 81.668873 81.25692 81.676702 C 81.292153 81.684532 81.350875 81.692362 81.382193 81.696276 C 81.417426 81.700191 81.468319 81.704106 81.499637 81.708021 C 81.527041 81.708021 81.574018 81.708021 81.601422 81.708021 C 81.628825 81.704106 81.667973 81.700191 81.691462 81.696276 C 81.71495 81.692362 81.750184 81.680617 81.769757 81.676702 C 81.789331 81.668873 81.82065 81.657128 81.836309 81.645384 C 81.851968 81.637555 81.875457 81.617981 81.887201 81.606236 C 81.898946 81.594492 81.914605 81.574918 81.922434 81.559259 C 81.930264 81.547514 81.942008 81.524026 81.945923 81.508366 C 81.953753 81.492707 81.957667 81.465304 81.957667 81.44573 C 81.957667 81.430071 81.957667 81.398752 81.953753 81.383093 C 81.953753 81.363519 81.945923 81.332201 81.942008 81.312627 C 81.934179 81.289138 81.922434 81.25782 81.914605 81.234331 C 81.90286 81.214757 81.887201 81.179524 81.875457 81.15995 C 81.863712 81.136461 81.840224 81.101228 81.824565 81.07774 C 81.808905 81.054251 81.785417 81.019018 81.765843 80.995529 C 81.750184 80.975955 81.718865 80.936807 81.699291 80.917233 C 81.679717 80.893745 81.644484 80.854597 81.620995 80.835023 C 81.601422 80.811534 81.562274 80.776301 81.538785 80.752812 C 81.515296 80.733238 81.476148 80.698005 81.448745 80.674516 C 81.425256 80.654942 81.382193 80.619709 81.35479 80.600135 C 81.331301 80.580561 81.284324 80.545328 81.25692 80.525754 C 81.229517 80.50618 81.186454 80.478777 81.15905 80.459203 C 81.131647 80.439629 81.084669 80.412225 81.057266 80.396566 C 81.029862 80.376992 80.982885 80.353504 80.955481 80.337844 C 80.928078 80.322185 80.8811 80.298697 80.853697 80.283037 C 80.826293 80.267378 80.779316 80.247804 80.751912 80.23606 C 80.724509 80.224316 80.681446 80.204742 80.657957 80.192997 C 80.630554 80.185168 80.587491 80.169508 80.564002 80.157764 C 80.536599 80.149935 80.497451 80.13819 80.473962 80.130361 C 80.450473 80.122531 80.411326 80.114701 80.387837 80.106872 C 80.364348 80.102957 80.329115 80.095127 80.309541 80.095127 C 80.289967 80.091213 80.254734 80.087298 80.239075 80.083383 C 80.219501 80.083383 80.188183 80.083383 80.172523 80.083383 C 80.152949 80.083383 80.129461 80.083383 80.113802 80.087298 C 80.098142 80.087298 80.074654 80.091213 80.062909 80.095127 C 80.051165 80.099042 80.031591 80.106872 80.019847 80.110787 C 80.012017 80.118616 79.996358 80.126446 79.984614 80.134275 C 79.976784 80.13819 79.96504 80.149935 79.961125 80.157764 C 79.953295 80.165594 79.945466 80.181253 79.941551 80.189082 C 79.937636 80.196912 79.933721 80.212571 79.933721 80.224316 C 79.929806 80.23606 79.929806 80.251719 79.929806 80.263463 C 79.929806 80.275208 79.933721 80.290867 79.937636 80.302611 C 79.941551 80.31827 79.945466 80.337844 79.94938 80.349589 C 79.95721 80.361333 79.96504 80.380907 79.972869 80.396566 C 79.976784 80.408311 79.988528 80.427885 80.000273 80.443544 C 80.008102 80.455288 80.019847 80.478777 80.031591 80.490521 C 80.043335 80.50618 80.058995 80.525754 80.070739 80.541413 C 80.082483 80.553158 80.102057 80.576647 80.113802 80.592306 C 80.129461 80.60405 80.149035 80.627539 80.164694 80.639283 C 80.176438 80.654942 80.199927 80.674516 80.215586 80.686261 C 80.231245 80.70192 80.254734 80.721494 80.270393 80.733238 C 80.286052 80.744983 80.313456 80.768471 80.329115 80.780216 C 80.344774 80.79196 80.372178 80.811534 80.387837 80.819364 C 80.407411 80.831108 80.434814 80.850682 80.450473 80.862426 C 80.466133 80.870256 80.493536 80.88983 80.51311 80.897659 C 80.528769 80.909404 80.556173 80.921148 80.575747 80.932892 C 80.591406 80.940722 80.618809 80.952466 80.634469 80.960296 C 80.650128 80.968126 80.677531 80.97987 80.69319 80.987699 C 80.70885 80.995529 80.736253 81.003359 80.751912 81.011188 C 80.767571 81.015103 80.79106 81.026847 80.806719 81.030762 C 80.822378 81.034677 80.845867 81.042507 80.861526 81.046421 C 80.873271 81.050336 80.896759 81.054251 80.908504 81.054251 C 80.920248 81.058166 80.943737 81.06208 80.955481 81.06208 C 80.967226 81.065995 80.982885 81.065995 80.994629 81.065995 C 81.006374 81.065995 81.022033 81.065995 81.033777 81.065995 C 81.041607 81.065995 81.057266 81.06208 81.065095 81.06208 C 81.072925 81.058166 81.084669 81.054251 81.092499 81.054251 C 81.100328 81.050336 81.112073 81.046421 81.115988 81.042507 C 81.119902 81.038592 81.127732 81.030762 81.135562 81.026847 C 81.139476 81.022933 81.143391 81.015103 81.147306 81.007273 C 81.151221 81.003359 81.151221 80.995529 81.155136 80.987699 C 81.155136 80.983785 81.15905 80.97204 81.15905 80.968126 C 81.15905 80.960296 81.155136 80.948552 81.155136 80.940722 C 81.155136 80.936807 81.151221 80.925063 81.151221 80.917233 C 81.147306 80.909404 81.143391 80.897659 81.139476 80.88983 C 81.135562 80.882 81.127732 80.866341 81.123817 80.858511 C 81.119902 80.850682 81.112073 80.838937 81.104243 80.831108 C 81.100328 80.823278 81.088584 80.807619 81.084669 80.79979 C 81.07684 80.79196 81.065095 80.780216 81.057266 80.772386 C 81.049436 80.760642 81.037692 80.748897 81.029862 80.741068 C 81.022033 80.733238 81.006374 80.721494 80.998544 80.709749 C 80.990714 80.70192 80.975055 80.690175 80.967226 80.682346 C 80.955481 80.674516 80.939822 80.662772 80.931993 80.654942 C 80.920248 80.647113 80.904589 80.635368 80.896759 80.627539 C 80.885015 80.623624 80.869356 80.61188 80.857612 80.60405 C 80.849782 80.596221 80.830208 80.588391 80.822378 80.580561 C 80.810634 80.576647 80.794975 80.564902 80.783231 80.560987 C 80.775401 80.553158 80.755827 80.545328 80.747997 80.541413 C 80.736253 80.537499 80.720594 80.529669 80.70885 80.525754 C 80.70102 80.517925 80.685361 80.51401 80.673616 80.510095 C 80.665787 80.50618 80.650128 80.498351 80.642298 80.498351 C 80.630554 80.494436 80.618809 80.490521 80.607065 80.486606 C 80.599235 80.482692 80.587491 80.482692 80.579662 80.478777 C 80.567917 80.478777 80.556173 80.474862 80.548343 80.474862 C 80.540514 80.474862 80.528769 80.470947 80.524854 80.470947 C 80.517025 80.470947 80.505281 80.470947 80.501366 80.470947 C 80.493536 80.470947 80.485707 80.470947 80.477877 80.470947 C 80.473962 80.474862 80.466133 80.474862 80.462218 80.474862 C 80.458303 80.478777 80.450473 80.482692 80.446559 80.482692 C 80.442644 80.486606 80.438729 80.486606 80.434814 80.490521 C 80.4309 80.494436 80.426985 80.498351 80.42307 80.502266 C 80.42307 80.502266 80.419155 80.510095 80.419155 80.51401 C 80.419155 80.51401 80.41524 80.52184 80.41524 80.525754 C 80.41524 80.529669 80.41524 80.533584 80.41524 80.537499 C 80.41524 80.545328 80.419155 80.549243 80.419155 80.553158 C 80.419155 80.560987 80.42307 80.564902 80.42307 80.572732 C 80.426985 80.576647 80.4309 80.584476 80.4309 80.588391 C 80.434814 80.592306 80.438729 80.600135 80.442644 80.60405 C 80.446559 80.61188 80.450473 80.619709 80.454388 80.623624 C 80.458303 80.627539 80.466133 80.635368 80.470047 80.643198 C 80.473962 80.647113 80.481792 80.654942 80.485707 80.658857 C 80.489621 80.666687 80.497451 80.674516 80.505281 80.678431 C 80.509195 80.682346 80.517025 80.690175 80.524854 80.69409 C 80.528769 80.70192 80.536599 80.705835 80.544428 80.713664 C 80.548343 80.717579 80.560088 80.725409 80.564002 80.729323 C 80.571832 80.733238 80.579662 80.741068 80.587491 80.744983 C 80.591406 80.748897 80.60315 80.752812 80.61098 80.756727 C 80.614895 80.760642 80.626639 80.768471 80.630554 80.772386 C 80.638383 80.776301 80.650128 80.780216 80.654043 80.78413 C 80.661872 80.788045 80.669702 80.79196 80.677531 80.795875 C 80.681446 80.795875 80.69319 80.79979 80.697105 80.803704 C 80.704935 80.807619 80.712764 80.811534 80.720594 80.811534 C 80.724509 80.815449 80.732338 80.815449 80.740168 80.819364 C 80.744083 80.819364 80.751912 80.823278 80.759742 80.823278 C 80.763657 80.823278 80.771486 80.827193 80.775401 80.827193 C 80.779316 80.827193 80.787145 80.831108 80.79106 80.831108 C 80.79889 80.831108 80.802805 80.831108 80.806719 80.831108 C 80.810634 80.831108 80.818464 80.831108 80.822378 80.831108 C 80.822378 80.831108 80.830208 80.827193 80.834123 80.827193 C 80.834123 80.827193 80.838038 80.827193 80.841952 80.823278 C 80.845867 80.823278 80.849782 80.823278 80.849782 80.819364 C 80.853697 80.819364 80.853697 80.815449 80.857612 80.815449 C 80.857612 80.815449 80.861526 80.811534 80.861526 80.807619 C 80.861526 80.807619 80.861526 80.803704 80.865441 80.79979 C 80.865441 80.79979 80.865441 80.795875 80.865441 80.79196 C 80.865441 80.79196 80.865441 80.788045 80.861526 80.78413 C 80.861526 80.780216 80.861526 80.776301 80.861526 80.772386 C 80.861526 80.772386 80.857612 80.768471 80.857612 80.764556 C 80.853697 80.760642 80.853697 80.756727 80.849782 80.752812 C 80.849782 80.748897 80.845867 80.744983 80.845867 80.741068 C 80.841952 80.741068 80.838038 80.733238 80.834123 80.733238 C 80.834123 80.729323 80.830208 80.725409 80.826293 80.721494 C 80.822378 80.717579 80.818464 80.713664 80.814549 80.709749 C 80.810634 80.705835 80.806719 80.70192 80.802805 80.698005 C 80.802805 80.698005 80.794975 80.690175 80.79106 80.690175 C 80.787145 80.686261 80.783231 80.682346 80.779316 80.678431 C 80.775401 80.674516 80.771486 80.670602 80.767571 80.670602 C 80.763657 80.666687 80.755827 80.662772 80.751912 80.658857 C 80.747997 80.658857 80.744083 80.654942 80.740168 80.651028 C 80.736253 80.651028 80.728424 80.647113 80.724509 80.643198 C 80.720594 80.643198 80.716679 80.639283 80.712764 80.639283 C 80.70885 80.635368 80.70102 80.631454 80.697105 80.631454 C 80.69319 80.631454 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C 80.747997 80.709749 80.747997 80.709749 80.744083 80.709749 C 80.744083 80.705835 80.744083 80.705835 80.744083 80.705835 C 80.740168 80.70192 80.740168 80.70192 80.740168 80.698005 C 80.736253 80.698005 80.736253 80.698005 80.736253 80.69409 C 80.732338 80.69409 80.732338 80.69409 80.728424 80.69409 C 80.728424 80.690175 80.724509 80.690175 80.724509 80.690175 C 80.724509 80.686261 80.720594 80.686261 80.720594 80.686261 C 80.720594 80.682346 80.716679 80.682346 80.716679 80.682346 C 80.712764 80.682346 80.712764 80.678431 80.70885 80.678431 C 80.70885 80.678431 80.704935 80.678431 80.704935 80.674516 C 80.704935 80.674516 80.70102 80.674516 80.70102 80.674516 C 80.697105 80.674516 80.697105 80.670602 80.697105 80.670602 C 80.69319 80.670602 80.69319 80.670602 80.689276 80.670602 C 80.689276 80.670602 80.689276 80.666687 80.685361 80.666687 C 80.685361 80.666687 80.681446 80.666687 80.681446 80.666687 C 80.681446 80.666687 80.677531 80.666687 80.677531 80.666687 C 80.677531 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80.681446 80.70192 80.681446 80.70192 C 80.681446 80.70192 80.685361 80.70192 80.685361 80.705835 C 80.685361 80.705835 80.685361 80.705835 80.689276 80.705835 C 80.69319 80.709749 80.69319 80.709749 80.69319 80.709749 C 80.69319 80.709749 80.697105 80.709749 80.697105 80.709749 C 80.697105 80.709749 80.697105 80.709749 80.70102 80.709749 C 80.70102 80.709749 80.70102 80.713664 80.70102 80.713664 C 80.70102 80.713664 80.704935 80.713664 80.704935 80.713664 C 80.70885 80.713664 80.70885 80.713664 80.70885 80.713664 C 80.712764 80.713664 80.712764 80.713664 80.712764 80.713664 C 80.712764 80.713664 80.716679 80.713664 80.716679 80.713664 C 80.716679 80.709749 80.716679 80.709749 80.716679 80.709749 C 80.716679 80.705835 80.716679 80.705835 80.716679 80.705835 C 80.716679 80.70192 80.716679 80.70192 80.716679 80.70192 C 80.716679 80.70192 80.716679 80.70192 80.712764 80.70192 C 80.712764 80.70192 80.712764 80.698005 80.712764 80.698005 C 80.712764 80.698005 80.712764 80.698005 80.70885 80.698005 C 80.70885 80.698005 80.70885 80.69409 80.70885 80.69409 C 80.70885 80.69409 80.704935 80.69409 80.704935 80.69409 C 80.704935 80.69409 80.704935 80.690175 80.704935 80.690175 C 80.704935 80.690175 80.70102 80.690175 80.70102 80.690175 C 80.70102 80.690175 80.697105 80.690175 80.697105 80.686261 C 80.697105 80.686261 80.697105 80.686261 80.69319 80.686261 C 80.689276 80.686261 80.689276 80.686261 80.689276 80.686261 C 80.689276 80.686261 80.689276 80.682346 80.689276 80.682346 C 80.685361 80.682346 80.685361 80.682346 80.685361 80.682346 C 80.685361 80.682346 80.685361 80.682346 80.685361 80.686261 C 80.685361 80.686261 80.685361 80.686261 80.681446 80.686261 C 80.681446 80.686261 80.681446 80.690175 80.681446 80.690175 C 80.681446 80.690175 80.685361 80.690175 80.685361 80.690175 C 80.685361 80.69409 80.685361 80.69409 80.685361 80.69409 C 80.689276 80.69409 80.689276 80.698005 80.689276 80.698005 C 80.689276 80.698005 80.689276 80.698005 80.69319 80.698005 C 80.69319 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80.69409 80.69319 80.69409 80.69319 80.69409 C 80.69319 80.69409 80.697105 80.69409 80.697105 80.69409 C 80.697105 80.69409 80.69319 80.69409 80.69319 80.69409 C 80.69319 80.69409 80.697105 80.69409 80.697105 80.69409 C 80.697105 80.69409 80.69319 80.69409 80.69319 80.69409 C 80.69319 80.69409 80.697105 80.69409 80.697105 80.69409 C 80.697105 80.69409 80.69319 80.69409 80.69319 80.69409 " transform="matrix(0.997819,0,0,-0.997819,199.943766,363.186038)"/>
+</g>
+<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,65.098572%,83.921814%);stroke-opacity:1;stroke-miterlimit:10;" d="M 79.185996 52.922574 L 76.273393 51.885155 " transform="matrix(0.997819,0,0,-0.997819,199.943766,363.186038)"/>
+<path style="fill-rule:nonzero;fill:rgb(0%,65.098572%,83.921814%);fill-opacity:1;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,65.098572%,83.921814%);stroke-opacity:1;stroke-miterlimit:10;" d="M 7.368191 0.0022922 L 1.285124 2.292942 L 3.291755 0.000947547 L 1.287528 -2.294923 Z M 7.368191 0.0022922 " transform="matrix(-0.940155,0.33412,0.33412,0.940155,278.957723,310.379738)"/>
+<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,65.098572%,83.921814%);stroke-opacity:1;stroke-miterlimit:10;" d="M 126.284828 137.239309 L 129.373597 137.117951 " transform="matrix(0.997819,0,0,-0.997819,199.943766,363.186038)"/>
+<path style="fill-rule:nonzero;fill:rgb(0%,65.098572%,83.921814%);fill-opacity:1;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,65.098572%,83.921814%);stroke-opacity:1;stroke-miterlimit:10;" d="M 7.368638 0.00158306 L 1.288176 2.294367 L 3.290118 -0.00150193 L 1.287716 -2.293632 Z M 7.368638 0.00158306 " transform="matrix(0.997,0.0390646,0.0390646,-0.997,325.954184,226.244975)"/>
+<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
+ <use xlink:href="#glyph0-2" x="365.094729" y="285.22447"/>
+</g>
+<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
+ <use xlink:href="#glyph0-3" x="277.413409" y="195.712153"/>
+</g>
+</g>
+</svg>
diff --git a/Chapter9/MarkovChains.md b/Chapter9/MarkovChains.md
index 0436b59..d99e6fd 100644
--- a/Chapter9/MarkovChains.md
+++ b/Chapter9/MarkovChains.md
@@ -69,7 +69,8 @@ Suppose $P$ is a stochastic matrix. If $\vect{x}$ is a probability vector, then
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:MarkovChains:StoMatPreservesProbVect`
+:class: myproof
Note first that every entry of a stochastic matrix or a probability vector is non-negative. Consequently, every entry of $P\vect{x}$ is also non-negative.
@@ -157,7 +158,8 @@ If $P$ is a stochastic matrix, then $1$ is an eigenvalue of $P$. Moreover, there
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Prop:MarkovChains:StoMat1EV`
+:class: myproof
Showing that $1$ is an eigenvalue is a very non-trivial exercise. Showing that there is a corresponding eigenvector without negative entries would lead us too far.
@@ -224,7 +226,8 @@ If $P$ is a regular stochastic matrix, then it has a unique steady state $\vect{
:::
-:::{prf:proof}
+:::{admonition} Proof of {prf:ref}`Thm:MarkovChains:PerronFrobenius`
+:class: myproof
The proof is quite complicated and falls outside the scope of this text.
@@ -239,6 +242,7 @@ Let us consider the following toy problem, illustrated in {numref}`Figure %s <Fi
:::{figure} Images/Fig-MarkovChains-MarkovChainonNodes.svg
:name: Fig:MarkovChains:FourNodesinCircle
+:class: dark-light
The problem from {prf:ref}`Ex:MarkovChains:MarkovChainonNodes` illustrated.
@@ -290,6 +294,7 @@ What makes this Markov chain behave so weirdly is the fact that any two nodes ar
:::{figure} Images/Fig-MarkovChains-ExtraLoop.svg
:name: Fig:MarkovChains:ExtraLoop
+:class: dark-light
The problem from {prf:ref}`Ex:MarkovChains:MarkovChainonNodes` with an extra loop. Note that the outgoing arrows from node 1 now have different probabilities.
diff --git a/Chapter9/PowerMethod.md b/Chapter9/PowerMethod.md
index 2f20f9b..d4167b5 100644
--- a/Chapter9/PowerMethod.md
+++ b/Chapter9/PowerMethod.md
@@ -126,8 +126,8 @@ Moreover, suppose $\vect{x}$ is the result after a (sufficiently) large number o
::::
-
-::::{dropdown} Informal proof of {prf:ref}`Prop:Powermethod:Powermed`.
+::::{admonition} Informal proof of {prf:ref}`Prop:Powermethod:Powermed`
+:class: myproof, dropdown
For the proof we assume that the matrix is diagonalizable, to be able to use {eq}`Eq:PowerMethod:GenSol`. For an 'arbitrary' matrix the odds are very small that it has a double eigenvalue, and as long as this is not the eigenvalue with the highest modulus the conclusion of the theorem is still valid. So we assume that $\vect{v}_1, \ldots, \vect{v}_n$ is a set of $n$ linearly independent eigenvectors for $A$.
@@ -424,8 +424,8 @@ Since $0$ is not an eigenvalue, we may conclude that $A$ is invertible.
Then the power method applied to $A^{-1}$ converges (apart from the usual exceptional cases) to an eigenvector $\vect{v}_n$ for the smallest eigenvalue $\lambda_n$.
::::
-
-::::{dropdown} Proof of the inverse power method
+::::{admonition} Proof of {prf:ref}`Prop:PowerMethod:SmallestEigenvalue` ({prf:ref}`Inverse Power Method <Prop:PowerMethod:SmallestEigenvalue>`)
+:class: myproof, dropdown
We make use of the property in {numref}`Exc:EigenValues:EigenvaluesInverse` in
@@ -768,7 +768,8 @@ As mentioned, starting from a complex initial vector $\vect{z}_0$ won't work e
The best is of course to think of a way out yourself. <BR>
If you do not see such a way out, but your curiosity has been aroused, you can open the 'workaround' below.
-::::{dropdown} Workaround to get to the dominant complex eigenvalues
+::::{admonition} Workaround to get to the dominant complex eigenvalues
+:class: note, dropdown
By definition, a dominant eigenvalue must be *unique* , i.e., $|\lambda_1| > |\lambda_2| \geq \ldots \, |\lambda_n|$. <BR>
The question is, how can we get rid of the dominant eigenvalue *pair* ?
diff --git a/README.md b/README.md
index adb7446..cee4ae6 100644
--- a/README.md
+++ b/README.md
@@ -61,7 +61,7 @@ jupyter-book build --all .
:url: lines_and_planes/normal_equation_plane_origin
:fig: Images/image_shown_in_print_version.svg
:name: name_that_is_used_to_refer_to_this_figure
-:status: approved
+:class: dark-light
:title: This title is shown when you full-screen the applet
A plane through the point $P$.
diff --git a/README.md.bak b/README.md.bak
index db5ef14..adb7446 100644
--- a/README.md.bak
+++ b/README.md.bak
@@ -72,22 +72,51 @@ A plane through the point $P$.
## Parameters for an applet
-Some parameters can be set for an applet. Only the `url` and `fig` parameters are required; the rest is optional.
+Some parameters can be set for an applet. Only the `url`, `fig` and `name` parameters are required; the rest is optional. It is recommended to add a `status` to the applet, which can be `unreviewed`, `in-review` or `reviewed`.
````md
```{applet}
:url: lines_and_planes/normal_equation_plane_origin # Required url
:fig: Images/lines_and_planes/normal_equation_plane_origin.svg # Image shown in print version
-:title: hello # a string that will be shown as the title of the applet when the aplet is in fullscreen mode
:status: reviewed # default is "unreviewed". Other options are "in-review" and "reviewed"
-:width: 100% # the width of the applet
-:height: 500px # the height of the applet
-:background: #ffffff # the background color of the applet
-:autoPlay: enabled # if the applet should start playing automatically
-:isPerspectiveCamera: disabled # if the camera should be a perspective or orthographic camera
-:position: 1,1,1 # the position of the camera related to the origin. Spaces not allowed
-:enablePan: disabled # if the user can pan the camera (right mouse drag on desktop, two finger drag on mobile)
-:distance: 30 # the distance of the camera from the origin for a *perspective* camera. Distance is a linear value, *higher* is further away.
-:zoom: 30 # the distance of the camera from the origin for a *orthographic* camera. Zoom is a logarithmic value, *lower* is further away.
+:name: Fig:InnerProduct:ProjectionVectorLine
+
+A title that describes the applet
```
````
+
+### Optional parameters
+
+| Parameter | Description | Default |
+| ----------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------- | ------------ |
+| `iframe` | This parameter is added by default and set to true for each applet in this book. |
+| \ Therefore, this parameter is not configurable for this book. When using an applet in different context will change the bahaviour. | false |
+| `title` | A string that will be shown as the title of the applet when the applet is in fullscreen mode | "" |
+| `status` | The status of the applet. Can be `unreviewed`, `in-review` or `reviewed` | `unreviewed` |
+| `width` | The width of the applet in pixels | 100% |
+| `height` | The height of the applet in pixels | 400px |
+
+### Control parameters
+
+> [!WARNING]
+> Work in progress
+
+### 2D Specific parameters
+
+> [!TIP]
+> You should add split-\* before the parameter to make it apply to the right scene
+
+| Parameter | Description | Default |
+| ---------- | -------------------------------------------- | ------- |
+| position2D | The position of the applet in the 2D plane | 0,0 |
+| zoom2D | The zoom level of the applet in the 2D plane | 1 |
+
+### 3D Specific parameters
+
+> [!TIP]
+> You should add split-\* before the parameter to make it apply to the right scene
+
+| Parameter | Description | Default |
+| ---------- | -------------------------------------------- | ------- |
+| position3D | The position of the applet in the 3D plane | 0,0,0 |
+| zoom3D | The zoom level of the applet in the 3D plane | 1 |
diff --git a/_ext/applet.py b/_ext/applet.py
index b71d662..1eb6b43 100644
--- a/_ext/applet.py
+++ b/_ext/applet.py
@@ -34,9 +34,19 @@ class AppletDirective(Figure):
def run(self):
url = self.options.get("url")
fig = self.options.get("fig")
+
assert url is not None
assert fig is not None
+ iframe_class = self.options.get("class") # expect a list/string of classes
+
+ if iframe_class is None:
+ iframe_class = ""
+ elif isinstance(iframe_class, list):
+ iframe_class = " ".join(iframe_class)
+ else:
+ iframe_class = str(iframe_class)
+
self.arguments = [fig]
self.options["class"] = ["applet-print-figure"]
(figure_node,) = Figure.run(self)
@@ -58,7 +68,7 @@ class AppletDirective(Figure):
applet_html = f"""
<div class="applet" style="{style}; ">
<noscript class="loading-lazy">
- <iframe src="{full_url}" allow="fullscreen" loading="lazy" frameborder="0"></iframe>
+ <iframe class="{iframe_class}" src="{full_url}" allow="fullscreen" loading="lazy" frameborder="0"></iframe>
</noscript>
</div>
"""
diff --git a/_static/extra_css_config.css b/_static/extra_css_config.css
index 4c28017..02123aa 100644
--- a/_static/extra_css_config.css
+++ b/_static/extra_css_config.css
@@ -1,3 +1,55 @@
+:root {
+ --raspberry: rgb(165,0,52);
+ --raspberry-min: rgb(165,0,52,3%);
+ --raspberry-mid: rgb(165,0,52,20%);
+ --yellow: #FFB81C;
+ --yellow-min: #FFB81C05;
+ --yellow-mid: #FFB81C20;
+ --darkGreen: #009B77;
+ --darkGreen-min: #009B7710;
+ --darkGreen-mid: #009B7725;
+ --blue: #0076C2;
+ --blue-min: #0076C210;
+ --blue-mid: #0076C225;
+ --black: var(--pst-color-text-base);
+ --black-min: #56565605;
+ --black-mid: #56565625;
+ --orange: #EC6842;
+ --orange-min: #EC684210;
+ --orange-mid: #EC684225;
+ --cyan: #00A6D6;
+ --gray: #7d7d7d;
+ --gray-min: #7d7d7d05;
+ --gray-mid: #7d7d7d20;
+ --purple: #6f1d77;
+ --purple-min: #6f1d7705;
+ --purple-mid: #6f1d7720;
+ --pink: #EF60A3;
+ --pink-min: #EF60A310;
+ --pink-mid: #EF60A325;
+ --red: #E03C31;
+ --red-min: #E03C3110;
+ --red-mid: #E03C3125;
+ --green: #6CC24A;
+ --green-min: #6CC24A10;
+ --green-mid: #6CC24A25;
+}
+
+/* Two div's to make default any other admonition a weird colour and symbol we just do not want to use*/
+div.admonition {
+ border-color: var(--pink);
+ background-color: var(--pink-min);
+}
+
+div.admonition > .admonition-title {
+ background-color: var(--pink-mid);
+}
+
+div.admonition > .admonition-title::after {
+ color: var(--pink);
+ content: "\f6e8";
+}
+
.applet {
height: 500px;
}
@@ -23,3 +75,214 @@
display: initial;
}
}
+
+
+div.grasple p.admonition-title::after {
+ content: "\f12e";
+ color: var(--raspberry);
+}
+div.grasple p.admonition-title::before {
+ content: "";
+}
+div.grasple {
+ border-color: var(--raspberry);
+ background-color: var(--raspberry-min);
+}
+div.grasple > p.admonition-title {
+ background-color: var(--raspberry-mid);
+}
+
+div.exercise p.admonition-title::after {
+ content: "\f12e";
+ color: var(--raspberry);
+}
+div.exercise p.admonition-title::before {
+ content: "";
+}
+div.exercise {
+ border-color: var(--raspberry);
+ background-color: var(--raspberry-min);
+}
+div.exercise > p.admonition-title {
+ background-color: var(--raspberry-mid);
+}
+
+
+div.definition p.admonition-title::after {
+ content: "\f02d";
+ color: var(--blue);
+}
+div.definition {
+ border-color: var(--blue);
+ background-color: var(--blue-min);
+}
+div.definition > p.admonition-title {
+ background-color: var(--blue-mid);
+}
+
+div.theorem p.admonition-title::after,
+div.proposition p.admonition-title::after,
+div.corollary p.admonition-title::after {
+ content: "\f51b";
+ color: var(--darkGreen);
+}
+div.theorem, div.proposition, div.corollary {
+ border-color: var(--darkGreen);
+ background-color: var(--darkGreen-min);
+}
+div.theorem > p.admonition-title,
+ div.proposition > p.admonition-title,
+ div.corollary >p.admonition-title {
+ background-color: var(--darkGreen-mid);
+}
+
+div.example {
+ border-color: var(--yellow);
+ background-color: var(--yellow-min);
+}
+div.example > p.admonition-title {
+ background-color: var(--yellow-mid);
+ /* color: white; */
+}
+div.example p.admonition-title::after {
+ content: "\f002";
+ transform: scale(-1, 1);
+ color: var(--yellow);
+}
+
+div.admonition.solution {
+ border-color: var(--purple);
+ background-color: var(--purple-min);
+}
+
+div.admonition.solution > .admonition-title {
+ background-color: var(--purple-mid);
+ /* color: white; */
+}
+
+div.admonition.solution > .admonition-title::after {
+ content: "\e5a0";
+ color: var(--purple);
+}
+
+div.admonition.myproof {
+ border-color: var(--darkGreen);
+ background-color: var(--gray-min);
+}
+
+div.admonition.myproof > .admonition-title {
+ background-color: var(--gray-mid);
+}
+
+div.admonition.myproof > .admonition-title::after {
+ content: "\f51c";
+ color: var(--gray);
+}
+
+div.algorithm > .admonition-title::after {
+ content: "\f051";
+ color: var(--black);
+}
+
+div.algorithm {
+ border-color: var(--black);
+ background-color: var(--black-min);
+}
+
+div.algorithm > .admonition-title {
+ background-color: var(--black-mid);
+}
+
+div.admonition.bonus {
+ border-color: var(--gray);
+ background-color: var(--gray-min);
+}
+
+div.admonition.bonus > .admonition-title {
+ background-color: var(--gray-mid);
+}
+
+div.admonition.bonus > .admonition-title::after {
+ content: "\f06b";
+ color: var(--gray);
+}
+
+div.admonition.caution, div.admonition.warning, div.admonition.attention, div.admonition.important {
+ border-color: var(--orange);
+ background-color: var(--orange-min);
+}
+
+div.admonition.caution > .admonition-title,
+ div.admonition.warning > .admonition-title,
+ div.admonition.attention > .admonition-title,
+ div.admonition.important > .admonition-title {
+ background-color: var(--orange-mid);
+}
+
+div.admonition.caution > .admonition-title::after,
+ div.admonition.warning > .admonition-title::after,
+ div.admonition.attention > .admonition-title::after,
+ div.admonition.important > .admonition-title::after {
+ color: var(--orange);
+}
+
+div.admonition.danger, div.admonition.error {
+ border-color: var(--red);
+ background-color: var(--red-min);
+}
+
+div.admonition.danger > .admonition-title,
+ div.admonition.error > .admonition-title {
+ background-color: var(--red-mid);
+}
+
+div.admonition.danger > .admonition-title::after,
+ div.admonition.error > .admonition-title::after {
+ color: var(--red);
+}
+
+div.admonition.hint, div.admonition.note, div.admonition.seealso, div.admonition.tip, div.admonition.remark, div.admonition.observation {
+ border-color: var(--green);
+ background-color: var(--green-min);
+}
+
+div.admonition.hint > .admonition-title,
+ div.admonition.note > .admonition-title,
+ div.admonition.seealso > .admonition-title,
+ div.admonition.tip > .admonition-title,
+ div.admonition.remark > .admonition-title,
+ div.admonition.observation > .admonition-title {
+ background-color: var(--green-mid);
+}
+
+div.admonition.hint > .admonition-title::after,
+ div.admonition.note > .admonition-title::after,
+ div.admonition.seealso > .admonition-title::after,
+ div.admonition.tip > .admonition-title::after,
+ div.admonition.remark > .admonition-title::after,
+ div.admonition.observation > .admonition-title::after {
+ color: var(--green);
+ content: "\f249";
+}
+
+html[data-theme="dark"] .dark-light {
+ filter: invert(1) hue-rotate(180deg) saturate(1.5);
+}
+
+html[data-theme="dark"] .dark-light-same {
+ background-color: transparent !important;
+}
+
+.red {color: var(--red)}
+.blue {color: var(--blue)}
+.green {color: var(--green)}
+
+html[data-theme="dark"] .red {
+ filter: invert(1) hue-rotate(180deg) saturate(1.5);
+}
+html[data-theme="dark"] .blue {
+ filter: invert(1) hue-rotate(180deg) saturate(1.5);
+}
+html[data-theme="dark"] .green {
+ filter: invert(1) hue-rotate(180deg) saturate(1.5);
+}
\ No newline at end of file
diff --git a/_toc.yml b/_toc.yml
index cad5186..b765505 100644
--- a/_toc.yml
+++ b/_toc.yml
@@ -70,7 +70,6 @@ parts:
numbered: True
chapters:
- file: 'Appendices/ComplexNumbers.md'
- # - file: 'Appendices/SVDProof.md'
- file: 'Appendices/InverseMatrixTheorem.md'
- caption: Colophon
numbered: False
diff --git a/add_darklight_grasple.py b/add_darklight_grasple.py
new file mode 100644
index 0000000..6e5badd
--- /dev/null
+++ b/add_darklight_grasple.py
@@ -0,0 +1,46 @@
+#!/usr/bin/env python
+# coding: utf-8
+
+# In[3]:
+
+
+import os
+import numpy as np
+files = []
+ignore = ['_build','_static','_ext','sphinx-grasple']
+for dirpath, dirnames, filenames in os.walk('.'):
+ if np.all([dirpath.find(x)<0 for x in ignore]):
+ for filename in filenames:
+ if filename.endswith('.md'):
+ files.append(os.path.join(dirpath, filename.replace('.md','')))
+
+for file in files:
+ # load the relevant notebook as a text file
+ with open(file+".md") as f:
+ new_data = None
+ if file not in ['.\\README']:
+ data = f.read()
+ datalines = np.array(data.split('\n'))
+ new_datalines = []
+ for li,line in enumerate(datalines):
+ # add line to new lines
+ new_datalines.append(line)
+ grasple = line.find('{grasple}')
+ if grasple>-1:
+ # grasple found
+ # check if next line is already dark-light
+ nextline = datalines[li+1]
+ if nextline!=':iframeclass: dark-light':
+ # add the line
+ new_datalines.append(':iframeclass: dark-light')
+ new_data = '\n'.join(new_datalines)
+ if not(new_data is None):
+ with open(file+".md",'w') as f:
+ f.write(new_data)
+
+
+# In[ ]:
+
+
+
+
diff --git a/install.sh b/install.sh
index 8a4a643..a28e31b 100644
--- a/install.sh
+++ b/install.sh
@@ -1,16 +1,8 @@
#!/bin/bash
echo "Installing dependencies..."
-pip install -r requirements.txt
-
-echo "Installing the new sphinx-grasple package"
-git clone https://github.com/dbalague/sphinx-grasple
-
-cd sphinx-grasple
-python3 setup.py install
-cd ..
-rm -rf sphinx-grasple
+pip install -r requirements.txt
+pip install sphinx-grasple/
-echo "sphinx-grasple package was installed!"
echo "Ready!"
diff --git a/sphinx-grasple b/sphinx-grasple
index ec36e44..5ad7a03 160000
--- a/sphinx-grasple
+++ b/sphinx-grasple
@@ -1 +1 @@
-Subproject commit ec36e444b3aeb9a2f840da910326ab844fea62d2
+Subproject commit 5ad7a03721f94b1f1e067a5396d6214430138406
--
GitLab