From 01395195aa059f4da50301b93b30f75a045b23c9 Mon Sep 17 00:00:00 2001
From: douden <d.denouden-vanderhorst@tudelft.nl>
Date: Thu, 18 Apr 2024 12:29:47 +0200
Subject: [PATCH] Some fixes and figures
---
Chapter3/MatrixOperations.md | 38 +++++------
Chapter5/DeterminantsExtras.md | 2 +-
.../Images/Fig-QuadForms-ConeWithPlanes.svg | 59 ------------------
.../Images/Fig-QuadForms-ConicSections.png | Bin 0 -> 54113 bytes
Chapter8/QuadraticForms.md | 6 +-
5 files changed, 22 insertions(+), 83 deletions(-)
delete mode 100644 Chapter8/Images/Fig-QuadForms-ConeWithPlanes.svg
create mode 100644 Chapter8/Images/Fig-QuadForms-ConicSections.png
diff --git a/Chapter3/MatrixOperations.md b/Chapter3/MatrixOperations.md
index ac889ab..f6cd0f3 100644
--- a/Chapter3/MatrixOperations.md
+++ b/Chapter3/MatrixOperations.md
@@ -452,13 +452,14 @@ $$
B = \mathbf{x} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},
$$
+
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}
-The product of an $m\times n$ matrix $A$ and an $n\times p$ matrix $B = [\,\mathbf{b_1}\quad \mathbf{b_2}\quad \ldots \quad \mathbf{b_p}]$ is defined by
+The product of an $m\times n$ matrix $A$ and an $n\times p$ matrix $B = [\,{\vect{b}_1}\quad {\vect{b}_2}\quad \ldots \quad {\mathbf{b}_p}]$ is defined by
$$
- AB = [\,A\mathbf{b_1}\quad A\mathbf{b_2}\quad \ldots \quad A\mathbf{b_p}].
+ AB = [\,A\mathbf{b}_1\quad A\mathbf{b}_2\quad \ldots \quad A\mathbf{b}_p].
$$
So we have
@@ -641,8 +642,8 @@ The $n \times n$ zero matrix $O_{nn}$ is a diagonal matrix.
::::::{exercise}
:label: Exc:MatrixOps:InterpretATB
-Suppose $A = \begin{bmatrix} \mathbf{a_1} & \mathbf{a_2} & \ldots & \mathbf{a_n} \end{bmatrix} $ is an $m\times n$ matrix and <br/>
-$B= \begin{bmatrix} \mathbf{b_1} & \mathbf{b_2} & \ldots & \mathbf{b_p} \end{bmatrix} $ an $n\times p$ matrix. Show that
+Suppose $A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \ldots & \mathbf{a}_n \end{bmatrix} $ is an $m\times n$ matrix and <br/>
+$B= \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \ldots & \mathbf{b}_p \end{bmatrix} $ an $n\times p$ matrix. Show that
$$
A^TB = %\begin{bmatrix} \vect{a_1}^T \vect{b_1} & \vect{a_1}^T \vect{b_2} & \ldots & \vect{a_1}^T \vect{b_p} \\
@@ -651,10 +652,10 @@ $$
% \vect{a_n}^T \vect{b_1} & \vect{a_n}^T \vect{b_2} & \ldots & \vect{a_n}^T \vect{b_p} \\
%\end{bmatrix}
% =
- \begin{bmatrix} \mathbf{a_1} \ip \mathbf{b_1} & \mathbf{a_1}\ip\mathbf{b_2} & \ldots & \mathbf{a_1}\ip \mathbf{b_p} \\
- \mathbf{a_2}\ip \mathbf{b_1} & \mathbf{a_2}\ip\mathbf{b_2} & \ldots & \mathbf{a_2}\ip \mathbf{b_p} \\
+ \begin{bmatrix} \mathbf{a}_1\ip \mathbf{b}_1 & \mathbf{a}_1\ip\mathbf{b}_2 & \ldots & \mathbf{a}_1\ip \mathbf{b}_p \\
+ \mathbf{a}_2\ip \mathbf{b}_1 & \mathbf{a}_2\ip\mathbf{b}_2 & \ldots & \mathbf{a}_2\ip \mathbf{b}_p \\
\vdots & \vdots & & \vdots \\
- \mathbf{a_n}\ip \mathbf{b_1} & \mathbf{a_n}\ip\mathbf{b_2} & \ldots & \mathbf{a_n}\ip \mathbf{b_p} \\
+ \mathbf{a}_n\ip \mathbf{b}_1 & \mathbf{a}_n\ip\mathbf{b}_2 & \ldots & \mathbf{a}_n\ip \mathbf{b}_p \\ \\
\end{bmatrix},
@@ -939,13 +940,13 @@ Rules i. and ii. are checked in a straightforward way. See {numref}`Exc:MatrixOp
<li>
We saw instances of this property already in {prf:ref}`Ex:MatrixOps:AtimesI` and {numref}`Exc:MatrixOps:I4timesA`.
-For the general case, one way to show validity of the first statement is to note that the $j$-th column of $AI_n$ is $A\mathbf{e_j}$ where $\mathbf{e_j}$ is the $j$-th column of the identity matrix $I_n$.
+For the general case, one way to show validity of the first statement is to note that the $j$-th column of $AI_n$ is $A\mathbf{e}_j$ where $\mathbf{e}_j$ is the $j$-th column of the identity matrix $I_n$.
This gives the linear combination
<BR>
$$
- A\mathbf{e_j} = 0\mathbf{a_1} + 0\mathbf{a_2} + \ldots + 1\mathbf{a_j} +\dots + 0\mathbf{a_n} = \mathbf{a_j}
+ A\mathbf{e}_j = 0\mathbf{a}_1 + 0\mathbf{a}_2 + \ldots + 1\mathbf{a}_j +\dots + 0\mathbf{a}_n = \mathbf{a}_j
$$
which shows that the $j$-th column of $AI_n$ is equal to the $j$-th column of $A$. And this holds for any column.
@@ -977,13 +978,13 @@ Now recall that (by definition)
<BR>
$$
- k\text{-th column of }BC = B\mathbf{c_k},
+ k\text{-th column of }BC = B\vect{c}_k,
$$
so
$$
- k\text{-th column of }A(BC) = A\,(B\mathbf{c_k})
+ k\text{-th column of }A(BC) = A\,(B\vect{c}_k)
$$
Making extensive use of the rule
@@ -998,11 +999,10 @@ we find
$$
\begin{array}{ccl}
- A\,(B\mathbf{c_k}) & = & A \,(c_{1k}\mathbf{b_1} +c_{2k}\mathbf{b_2} + \ldots + c_{pk}\mathbf{b_p})\\
- %& = & c_{1k}A\vect{b_1} +c_{2k}A\vect{b_2} + \ldots + c_{pk}A\vect{b_p}\\
- & = & c_{1k}(A\mathbf{b_1}) +c_{2k}(A\mathbf{b_2}) + \ldots + c_{pk}(A\mathbf{b_p})\\
- & = & \begin{bmatrix} A\mathbf{b_1} & A\mathbf{b_2} & \ldots & A\mathbf{b_p} \end{bmatrix} \begin{bmatrix} c_{1k} \\ \vdots \\ c_{pk} \end{bmatrix} \\
- & = & (AB)\mathbf{c_k}.
+ A\,(B\mathbf{c_k}) & = & A \,(c_{1k}\mathbf{b}_1 +c_{2k}\mathbf{b}_2 + \ldots + c_{pk}\mathbf{b}_p)\\
+ & = & c_{1k}(A\mathbf{b}_1) +c_{2k}(A\mathbf{b}_2) + \ldots + c_{pk}(A\mathbf{b}_p)\\
+ & = & \begin{bmatrix} A\mathbf{b}_1 & A\mathbf{b}_2 & \ldots & A\mathbf{b}_p \end{bmatrix} \begin{bmatrix} c_{1k} \\ \vdots \\ c_{pk} \end{bmatrix} \\
+ & = & (AB)\mathbf{c}_k.
\end{array}
$$
@@ -1035,19 +1035,19 @@ The definition of the product of two matrices then precisely matches the composi
if $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix
$$
- \mathbf{x}\in\mathbb{R}^p \,\,\stackrel{B}{\longrightarrow}\,\, \mathbf{y_1} = B\mathbf{x}\in\mathbb{R}^n \,\, \stackrel{A}{\longrightarrow} \,\,\, \mathbf{y_2} = A(B\mathbf{x}) \in \mathbb{R}^m\,
+ \mathbf{x}\in\mathbb{R}^p \,\,\stackrel{B}{\longrightarrow}\,\, \vect{y}_1 = B\vect{x}\in\mathbb{R}^n \,\, \stackrel{A}{\longrightarrow} \,\,\, \vect{y}_2 = A(B\mathbf{x}) \in \mathbb{R}^m\,
$$
and
$$
- \mathbf{x}\in\mathbb{R}^p\,\,\,\stackrel{AB}{\longrightarrow} \,\,\,\mathbf{y_3} \,=\, (AB)\mathbf{x} \in \mathbb{R}^m
+ \mathbf{x}\in\mathbb{R}^p\,\,\,\stackrel{AB}{\longrightarrow} \,\,\,\vect{y}_3 \,=\, (AB)\mathbf{x} \in \mathbb{R}^m
$$
yield the same vector:
$$
-\mathbf{y_2} = \mathbf{y_3}.
+\vect{y}_2 = \vect{y}_3.
$$
%
diff --git a/Chapter5/DeterminantsExtras.md b/Chapter5/DeterminantsExtras.md
index e5a9613..72855d1 100644
--- a/Chapter5/DeterminantsExtras.md
+++ b/Chapter5/DeterminantsExtras.md
@@ -1137,7 +1137,7 @@ $$
::::::{grasple}
-:url: https://embed.grasple.com/exercises/
+: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>`
diff --git a/Chapter8/Images/Fig-QuadForms-ConeWithPlanes.svg b/Chapter8/Images/Fig-QuadForms-ConeWithPlanes.svg
deleted file mode 100644
index 877bf90..0000000
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+++ /dev/null
@@ -1,59 +0,0 @@
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diff --git a/Chapter8/QuadraticForms.md b/Chapter8/QuadraticForms.md
index 3de9ddf..cb18bf1 100644
--- a/Chapter8/QuadraticForms.md
+++ b/Chapter8/QuadraticForms.md
@@ -734,12 +734,10 @@ Check that the vector you found in iii. is an eigenvector of the matrix of the q
A _conic section_ or _conic_ is a curve that results when a circular cone is intersected with a plane.
{numref}`Figure %s <Fig:QuadForms:ConeWithPlanes>` shows the different shapes when the plane is *not* going through the apex..
-:::{figure} Images/Fig-QuadForms-ConeWithPlanes.svg
+:::{figure} Images/Fig-QuadForms-ConicSections.png
:name: Fig:QuadForms:ConeWithPlanes
-Intersections of a cone with planes (not going through the apex).
-
-<a href="https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg">Pbroks13</a>, <a href="https://creativecommons.org/licenses/by/3.0">CC BY 3.0</a>, via Wikimedia Commons
+Intersections of a cone with several planes (not going through the apex).
:::
--
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