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(Sec:LinearCombinations)=
# Linear Combinations
::::{prf:definition}
......@@ -13,23 +14,23 @@ where $x_1, \ldots, x_n$ are real numbers, is called a **linear combination** of
::::
::::{prf:example}
The vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ are two vectors in the plane $\mathbb{R}^2$. As we can see in {numref}`Figure %s <Fig:LinearCombinations:LinearCombinations>`, the vector $\mathbf{u}$ is a linear combination of $\mathbf{v}_1$ and $\mathbf{v}_2$ since it can be written as $\mathbf{u}=2\mathbf{v}_1+\mathbf{v}_2$. The vector $\mathbf{w}$ is a linear combination of these two vectors as well. It can be written as $\mathbf{w}=-3\mathbf{v}_1+2\mathbf{v}_2$.
:::{figure} Images/Fig-LinearCombinations-LinComb.svg
```{applet}
:url: linear_combinations/linearcombinations
:fig: Images/Fig-LinearCombinations-LinComb.svg
:name: Fig:LinearCombinations:LinearCombinations
:status: approved
Linear combinations of vectors in the plane.
:::
```
::::
If we want to determine whether a given vector is a linear combination of other vectors, then we can do that using systems of equations.
::::{prf:example}
$$
......@@ -58,7 +59,6 @@ $$
The augmented matrix of this system of equations is equal to
$$
\left[\begin{array}{cc|c} 1 & 3 & -1 \\ 2 & 1 & 3 \\ 1 & 2 & 0 \end{array}\right]
$$
......@@ -82,7 +82,6 @@ We have found that $\mathbf{b}$ can be written as $2\mathbf{v}_1-\mathbf{v_2}$.
::::
::::{prf:example}
$$
......@@ -96,8 +95,6 @@ In this case it is a lot easier to decide whether $\mathbf{b}$ is a linear combi
::::
::::{grasple}
:url: https://embed.grasple.com/exercises/ac63b286-09e1-46e5-91fc-952b54436293?id=78560
:label: grasple_exercise_2_2_A
......@@ -114,13 +111,10 @@ In this case it is a lot easier to decide whether $\mathbf{b}$ is a linear combi
::::
## Span
In linear algebra it is often important to know whether each vector in $\mathbb{R}^n$ can be written as a linear combination of a set of given vectors. In order to investigate when it is possible to write any given vector as a linear combination of a set of given vectors we introduce the notion of a **span**.
::::{prf:definition}
:label: Dfn:LinearCombinations:Span
......@@ -132,7 +126,6 @@ The span of an empty collection of vectors will be defined as the set that only
::::
::::{prf:remark}
The collection $\Span{\mathbf{v}_1, \ldots, \mathbf{v}_k}$ always contains all of the vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$. This is true since each vector $\mathbf{v}_i$ can be written as the linear combination
......@@ -151,82 +144,94 @@ $$
The following examples will give us a bit of an idea what spans look like.
::::{prf:example}
:label: Ex:LinearCombinations:SpanOfOneVector
What does the span of a single non-zero vector look like? A linear combination of a vector $\mathbf{v}$ is of the form $x\mathbf{v}$, where $x$ is some real number. Linear combinations of a single vector $\mathbf{v}$ are thus just multiples of that vector. This means that $\Span{\mathbf{v}}$ is simply the collection of all vectors on the line through the origin and with directional vector $\mathbf{v}$ as we can see in {numref}`Figure %s <Fig:LinearCombinations:SpanOneVectors>`.
:::{figure} Images/Fig-LinearCombinations-SpanOne.svg
```{applet}
:url: linear_combinations/span_one
:fig: Images/Fig-LinearCombinations-SpanOne.svg
:name: Fig:LinearCombinations:SpanOneVectors
:status: approved
The span of a single non-zero vector.
:::
```
::::
::::{prf:example}
:label: Ex:LinearCombinations:SpanOfTwoVectors
Let $\mathbf{u}$ and $\mathbf{v}$ be two non-zero vectors in $\mathbb{R}^3$, as depicted in {numref}`Figure %s <Fig:LinearCombinations:SpanTwoVectors>`. What does the span of these vectors look like? By definition, $\Span{\mathbf{u}, \mathbf{v}}$ contains all linear combinations of $\mathbf{u}$ and $\mathbf{v}$. Each of these linear combinations is of the form
$$
x_1\mathbf{u}+x_2\mathbf{v} \quad \textrm{$x_1$, $x_2$ in $\mathbb{R}$}.
$$
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:
:::
:::{figure} Images/Fig-LinearCombinations-SpanTwoPlane.svg
```{applet}
:url: linear_combinations/span_two_plane
:fig: Images/Fig-LinearCombinations-SpanTwoPlane.svg
:name: Fig:LinearCombinations:SpanTwoVectors
:status: approved
The span of two non-zero, non-parallel vectors.
:::
```
::::
::::{prf:example}
The span of two non-zero vectors does not need to be a plane through the origin. If $\mathbf{u}$ and $\mathbf{v}$ are parallel, as in {numref}`Figure %s <Fig:LinearCombinations:SpanTwoParallelVectors>`, then the span is actually a line through the origin.
:::{figure} Images/Fig-LinearCombinations-SpanTwoLine.svg
```{applet}
:url: linear_combinations/span_two_line
:fig: Images/Fig-LinearCombinations-SpanTwoLine.svg
:name: Fig:LinearCombinations:SpanTwoParallelVectors
:status: approved
The span of two non-zero, parallel vectors.
:::
```
If two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel, then $\mathbf{v}$ can be written as a multiple of $\mathbf{u}$. Assume for example that $\mathbf{v}=2\mathbf{u}$. Any linear combination $x_1\mathbf{u}+x_2\mathbf{v}$ can then be written as $x_1\mathbf{u}+2x_2\mathbf{u}$ or $(x_1+2x_2)\mathbf{u}$. This means that in this case each vector in the span of $\mathbf{u}$ and $\mathbf{v}$ is a multiple of $\mathbf{u}$. Therefore, the span will be a line through the origin.
::::
::::{prf:example}
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} Images/Fig-LinearCombinations-SpanThreeR3.svg
:::{figure}
:name:
:::
```{applet}
:url: linear_combinations/span_three
:fig: Images/Fig-LinearCombinations-SpanThreeR3.svg
:name: Fig:LinearCombinations:SpanThreeVectors1
:status: approved
The span of three vectors.
:::
```
On the other hand, if we start with the three vectors that you can see in {numref}`Figure %s <Fig:LinearCombinations:SpanThreeVectors2>`, then the span is equal to a plane through the origin.
:::{figure} Images/Fig-LinearCombinations-SpanThreePlane.svg
```{applet}
:url: linear_combinations/span_three_plane
:fig: Images/Fig-LinearCombinations-SpanThreePlane.svg
:name: Fig:LinearCombinations:SpanThreeVectors2
:status: approved
The span of three vectors lying in the same plane.
:::
```
There is also a possibility where the span of three non-zero vectors in $\mathbb{R}^3$ is equal to a line through the origin. Can you figure out when this happens?
......@@ -240,20 +245,14 @@ There is also a possibility where the span of three non-zero vectors in $\mathbb
::::
We will now look at a very specific set of vectors in $\mathbb{R}^n$ of which the span is always the entire space $\mathbb{R}^n$.
::::{prf:definition}
Suppose we are working in $\mathbb{R}^n$. Let $\mathbf{e}_k$ be the vector of which all components are equal to 0, with the exception that the entry on place $k$ is equal to 1. The vectors $(\mathbf{e}_1, \ldots, \mathbf{e}_n)$ will be called the **standard basis** of $\mathbb{R}^n$.
::::
::::{prf:example}
The following vectors form the standard basis for $\mathbb{R}^2$.
......@@ -273,7 +272,6 @@ $$
then clearly we have that
$$
\mathbf{v}=a
\begin{bmatrix} 1 \\ 0 \end{bmatrix}+b
......@@ -284,7 +282,6 @@ It is easy to see that this is the only linear combination of $\mathbf{e}_1$ and
::::
::::{prf:example}
The three vectors below form the standard basis for $\mathbb{R}^3$.
......@@ -300,7 +297,6 @@ Here too, it is true that each vector in $\mathbb{R}^3$ can be written as a uniq
::::
::::{prf:proposition}
:label: Prop:LinearCombinations:SpanStandardBasis
......@@ -324,7 +320,7 @@ The vector $\mathbf{v}$ can be written as
\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}+a_2
\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}+ \ldots a_n
\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix} \\
&= a_n\mathbf{e}_1+a_2\mathbf{v}_2+\ldots +a_n\mathbf{e}_n.
&= a_n\mathbf{e}\_1+a_2\mathbf{v}\_2+\ldots +a_n\mathbf{e}\_n.
\end{align*}
This means that $\mathbf{v}$ is in the span of $\mathbf{e}_1, \ldots, \mathbf{e}_n$.
......@@ -371,7 +367,6 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
:url: https://embed.grasple.com/exercises/fab5c526-91ed-407b-9faa-645f40c22b8b?id=70169
:label: grasple_exercise_2_2_5
......@@ -388,7 +383,6 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
% ------------------------------------------------
::::{grasple}
......@@ -399,7 +393,6 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
:url: https://embed.grasple.com/exercises/c008320d-9d0e-463f-8bb7-344988f10438?id=70176
:label: grasple_exercise_2_2_8
......@@ -408,7 +401,6 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
::::
::::{grasple}
:url: https://embed.grasple.com/exercises/b4f4dc1f-4f56-41e8-b16d-a2694e90890c?id=70181
:label: grasple_exercise_2_2_9
......@@ -435,5 +427,4 @@ In {prf:ref}`Prop:LinearCombinations:SpanStandardBasis` we saw that the span of
:dropdown:
:description: Conversion between vector equation and linear system.
::::
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