ObsidianVault/College/AU 26/MATH 208 (Matrix Algebra)/Chapter 2.2 - Span.md

#uw/class/math208 #uw/notes

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[!abstract] Definition if \{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\} is a set of vectors in \mathbf{R}^n then the span of the set is given as \text{span}\{\mathbf{u_{1}}, \mathbf{u_{2}}, \dots, \mathbf{u_{m}}\} and represents the set of all linear combinations

x_{1}\mathbf{u_{1}}+x_{2}\mathbf{u_{2}}+\dots+x_{m}\mathbf{u_{m}}

where x_{1}, x_{2},\dots, x_{m} can be any real numbers

Span represents all points in n-dimensional space that a set of vectors could reach when combined in a certain linear combination.

if \text{span}\{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\} = \mathbf{R^n} then the vectors \{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\} spans \mathbf{R}^n

In three dimensions the span of two vectors can be visualized by the plane containing both of them. Any point on this plane can be reached by a linear combination of the vectors but any other point that does not lie on the plane cannot.

An augmented matrix can be used to find if a point (a,b,c) can be reached by a set of vectors

\mathbf{u}_{1}=\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}, \mathbf{u}_{2}=\begin{bmatrix}4 \\ 5 \\ 6\end{bmatrix}, \mathbf{u}_{3}=\begin{bmatrix}7 \\ 8 \\ 9\end{bmatrix}, \mathbf{v}=\begin{bmatrix}a \\ b \\ c\end{bmatrix} \implies \begin{bmatrix} 1 & 4 & 7 & a \\ 2 & 5 & 8 & b \\ 3 & 6 & 9 & c\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{1} & \mathbf{u}_{2} & \mathbf{u}_{3} & \mathbf{v}\end{bmatrix}

Based on this the point \mathbf{v} can only be reached by \mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m} if the augmented matrix $$\begin{bmatrix} \mathbf{u}{1} & \mathbf{u}{2} & \dots & \mathbf{u}_{m} & \mathbf{v} \end{bmatrix}$$ has a solution

for the set of vectors \mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m} in \mathbf{R}^n if m<n the set does not span \mathbf{R}^n. If m \ge n the set may span \mathbf{R}^n.

[!abstract] Definition Let \mathbf{a}_{1}, \mathbf{a}_{2}, \dots, \mathbf{a}_{m} be vectors in \mathbf{R}^n. If $$A = \begin{bmatrix} \mathbf{a}{1} & \mathbf{a}{2} & \dots & \mathbf{a}{m} \end{bmatrix} \quad \text{and} \quad \mathbf{x} = \begin{bmatrix} x{1} \ x_{2} \ \vdots \ x_{m} \end{bmatrix}$$ then A\mathbf{x} = x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2} + \dots + x_{m}\mathbf{a}_{m} (only if the number of columns of A equals the number of components of \mathbf{x})

This means that A\mathbf{x} is the linear combination of the columns of A where the scalars are the components of the vector \mathbf{x}.

This is useful to represent vector equations in a more compact way in the form A\mathbf{x} = \mathbf{b}

A\mathbf{x}=\mathbf{b} \quad \sim \quad x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2}+\dots+x_{m}\mathbf{a}_{m}=\mathbf{b} \quad \sim \quad \begin{flalign} a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1m}x_{m} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2m}x_{m} = b_{2} \\ \vdots \ \\ a_{n1}x_{1} + a_{n2}x_{2} + \dots + a_{nm}x_{m} = b_{n} \end{flalign}

[!example] Example if \mathbf{a}_{1} = \begin{bmatrix}10 \\ 5 \\ 7\end{bmatrix}, \ \mathbf{a}_{2} = \begin{bmatrix}8 \\ 6 \\ -1\end{bmatrix}, \ \mathbf{x} = \begin{bmatrix}x_{1} \\ x_{2}\end{bmatrix}, \ \mathbf{b} = \begin{bmatrix}18 \\ 31 \\ 3\end{bmatrix} then $$A = \begin{bmatrix} \mathbf{a}{1} & \mathbf{a}{2} \end{bmatrix} = \begin{bmatrix} 10 & 8 \ 5 & 6 \ 7 & -1 \end{bmatrix}$$ We can use these vectors to represent the linear system of equations A\mathbf{x} = \mathbf{b} which when expanded is $$\begin{gather} 10x_{1} + 8x_{2} = 18 \ 5x_{1} + 6x_{2} = 31 \ 7x_{1} - x_{2} = 3 \end{gather}$$