52 lines
3.6 KiB
Markdown
52 lines
3.6 KiB
Markdown
#uw/class/math208 #uw/notes
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> [!abstract] Definition
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> 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
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> $$x_{1}\mathbf{u_{1}}+x_{2}\mathbf{u_{2}}+\dots+x_{m}\mathbf{u_{m}}$$
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> where $x_{1}, x_{2},\dots, x_{m}$ can be any real numbers
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Span represents all points in n-dimensional space that a set of vectors could reach when combined in a certain linear combination.
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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 $\mathbf{R}^n$
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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.
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An augmented matrix can be used to find if a point $(a,b,c)$ can be reached by a set of vectors
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$$\mathbf{u}_{1}=\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}, \mathbf{u}_{2}=\begin{bmatrix}4 \\ 5 \\ 6\end{bmatrix}, \mathbf{u}_{3}=\begin{bmatrix}7 \\ 8 \\ 9\end{bmatrix}, \mathbf{v}=\begin{bmatrix}a \\ b \\ c\end{bmatrix} \implies \begin{bmatrix} 1 & 4 & 7 & a \\ 2 & 5 & 8 & b \\ 3 & 6 & 9 & c\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{1} & \mathbf{u}_{2} & \mathbf{u}_{3} & \mathbf{v}\end{bmatrix}$$
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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
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$$\begin{bmatrix}
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\mathbf{u}_{1} & \mathbf{u}_{2} & \dots & \mathbf{u}_{m} & \mathbf{v}
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\end{bmatrix}$$
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has a solution
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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 $\mathbf{R}^n$.
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> [!abstract] Definition
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> Let $\mathbf{a}_{1}, \mathbf{a}_{2}, \dots, \mathbf{a}_{m}$ be vectors in $\mathbf{R}^n$. If
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> $$A = \begin{bmatrix}
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> \mathbf{a}_{1} & \mathbf{a}_{2} & \dots & \mathbf{a}_{m}
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> \end{bmatrix} \quad \text{and} \quad \mathbf{x} = \begin{bmatrix}
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> x_{1} \\ x_{2} \\ \vdots \\ x_{m}
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> \end{bmatrix}$$
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> 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 $\mathbf{x}$)
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This means that $A\mathbf{x}$ is the linear combination of the columns of $A$ where the scalars are the components of the vector $\mathbf{x}$.
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This is useful to represent vector equations in a more compact way in the form $A\mathbf{x} = \mathbf{b}$
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$$A\mathbf{x}=\mathbf{b} \quad \sim \quad x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2}+\dots+x_{m}\mathbf{a}_{m}=\mathbf{b} \quad \sim \quad \begin{flalign} a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1m}x_{m} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2m}x_{m} = b_{2} \\ \vdots \ \\ a_{n1}x_{1} + a_{n2}x_{2} + \dots + a_{nm}x_{m} = b_{n} \end{flalign}$$
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> [!example] Example
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> 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
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> $$A = \begin{bmatrix}
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> \mathbf{a}_{1} & \mathbf{a}_{2}
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> \end{bmatrix} = \begin{bmatrix}
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> 10 & 8 \\ 5 & 6 \\ 7 & -1
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> \end{bmatrix}$$
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> We can use these vectors to represent the linear system of equations $A\mathbf{x} = \mathbf{b}$ which when expanded is
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> $$\begin{gather}
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> 10x_{1} + 8x_{2} = 18 \\ 5x_{1} + 6x_{2} = 31 \\ 7x_{1} - x_{2} = 3
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> \end{gather}$$
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testing |