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

@@ -37,4 +37,15 @@ This means that $A\mathbf{x}$ is the linear combination of the columns of $A$ wh
 This is useful to represent vector equations in a more compact way in the form $A\mathbf{x} = \mathbf{b}$ 
 $$A\mathbf{x}=\mathbf{b} \quad \sim \quad x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2}+\dots+x_{m}\mathbf{a}_{m}=\mathbf{b} \quad \sim \quad \begin{flalign} a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1m}x_{m} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2m}x_{m} = b_{2} \\ \vdots \ \\ a_{n1}x_{1} + a_{n2}x_{2} + \dots + a_{nm}x_{m} = b_{n} \end{flalign}$$
 
-if $\mathbf{a}_{1} = \begin{bmatrix}10 \\ 5 \\ 7\end{bmatrix}, \mathbf{a}_{2} = \begin{bmatrix}8 \\ 6 \\ -1\end{bmatrix}, \mathbf{x} = \begin{bmatrix}x_{1} \ x_{2}\end{bmatrix}, \mathbf{b} = \begin{bmatrix}18 \\end{bmatrix}$
+> [!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 
+> $$\begin{gather}
+> 	10x_{1} + 8x_{2} = 18 \\ 5x_{1} + 6x_{2} = 31 \\ 7x_{1} - x_{2} = 3
+> \end{gather}$$
+