 # Find the general form of the span of th e indicated matrices, span(A1, A2) B=begin{bmatrix}2 & 5 0 & 3 end{bmatrix} , A_1=begin{bmatrix}1 & 2 -1 & 1 end{bmatrix} , A_2=begin{bmatrix}0 & 1 2 & 1 end{bmatrix} waigaK 2021-02-25 Answered
Find the general form of the span of th e indicated matrices, span(A1, A2)
$B=\left[\begin{array}{cc}2& 5\\ 0& 3\end{array}\right],{A}_{1}=\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right],{A}_{2}=\left[\begin{array}{cc}0& 1\\ 2& 1\end{array}\right]$
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Step 1
We have to find the general form of the span of the indicated matrices, span(A1, A2) ${A}_{1}=\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right],{A}_{2}=\left[\begin{array}{cc}0& 1\\ 2& 1\end{array}\right]$
Step 2
$\left({A}_{1},{A}_{2}\right)={c}_{1}\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]+{c}_{2}\left[\begin{array}{cc}0& 1\\ 2& 1\end{array}\right]$
$=\left[\begin{array}{cc}{c}_{1}& 2{c}_{1}+{c}_{2}\\ -{c}_{1}+2{c}_{2}& {c}_{1}+{c}_{2}\end{array}\right]$
$=\left[\begin{array}{cc}w& x\\ y& z\end{array}\right]$
$\left[\begin{array}{ccc}1& 0& |w\\ 2& 1& |x\\ -1& 2& |y\\ 1& 1& |z\end{array}\right]=\left[\begin{array}{ccc}1& 0& w\\ 1& 0& x-z\\ 0& 2& y+w\\ 1& 1& z\end{array}\right]$
$=\left[\begin{array}{ccc}1& 0& w\\ 1& 0& x-z\\ 0& 1& \frac{y+w}{2}\\ 1& 1& z-w\end{array}\right]$
Step 3
$=\left[\begin{array}{ccc}1& 0& w\\ 0& 0& x-z-w\\ 0& 1& \frac{y+w}{2}\\ 0& 0& z-w-\frac{y+w}{2}\end{array}\right]$
$w=x-z$
$z-w=\frac{y+w}{2}$
$2\left(z-w\right)=y+w$
$2z-3w=y$
$2z-3\left(x-z\right)=y$
$5z-3x=y$
$\left({A}_{1},{A}_{2}\right)=\left[\begin{array}{cc}x-z& x\\ 5z-3x& z\end{array}\right]$
Step 4
$\left({A}_{1},{A}_{2}\right)=\left[\begin{array}{cc}x-z& x\\ 5z-3x& z\end{array}\right]$

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