# Consider the bases B = left(begin{array}{c}begin{bmatrix}2

Consider the bases .
and the linear maps  given given (with respect to the standard bases) by Find each of the following coordinate representations. $\left(b\right){\left[S\right]}_{E,C}$
$\left(c\right){\left[S\right]}_{B,C}$

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Caren

$B=\left(\begin{array}{c}\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 5\end{array}\right]\end{array}\right)of{R}^{2},C=\left(\begin{array}{c}\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]\end{array},\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]\right)of{R}^{3}$.

$S\in L\left({R}^{2},{R}^{3}\right),T\in L\left({R}^{3},{R}^{2}\right)$
$\left[S{\right]}_{E,E}=\left[\begin{array}{cc}2& -1\\ 5& -3\\ -3& 2\end{array}\right],\left[T{\right]}_{E,E}=\left[\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\end{array}\right]$

(b) To find ${\left[S\right]}_{E,C}:$
$S\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\right)=\left[\begin{array}{c}2\\ 5\\ -3\end{array}\right]=a\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]+b\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]+c\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}a+b\\ a+c\\ b+c\end{array}\right]$
$\therefore a+b=2,a+c=5,b+c=-3$
$b=2-a⇒b+c=2-a+c=-3⇒-a+c=-5$
$\begin{array}{c}\therefore a+c=5\\ +-a+c=-5\\ 2c=0\end{array}\begin{array}{c}⇒c=0,b=-3,a=5\\ ⇒a=5,b=-3,c=0\end{array}$
$S\left(\left[\begin{array}{c}0\\ 1\end{array}\right]\right)=\left[\begin{array}{c}-1\\ -3\\ 2\end{array}\right]=a\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]+b\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]+c\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}a+b\\ a+c\\ b+c\end{array}\right]$
$\therefore a+b=-1,a+c=-3,b+c=2$
$\therefore b=-1-a⇒b+c=-1-a+c=2⇒-a+c=3$
$\begin{array}{c}\therefore a+c=-3\\ +-a+c=3\\ 2c=0\end{array}\begin{array}{c}⇒c=0,a=-3,b=2\\ ⇒a=-3,b=2,c=0\end{array}$
$\therefore \left[S{\right]}_{E,C}=\left[\begin{array}{cc}5& -3\\ -3& 2\\ 0& 0\end{array}\right]$

c) To find ${\left[S\right]}_{B,C}$