# Let B = left{ begin{bmatrix} 1 -2 end{bmatrix}

Let
be bases for${R}^{2}.$ Change-of-coordinate matrix from C to B.

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Given that Let x, y are scalare such that $\left[\begin{array}{c}2\\ 1\end{array}\right]=x\left[\begin{array}{c}1\\ -2\end{array}\right]+y\left[\begin{array}{c}2\\ 1\end{array}\right]$ clearly $x=1,y=0$
$\left[\begin{array}{c}2\\ 1\end{array}\right]=0\left[\begin{array}{c}1\\ -2\end{array}\right]+1\left[\begin{array}{c}2\\ 1\end{array}\right]\to \left(c\right)$ and $\left[\begin{array}{c}1\\ 3\end{array}\right]=2\left[\begin{array}{c}1\\ -2\end{array}\right]+y\left[\begin{array}{c}2\\ 1\end{array}\right]$
$=1\left[\begin{array}{c}1\\ 3\end{array}\right]=\left[\begin{array}{c}x+2y\\ -2x+y\end{array}\right]$
$x+2y=1,-2x+y=3$ Solving above equation $\begin{array}{c}2\text{⧸}x+4y=2\\ -\text{⧸}2x+y=3\\ 5y=5⇒y=1\end{array}$ pul $-y=1\in x+2y=1$
$⇒x+2=1$
$⇒x=-1$
$\therefore \left[\begin{array}{c}1\\ 3\end{array}\right]=\left(-1\right)\left[\begin{array}{c}1\\ -2\end{array}\right]+1\left[\begin{array}{c}2\\ 1\end{array}\right]\to \left(d\right)$ from c, d Transition matrix of C over D $\left[T{\right]}_{C,B}=\left[\begin{array}{cc}0& -1\\ 1& 1\end{array}\right]$