Consider the ordered bases $B=(\left[\begin{array}{cc} 5 &0\cr -3 &-5 \end{array}\right],\left[\begin{array}{cc} 2 &0\cr -1 &-2 \end{array}\right])$ and $C=(\left[\begin{array}{cc} 1 &0\cr -1 &-1 \end{array}\right],\left[\begin{array}{cc} 2 &0\cr 3 &-2 \end{array}\right])$ for the vector space $V$ of lower triangular $2\times 2$ matrices with zero trace.
a. Find the transition matrix from $C$ to $B$. Hint: use $F =(\left[\begin{array}{cc} 1 &0\cr 0 &-1 \end{array}\right],\left[\begin{array}{cc} 0 &0\cr 1 &0 \end{array}\right])$.
$T_C^B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
b. Find the coordinates of $M$ in the ordered basis $B$ if the coordinate vector of $M$ in $C$ is $[M]_C = \left[\begin{array}{c} 1\cr -2 \end{array}\right]$.
$[M]_B =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
c. Find $M$.
$M =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$