Consider the ordered bases $B=(\left[\begin{array}{cc} 3 &-2\cr 0 &-2 \end{array}\right],\left[\begin{array}{cc} 0 &1\cr 0 &-1 \end{array}\right],\left[\begin{array}{cc} 4 &-3\cr 0 &-2 \end{array}\right])$ and $C=(\left[\begin{array}{cc} 4 &0\cr 0 &3 \end{array}\right],\left[\begin{array}{cc} 0 &2\cr 0 &4 \end{array}\right],\left[\begin{array}{cc} 0 &-1\cr 0 &-1 \end{array}\right])$ for the vector space $V$ of upper triangular $2\times 2$ matrices.
a. Find the transition matrix from $C$ to $B$.
$T_C^B =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$
b. Find the coordinates of $M$ in the ordered basis $B$ if the coordinate vector of $M$ in $C$ is $[M]_C = \left[\begin{array}{c} 2\cr -2\cr -1 \end{array}\right]$.
$[M]_B =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$
c. Find $M$.
$M =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$