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 fromC to B .

T_C^B = \left[\Rule{0pt}{3.6em}{0pt}\right. \left]\Rule{0pt}{3.6em}{0pt}\right.

b. Find the coordinates ofM 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. FindM .

M = \left[\Rule{0pt}{2.4em}{0pt}\right. \left]\Rule{0pt}{2.4em}{0pt}\right.

a. Find the transition matrix from

b. Find the coordinates of

c. Find

You can earn partial credit on this problem.