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 fromC 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 ofM 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. 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.