Consider the ordered bases $B = \left( \begin{bmatrix} 0 & 2 \\ 0 & 3 \end{bmatrix}, \begin{bmatrix} -1 & -1 \\ 0 & -2 \end{bmatrix}, \begin{bmatrix} -1 & 2 \\ 0 & 3 \end{bmatrix} \right)$ and $C = \left( \begin{bmatrix} 2 & -3 \\ 0 & -2 \end{bmatrix}, \begin{bmatrix} -3 & 4 \\ 0 & 4 \end{bmatrix}, \begin{bmatrix} 4 & 2 \\ 0 & 1 \end{bmatrix} \right)$ for the vector space $V$ of upper triangular $2 \times 2$ matrices.
a. Find the change of coordinates matrix from $C$-coordinates to $B$-coordinates.
${}_B[Id]_C = \begin{bmatrix} -7 & 5 & 11 \\ -5 & 4 & 4 \\ 3 & -1 & -8 \end{bmatrix}$
b. Find the coordinates of $M$ in the ordered basis $B$ if the coordinate vector of $M$ relative to basis $C$ is $[M]_C = \begin{bmatrix} -1 \\ 2 \\ -1 \end{bmatrix}$.
$[M]_B = \begin{bmatrix} 6 \\ 9 \\ 3 \end{bmatrix}$
c. Find $M$.
$M = \begin{bmatrix} -12 & 9 \\ 0 & 9 \end{bmatrix}$
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