How to find the transition matrix for ordered basis of 2x2 diagonal matrices The problem:
For the vector space of lower triangular matrices with zero trace,
given ordered basis:
$B=${$$
        \begin{bmatrix}
        -5 & 0 \\
        4 & 5 \\
        \end{bmatrix},
$$
\begin{bmatrix}
    -1 & 0 \\
    1 & 1 \\
    \end{bmatrix}}
and $C=${$$
    \begin{bmatrix}
    -5 & 0 \\
    -4 & 5 \\
    \end{bmatrix},
$$
\begin{bmatrix}
    -1 & 0 \\
    5 & 1 \\
    \end{bmatrix}}
find the transition matrix $C$ to $B$.
I know how to find a transition matrix when the basis consists of $n \times 1 $ vectors, but my textbook doesn't address this scenario where the basis consists of a set of $2 \times 2$ matrices and haven't found applicable guidance online.

Thank you for your direction.  I was able to use your ideas to find the correct solution to the problem.  First I expressed B and C in terms of the basis 
$\left[ \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} \right]$,$\left[ \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right]$
The basis B is thus equivalent to $\left[ \begin{matrix} -5\\ 4 \end{matrix} \right]$,$\left[ \begin{matrix} -1 \\ 1\end{matrix} \right]$
And the basis C to   $\left[ \begin{matrix} -5\\ -4 \end{matrix} \right]$,$\left[ \begin{matrix} -1 \\ 5\end{matrix} \right]$
Then I computed the transition matrix from C to B by multiplying the inverse of B times C: 
$\left[ \begin{matrix} -5 & -1 \\ 4 & 1 \end{matrix} \right]^{-1}$$\left[ \begin{matrix} -5 & -1 \\ -4 & 5 \end{matrix} \right]$=$\left[ \begin{matrix} 9 & -4 \\ -40 & 21 \end{matrix} \right]$
