Find the matrix of the following linear transformation in the standard bases $\mathcal{B}$ and $\mathcal{C}$ for the domain and codomain, respectively.
$T: \mathcal{M}_2 \to \mathcal{M}_2; T(B) = BC-CB$, where $B=\begin{pmatrix}1&2\\ \:\:3&4\end{pmatrix}, C = \begin{pmatrix}1&1\\ \:\:0&1\end{pmatrix}$.
I found that: $T(B) = \begin{pmatrix}-3&-3\\ 0&3\end{pmatrix} = T\begin{pmatrix}1&0\\ 0&0\end{pmatrix} + 2T\begin{pmatrix}0&1\\ 0&0\end{pmatrix} + 3T\begin{pmatrix}0&0\\ 1&0\end{pmatrix} + 4T\begin{pmatrix}0&0\\ 0&1\end{pmatrix}$. How do I proceed from here? I could just say that $T\begin{pmatrix}1&0\\ 0&0\end{pmatrix} = \begin{pmatrix}-3&0\\ 0&0\end{pmatrix}$, etc, but this would be just one possible solution? It could very well be many other solutions from what I can see.