Finding a basis for a set of $2\times2$ matrices 
Find a basis for $M_{2\times2}$ containing the matrices $$\begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}$$ and $$\begin{pmatrix} 1 & 1 \\ 3 & 2 \end{pmatrix}$$

I know that every $2\times2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ can be written as: 
$$a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + b \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}  + c \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}  + d \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$ 
so those matrices are a basis for the vector space of $2\times2$ matrices, but how do I apply this to specific matrices? I know how to find a basis for a set of vectors, but matrices confuse me. 

Here's a systematic approach: to begin, find a basis for $\Bbb R^4$ containing the vectors $(1,1,2,3)$ and $(1,1,3,2)$.  Following the method that I outline here, we find that $\{(1,1,2,3),(1,1,3,2),(1,0,0,0),(0,1,0,0)\}$ is a basis for $\Bbb R^4$.  Correspondingly,
$$
\left\{\pmatrix{1&1\\2&3},
\pmatrix{1&1\\3&2},
\pmatrix{1&0\\0&0},
\pmatrix{0&1\\0&0}
\right\}
$$
is a basis for $M_{2 \times 2}$
