<s>[INST] 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. 
 [/INST] The two matrices are linearly independent, so they are a basis for the  two dimensional vector space spanned by them.
If you want a basis for $M_{2\times2}$ add two linearly independent matrices that are linearly independent from them. As an example you can chose:
$$
\begin{pmatrix} 0 & 0 \\ 2 & 3 \end{pmatrix}
\qquad
\begin{pmatrix} 3 & 2 \\ 0 & 0 \end{pmatrix}
$$ 
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