How to show a particular set $S$ is a basis for $M_{2\times 2}$? I'm given a set $S$ of four $2\times 2$ matrices with numbers in them and need to show they are a basis for $M_{2\times 2}$. all I know is that the linear combination of these matrices $= [0,0,0,0]$ must only have a solution of a scalar * each of them being 0. All I can think of is making a set of $4$ vectors and using that to test LI somehow. but how exactly am I allowed to make these vectors? Is this even the right method? 
also, given another particular $2\times 2$ matrix, how would I figure out how to write it as a linear combination of the $4$ matrices in the set $S$?

You can make the transformation of a matrix $$
\begin{pmatrix}
a&b\\
c&d\\
\end{pmatrix}
$$
to the vector in $\mathbb{R^4}$ 
\begin{pmatrix}
a\\
b\\
c\\
d\\
\end{pmatrix}
You can then determine if the four vectors are linearly independent.
