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 need to transform them to coordinate vectors and then your matrix will be converted in form (a,b,c,d) (i,e vector). then procceed with usual method of checking L.I ,by checking determinant or whatever
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https://www.youtube.com/playlist?list=PLGqzsq0erqU7w7ZrTZ-pWWk4-AOkiGEGp


