How to prove a set is a basis of a matrix I am unsure how to go about proving this question, a gentle push in
the right direction would be much appreciated. 
Recall the standard basis of $M_{2}(\mathbb{R})$ is $S=\{E_{11},E_{12},E_{21},E_{22}\}$
Where 
E$_{11}=\left(\begin{array}[t]{cc}
1 & 0\\
0 & 0
\end{array}\right)$
E$_{22}=\left(\begin{array}[t]{cc}
0 & 1\\
0 & 0
\end{array}\right)$
E$_{21}$ = $\left(\begin{array}[t]{cc}
0 & 0\\
1 & 0
\end{array}\right)$
E$_{22}$=$\left(\begin{array}[t]{cc}
0 & 0\\
0 & 1
\end{array}\right)$
1) Prove the set $B= \{E_{11}+E_{12},\ E_{11}-E_{12},\ E_{21}+E_{22},\ E_{21}-E_{22}\}$
is a basis for $M_{2}(\mathbb{R}) $ ?
2) Write down a change of basis matrix $ P $ from the basis $B$ to the basis $ S$ ?. 
 A: You need to show that these four matrices are linearly indep and spanning. 
Depending on what you have shown in class, you might be able to cite the fact that (1,1) and (1,-1) spans (x,y) so it is easy to see that these matrices would span (x,y,z,a) linearly. (The x,y pair doesn't affect the z,a pair)
To show that (1,1) and (1,-1) are linearly indep note that $\forall a,b \in \mathbb Z a*(1,1)+b*(1,-1) = (a+b,a-b)$ Immediately note that for the second part of the tuple to be 0, $a=b$, but then $a+b \neq 0$. Now we must show that this is spanning. Suppose we want the arbitrary point $(x,y) = (a+b, a-b)$ So $x =a+b, y = a-b$ Thus, $x = y + 2*b$ by substitution. But the only variable we get to play with here is b, so since b is also in our field, it must always be possible to find (or create) such a b for each x,y. Thus it is spanning. Therefore, we can return to what I said earlier about this matrix question really just being a question about a 4-vector (since we are not using the matrix properties really). Hope this helps.
