$\displaystyle S=\lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace$.

I.e. it is a singleton.

I've already showed that it is a subspace of $\displaystyle M_2(R)$ (hopefully you'll agree that it is indeed a vector space).

But the problem is:

How can I find a basis for it?!

A basis must span the set S, but it must also be a linearly independent set!

So if I understand correctly, $\displaystyle \lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace$ spans S, but it isnotlinearly independent (since any singleton set containing just the 0 vector is linearlydependent).

What am I to do?