Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$. Throughout the following, we make the following hypothesis: every element of $V$ is a matrix of rank at most $r$.\\
Show that we can assume that $V$ contains the block matrix:
$$A = \left( \begin{array} { c c }
I _ { r } & 0 \\
0 & 0
\end{array} \right)$$