a) Let $r , m , n$ be strictly positive integers such that $r \leq n \leq m$. Show that if $E$ is a subspace of $M _ { m , n } ( \mathbf { C } )$ such that every element of $E$ is a matrix of rank at most $r$, then $\operatorname { dim } E \leq m r$.\\
b) Give an example of a subspace $E$ of $M _ { m , n } ( \mathbf { C } )$ satisfying $\operatorname { dim } E = m r$ and such that every element of $E$ is a matrix of rank at most $r$.