Let $V$ be a finite-dimensional vector space over a field $K$. Let $U$ be a linear subspace of $\mathrm{End}(V)$. Write $UV$ for the span of all $Av$ where $A\in U$ and $v\in V$. Suppose that $$ \ker(U)=\bigcap_{A\in U}\ker(A) $$ is zero and that $$ \mathrm{coker}(U)=V/UV $$ is zero. Is it true that $U$ must contain an invertible element of $\mathrm{End}(V)\ $?