Why is it true that, if $f:V\to V$ is a linear map and $V$ is finite dimensional, then there must be some $n$ such that $\operatorname{Span}(\operatorname{nullspace}(f^n),\operatorname{image}(f^n))=V?$
The first thing that came to mind is the rank-nullity theorem, but I don't suppose it is of much help here. Maybe we can consider the map as a matrix, $F$, since it is a linear map? Would that help?