Let $k$ be a field and let $V$ be a vector space over $k$. Then $V$ is finite dimensional if and only if for every $\phi\in End_k(V)$, there are $a_0,\dots,a_{m-1}\in k$ such that $\phi^m+a_{m-1}\phi^{m-1}+\cdots+a_1\phi+a_0id_V=0.$
I have no idea on how to prove this statement. I was trying to use the fact that $V$ is finite dimensional if and only if $End_k(V)$ is finite dimensional... Could you help me with this? Thanks!