Let $\displaystyle V$ be a finite dimensional vector space over $\displaystyle \mathbb C$. Let $\displaystyle T$ be a linear transformation

on $\displaystyle V$ all of whose eigenvalues are zero. Show that $\displaystyle T^{n} = 0$ for some $\displaystyle n$, i.e.

that $\displaystyle T$ is nilpotent.