I'm trying to figure out if I need to use traits of Characteristic and/or Minimal Polynomials.

Suppose that $\displaystyle dim_F(V) = n$ and $\displaystyle \theta \epsilon Hom(V,V)$. Prove that if $\displaystyle ker(\theta^{n-1}) \neq ker(\theta^n)$, then $\displaystyle \theta$ is nilpotent, and $\displaystyle dim(ker(\theta^j)) = j$ for all $\displaystyle j \epsilon \{1, 2, ..., n\}$

I'm not looking for a solution, I simply want a head start.