I've been trying to solve this problem, but to be honest I have no clue on how to tackle it.
Let $V$ a finite dimensional vector space over a field $K$, and a endomorphism $T: V \to V$. Prove that $V$ can be uniquely split as $V_0+V_1$ such that $T(V_0)\subset V_0$, $T(V_1)\subset V_1$ , $T|_{V_0}$ nilpotent and $T|_{V_1}$ invertible.
Found something similar,but don't know if is the same.
Like the straightforward way is to consider $\beta_i$ a basis for $V_i$, $i=0,1$, and show that $\beta=\beta_0\cup \beta _1$ is a basis for $V$ and somehow use the properties of the restrictions of $T$ to each subspace. Although can't figure how.