Decomposition of vector space to produce nilpotent and invertible transformations

Question

Let $V$ be a finite-dimensional vector space over a field $K$. Let $T$ be a linear operator on $V$. Prove that there exists a unique sum $V=V_{0}+V_{1}$ such $T(V_{0}) \subseteq V_{0}$, $T(V_{1}) \subseteq V_{1}$, $T|_{V_{0}}$ is nilpotent, and $T|_{V_{1}}$ is invertible.

Is this exercise related to invariant spaces and direct sums? I was reading the theory, and I can't see how start the problem.