I am required to prove the following claim
Proposition. Given that $T:V\to V$ is a linear map prove that for some positive integer $k$ we have $V = \operatorname{null }(T^k)\oplus\operatorname{range}(T^k)$.
I have spent a considerable amount of time thinking about this and so far have not been able to come up with much Any Hints on how to get started?