Let $f: V\to V$ be linear map and an isomorphism. Does this condition automatically yield a direct sum in the form of
$$V = \ker T \oplus \operatorname{Im}( T)$$
I am working on a problem right now where $f$ is not a given isomorphism, but I've managed to reduce $f$ to an isomorphism. I am not sure if this immediately implies Rank-Nullity (other than a dimension argument).
The map is \begin{align} f^2 = f. \end{align} But I just want to know technique.