Proof of nilpotent matrix in Complex Numbers

Question

n $\in \mathbb N$ V is a n-dimensional Vectorspace of $ \mathbb C$ and $\phi$ is a endomorphism of V with $$ im(\phi) \subseteq ker(\phi)$$

Prove that $\phi$ is nilpotent.

Additionally find dependent of $rank(\phi)$ :

Characterstic polynomial, minimal polynomial and jordan normal form of $\phi$.

Thanks!

Answer 1

Hint: We can deduce from the information given that $\phi^2=0$