Linear mapping/Criterion for trivial intersection of image and kernel/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and ${}f\colon V\rightarrow V$ be an endomorphism. Show that the following statements are equivalent.

1. ${}\operatorname {kern} f=\operatorname {kern} {\left(f\circ f\right)}$,
2. ${}\operatorname {kern} f\cap \operatorname {Im} f=\{0\}$,
3. ${}V=\operatorname {kern} f\oplus \operatorname {Im} f$,
4. ${}\operatorname {Im} f=\operatorname {Im} {\left(f\circ f\right)}$.

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