Vector space and linear map problem

Question

I need to solve the following problem: $V$ is a Vectorspace, and $\phi$ is a linear map from $V \rightarrow V$ so that $\phi \cdot \phi = \phi$. I need to prove that $$Ker(\phi) \cap Im(\phi)=\{0\}$$ and $$Ker(\phi) + Im(\phi)=V$$

I solved it saying that from definition $\phi(\phi(v))=\phi(v)$, so is $\phi$ the identity function, an so follows the observation, but I am not sure that it is correct, can someone help me? Thanks a lot :)

Answer 1

Any element in $Im \phi$ is of the form $\phi(v)$. If this element is also in the kernel, then $0 = \phi(\phi(v)) = \phi(v)$.

For the second part, write $v = v - \phi(v) + \phi(v)$.

Answer 2

If $v \in \ker \phi \cap {\rm Im}\,\phi$, then $\phi(v) = 0$ but also $v = \phi(w)$ for some $w$, so $0 = \phi(v) = \phi(\phi(w)) = \phi(w) = v$ and the first part is done.

For the second part, $v = v - \phi(v) + \phi(v)$ does it, since $\phi(v-\phi(v)) = \phi(v) - \phi(\phi(v)) = 0$.