Let $V$ be a finite-dimensional vector space over a field. Show that an endomorphism $\alpha$ of $V$ is idempotent, i.e. $\alpha^{2}=\alpha$, if and only if $\alpha$ is a projection onto its image.

Determine whether the following statements are true or false, giving a proof or counterexample as appropriate:

(i) If $\alpha^{3}=\alpha^{2}$, then $\alpha$ is idempotent.

(ii) The condition $\alpha(1-\alpha)^{2}=0$ is equivalent to $\alpha$ being idempotent.

(iii) If $\alpha$ and $\beta$ are idempotent and such that $\alpha+\beta$ is also idempotent, then $\alpha \beta=0$.

(iv) If $\alpha$ and $\beta$ are idempotent and $\alpha \beta=0$, then $\alpha+\beta$ is also idempotent.