Let $V$ be a finite-dimensional vector space and let $T: V \rightarrow V$ be an endomorphism of $V$. Show that there is a positive integer $l$ such that $V=\operatorname{ker}\left(T^{l}\right) \oplus \operatorname{im}\left(T^{l}\right)$. Hence, or otherwise, show that if $T$ has zero determinant there is some non-zero endomorphism $S$ with $T S=0=S T$.

Suppose $T_{1}$ and $T_{2}$ are endomorphisms of $V$ for which $T_{i}^{2}=T_{i}, i=1,2$. Show that $T_{1}$ is similar to $T_{2}$ if and only if they have the same rank.