The full question:
$T^2$ means the composition, i.e. $T(T(x))$
Let $V$ be a vector space and let $T: V \rightarrow V$ be linear. Prove that $T^2$ is the zero transformation if and only if $range(T) \subseteq \ker(T)$.
So far I know that for ($\implies$). Assume $T^2$ is the zero transformation. Then $ker(T^2)=\{0 \in V: T^2(v)=0\}$. Note that this is $\forall v \in V$. We also know that the $range(T)=\{0\in V: \exists v \in V: T^2(v)=0\}$.
Not sure where to go from here. Also for the other way, I don't know where to go.