Let $T:V\to V$ be a linear transformation from a finite dimensional vector space $V$. We are asked to show that $\textrm{rank}(T)=\textrm{rank}(T^2)\implies \textrm{N}(T)\cap\textrm{R}(T)={0}$.
My idea was as follows:
Suppose $\textrm{rank}(T)=\textrm{rank}(T^2)$. Define the transformation $T':\textrm R(T)\to V$ by $v\mapsto T(v)$. This is obviously a linear transformation. By the dimension theorem for linear transformations we have $\textrm{dim}(N(T'))+\textrm{rank}(T^2)=\textrm{dim}(N(T'))+\textrm{dim}(R(T'))=\textrm{dim}(R(T))=\textrm{rank}(T)$, so $\textrm{dim}(N(T'))=0$ which implies that $T'$ is injective.
I was hoping to be able to use this to show what I was meant to show, but I can't find how. Can someone help me with this or suggest another way of getting the required result?