Call a linear transformation $\rho: V \to V$ ($V$ is a vector space) idempotent if $\rho^2 = \rho$. Prove that if $\rho$ is idempotent, then it acts as the identity on $\rho(V)$.
If I understand the question, I think that the methodology will be clear to me. What exactly does the phrase "it acts as the identity on $\rho(V)$" mean? Can someone formulate it in a more mathematical fashion?