An explanation about terminology in vector spaces

Question

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?

Answer 1

It means that for every $y \in \rho(V)$, $\rho(y) = y$.

Answer 2

Identity means $\rho(\mathbf a) = \mathbf a$ for any $\mathbf a \in \mathbf V$