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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?

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up vote 4 down vote accepted

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

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Thank you very much! – user44069 Feb 13 '13 at 3:02

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

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