Prove that linear operator from $V$ to itself with property $\rho = \rho^2$ is a projection operator. It is just a standard result on textbook,what I don't understand is why the proof on the textbook first prove that $V$ is direct sum of $\ker \rho$ and $Img\ \rho$? By the property of $\rho = \rho^2$ .
Since $V$ is direct sum of $\ker \rho$ and $Img\ \rho$ is just a general result for linear transformation,why we need to show this result again by idempotent?