I need help with part b in this problem:
Let $V$ be a vector space and $π$ an endomorphism of $V$.
a. Prove that $π$ is a projection if and only if $id_V − π$ is a projection, where $id_V$ is the identity endomorphism on $V$.
b. Assume now that $π$ is a projection. Calculate $Im(id_V −π)$ and $ker(id_V −π)$ as a function of $Im(π)$ and $ker(π)$.
I am familiar with identity automorphism but what does identity endomorphism mean and what is it's transformation matrix?
I assumed it's $I$ (the identity matrix) so I used the condition for a projection: $(I-P)^2=I-P$
Where $P$ is transformation matrix for $π$, and from this we obtain:
$P^2=P$
So π is indeed a projection, but now for part b I know that $Im(π)=span(P_i)$ (span of columns of P) and I don't know how to continue from here...