Let $V$ be a finite dimensional vector space over the reals and $P:V \rightarrow V$ be a linear transformation. Let
$V_1=\ker(P)$
$V_2=\ker(I-P)$
and $V=V_1 \oplus V_2$
How would you prove $P^2=P$? I've been trying to show that $\mathrm{Im}(P)=\ker(I-p)$ as that would be equivalent.