 Kernel-image identities for complementary projectors

For an idempotent linear map $P$, $I-P$ is also idempotent andFor the first identity, $P(I-P)=0$ proves one inclusion. If $Pv=0$, then $v=(I-P)v$, proving the reverse inclusion. Apply the same argument to $I-P$ for the second identity. Consequently the ambient vector space is the direct sum of the image and kernel of a linear map of $P$. Orthogonality is not required; it is a separate property of an orthogonal projection.