= Kernel-image identities for complementary projectors
{title2=$\ker P=\operatorname{im}(I-P)$}
For an <idempotent> <linear map> $P$, $I-P$ is also <idempotent> and
$$
\ker P=\operatorname{im}(I-P),\qquad \ker(I-P)=\operatorname{im}P.
$$
For 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>.
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