= Inner product adapted to an idempotent linear map
{title2=$\langle u,v\rangle_P=\langle Pu,Pv\rangle+\langle(I-P)u,(I-P)v\rangle$}
For an <idempotent linear map> $P$ on a finite-dimensional real <vector space>, take any positive definite <inner product> on that space. The displayed new <inner product> is positive definite since its squared norm vanishes only if both $Pu$ and $(I-P)u$ vanish. It makes the <kernel of a linear map> and <image of a linear map> orthogonal. Since $P^2=P$, one has $\langle Pu,v\rangle_P=\langle u,Pv\rangle_P$; thus $P$ becomes an <orthogonal projection>.
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