Orthogonal Complement

linear-algebra

Definition

Orthogonal Complement

Let $V$ be an inner product space and $U\subseteq V$ a subspace. The orthogonal complement of $U$ is

$$U^{\perp }=\{v\in V\mid ⟨v,u⟩=0\forall u\in U\}.$$

Properties

Subspace

$U^{\perp }$ is itself a subspace of $V$.

Direct sum

For finite-dimensional $V$,

$$V=U\oplus U^{\perp }$$

meaning every $x\in V$ decomposes uniquely as $x=u+w$ with $u\in U$, $w\in U^{\perp }$.

Double complement

$(U^{\perp }{)}^{\perp }=U$ for finite-dimensional $V$.