Orthogonal Complement

Topics: Algebra - Orthonormal and Orthogonal Set - Vector

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(definition)

Let $H\in {\mathbb{R}}^n$. We define the orthogonal complement of $H$, denoted as $H^{\perp }$, as:

$$H^{\perp }=\{x\in {\mathbb{R}}^n|⟨x,h⟩=0\forall h\in H\}$$

Properties §

(theorem)

If $H\le {\mathbb{R}}^n$, then:

1. $H^{\perp }\le {\mathbb{R}}^n$
2. $H\cap H^{\perp }=\{0\}$
3. $\dim (H^{\perp })=n-\dim (H)$

Orthogonal Complement and Orthogonal Projection §

(theorem) (relative to orthogonal projection)

Let $H\le {\mathbb{R}}^n$ and let $v\in {\mathbb{R}}^n$.

Then, there exists a pair of vectors $h$ and $p$ such that $h\in H$, $p\in H^{\perp }$ and $v=h+p$.

In particular, if $h={\text{proj}}_{H}v$ and $p={\text{proj}}_{H}v$, then:

$$v=h+p={\text{proj}}_{H}v+{\text{proj}}_{H^{\perp }}v$$