Orthogonal Projection of a Vector

Topics: Algebra - Orthonormal and Orthogonal Set - Vector

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(definition)

Let $H\le {\mathbb{R}}^n$ with an orthonormal basis $\{u_1,\dots ,u_k\}$.

If $v\in {\mathbb{R}}^n$, then the orthogonal projection of $v$ on $H$, denoted as ${\text{proj}}_{H}v$, is defined as:

$${\text{proj}}_{H}v:=⟨v,u_1⟩u_1+⟨v,u_2⟩u_2+\cdots +⟨v,u_k⟩u_k$$

Note

The orthogonal projection is well-defined and it does not depend on the basis choice.

Projections and Orthonormal Bases §

(theorem)

Let $\beta =\{u_1,\dots ,u_n\}$ be an orthonormal basis for ${\mathbb{R}}^n$. , and let $v\in R$. Then:

$$v=⟨v,u_1⟩u_1+\cdots +⟨v,u_n⟩u_n$$

(theorem)

Let $H\le {\mathbb{R}}^n$ have two orthonormal bases $\{u_1,\dots ,u_k\}$ and $\{w_1,\dots ,w_k\}$.

If $v\in {\mathbb{R}}^n$, then:

$$⟨v,u_1⟩u_1+\cdots +⟨v,u_k⟩u_k=⟨v,w_1⟩w_1+\cdots +⟨v,w_k⟩w_k$$