Orthonormal Basis

linear-algebra

Definition

Orthonormal Basis

Let $(V,⟨\cdot |\cdot ⟩)$ be an inner product space. A family of vectors $\{e_i{\}}_{i\in I}\subseteq V$ is an orthonormal basis if it is orthonormal and

$$\overline{\mathrm{span}\{e_i:i\in I\}}=V,$$

where $\overline{\cdot }$ denotes topological closure (in finite dimensions this reduces to $\mathrm{span}\{e_i\}=V$).

Every $v\in V$ then admits the expansion

$$v=\sum _i⟨e_i|v⟩e_i.$$

Parseval Identity

For $c_i=⟨e_i|v⟩$, the coefficients satisfy

$$\Vert v{\Vert }^2=\sum _i|c_i{|}^2,$$

i.e. Parseval’s identity.

Gram-Schmidt

Gram-Schmidt

Given a linearly independent basis, we can form linear combinations of the basis vectors to obtain an orthonormal basis.

Examples

Trivial Orthonormal Basis of $\mathbb{R}$

The trivial orthonormal basis of $\mathbb{R}$ is the family $\{1\}$. With the usual inner product on $\mathbb{R}$,

$$⟨1,1⟩=1.$$

Every $x\in \mathbb{R}$ can then be written as

$$x=⟨1,x⟩\cdot 1=x\cdot 1=x.$$

Orthonormal Basis of ${\mathbb{R}}^2$

The standard basis of ${\mathbb{R}}^2$ is the family

$$\{e_1,e_2\}=\left\{\left(\begin{array}{c}1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right)\right\}.$$

With the usual inner product,

$$⟨e_1,e_1⟩=1,⟨e_2,e_2⟩=1,⟨e_1,e_2⟩=0.$$

So every vector

$$v=\left(\begin{array}{c}x\\ y\end{array}\right)$$

can be written as

$$v=⟨e_1,v⟩e_1+⟨e_2,v⟩e_2=x{e}_1+y{e}_2.$$