Orthonormal Basis

Definition: Orthonormal Basis

An orthonormal basis of an inner product space is an orthogonal basis $B$, where the canonical norm of each basis element is equal to one:

$$||\mathbf{v}||=1\forall \mathbf{v}\in B$$

Theorem: Vector Representation through an Orthonormal Basis

If $B=\{{\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n\}$ is an orthonormal basis of an inner product space $(V,F,+,\cdot )$, then the $i$-th coefficient $c_i$ in the basis representation $\mathbf{v}={\sum }_{i=1}^n{c}_i{\mathbf{b}}_i$ of any $\mathbf{v}\in V$ is the inner product of $\mathbf{v}$ with the $i$-th basis vector ${\mathbf{b}}_i$:

$$c_i=⟨\mathbf{v},{\mathbf{b}}_i⟩$$

PROOF

$$⟨\mathbf{v},{\mathbf{b}}_i⟩=c_1⟨{\mathbf{b}}_1,{\mathbf{b}}_i⟩+\cdots +c_i⟨{\mathbf{b}}_i,{\mathbf{b}}_i⟩+\cdots +c_n⟨{\mathbf{b}}_n,{\mathbf{b}}_i⟩$$

Since the basis is orthogonal, $⟨{\mathbf{b}}_j,{\mathbf{b}}_i⟩=0$ for all $j\ne i$. Moreover, the basis is orthonormal and so $⟨{\mathbf{b}}_i,{\mathbf{b}}_i⟩=||{\mathbf{b}}_i|{|}^2=1^2=1$ which means that

$$⟨\mathbf{v},{\mathbf{b}}_i⟩=c_1\underset{0}{\underbrace{⟨{\mathbf{b}}_1,{\mathbf{b}}_i⟩}}+\cdots +c_i\underset{1}{\underbrace{⟨{\mathbf{b}}_i,{\mathbf{b}}_i⟩}}+\cdots +c_n\underset{0}{\underbrace{⟨{\mathbf{b}}_n,{\mathbf{b}}_i⟩}}=c_i$$