Orthonormal bases.

Definition 12.4.1   Let $e= \left\{ \overrightarrow{e_i} , \; i \in I \right\}$ be a basis for the vector space $\mathcal{V}$. If $\forall i,j \in I, \; \langle \overrightarrow{e_i} ,\overrightarrow{e_j}\rangle=0$, the basis is called an orthogonal basis for $\mathcal{V}$.

Definition 12.4.2   An orthonormal basis of the vector space $\mathcal{V}$ is an orthogonal basis, all of whose vectors being unit vectors.

This means that in an orthonormal basis, the vectors are orthogonal and of length 1.

Example 12.4.3   Some orthonormal bases for $\mathbb{R} ^2$:

• $e= \left\{ \overrightarrow{e_1} =\begin{pmatrix}1 \\ 0 \end{pmatrix}, \overrightarrow{e_2} =\begin{pmatrix}0 \\ 1 \end{pmatrix} \right\}$.
• $f= \left\{ \overrightarrow{f_1} =\begin{pmatrix}0 \\ 1 \end{pmatrix}, \overrightarrow{f_2} =\begin{pmatrix}1 \\ 0 \end{pmatrix} \right\}$.
• $g=\left\{ \overrightarrow{g_1} =\begin{pmatrix}\frac 12 \\ \frac {\sqrt{3}}{2} ... ...row{g_2} =\begin{pmatrix}\frac {\sqrt{3}}{2} \\ -\frac 12 \end{pmatrix}\right\}$
• $h=\left\{ \overrightarrow{h_1} =\begin{pmatrix}\frac {\sqrt{2}}2 \\ \frac {\sq... ...egin{pmatrix}-\frac {\sqrt{2}}{2} \\ \frac {\sqrt{2}}{2} \end{pmatrix} \right\}$

Please check them!

Example 12.4.4   Let $\mathcal{V}= \text{Span}( \cos x, \sin x )$, with the inner product

$$\langle f ,g \rangle = \frac {1}{\pi} \int_{- \pi}^{\pi } f(x)g(x) dx$$

Then the sine and the cosine function form an orthonormal basis of $\mathcal{V}$.