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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 $:

Please check them!

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

\begin{displaymath}\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 $\mathcal{V}$.

Noah Dana-Picard