Editing Orthonormal basis (section)

==Examples==

* For <math>\mathbb{R}^3</math>, the set of vectors <math>\left\{ \mathbf{e_1} =
\begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \ ,

\ \mathbf{e_2} =
\begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \ ,

\ \mathbf{e_3} =
\begin{pmatrix} 0 & 0 & 1 \end{pmatrix}

\right\},</math> is called the '''standard basis''' and forms an orthonormal basis of <math>\mathbb{R}^3</math> with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing <math>\mathbb{R}^3</math> as the [[Cartesian product]] <math>\mathbb{R}\times\mathbb{R}\times\mathbb{R}</math>
*:'''Proof:''' A straightforward computation shows that the inner products of these vectors equals zero, <math>\left\langle \mathbf{e_1}, \mathbf{e_2} \right\rangle = \left\langle \mathbf{e_1}, \mathbf{e_3} \right\rangle = \left\langle \mathbf{e_2}, \mathbf{e_3} \right\rangle = 0</math> and that each of their magnitudes equals one, <math>\left\|\mathbf{e_1}\right\| = \left\|\mathbf{e_2}\right\| = \left\|\mathbf{e_3}\right\| = 1.</math> This means that <math>\left\{\mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3}\right\}</math> is an orthonormal set. All vectors <math>(\mathbf{x}, \mathbf{y}, \mathbf{z}) \in \R^3</math> can be expressed as a sum of the basis vectors scaled <math display="block"> (\mathbf{x},\mathbf{y},\mathbf{z}) = \mathbf{x e_1} + \mathbf{y e_2} + \mathbf{z e_3},</math> so <math>\left\{\mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3}\right\}</math> spans <math>\R^3</math> and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin also forms an orthonormal basis of <math>\R^3</math>.
* For <math>\mathbb{R}^n</math>, the standard basis and inner product are similarly defined. Any other orthonormal basis is related to the standard basis by an [[orthogonal transformation]] in the group O(n).
* For [[pseudo-Euclidean space]] <math>\mathbb{R}^{p,q},</math>, an orthogonal basis <math>\{e_\mu\}</math> with metric <math>\eta</math> instead satisfies <math>\eta(e_\mu,e_\nu) = 0</math> if <math>\mu\neq \nu</math>, <math>\eta(e_\mu,e_\mu) = +1</math> if <math>1\leq\mu\leq p</math>, and <math>\eta(e_\mu,e_\mu) =-1</math> if <math>p+1\leq\mu\leq p+q</math>. Any two orthonormal bases are related by a pseudo-orthogonal transformation. In the case <math>(p,q) = (1,3)</math>, these are Lorentz transformations.
* The set <math>\left\{f_n : n \in \Z\right\}</math> with <math>f_n(x) = \exp(2 \pi inx),</math> where <math>\exp</math> denotes the [[exponential function]], forms an orthonormal basis of the space of functions with finite Lebesgue integrals, <math>L^2([0,1]),</math> with respect to the [[2-norm]]. This is fundamental to the study of [[Fourier series]].
* The set <math>\left\{e_b : b \in B\right\}</math> with <math>e_b(c) = 1</math> if <math>b = c</math> and <math>e_b(c) = 0</math> otherwise forms an orthonormal basis of <math>\ell^2(B).</math>
* [[Eigenfunction|Eigenfunctions]] of a [[Sturm–Liouville eigenproblem]].
* The [[column vectors]] of an [[orthogonal matrix]] form an orthonormal set.