# Orthonormal basis

﻿
Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal.[1][2][3] For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for Rn arises in this fashion.

For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of Rn under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces (or pre-Hilbert spaces).[4] Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H, but it may not be the entire space.

## Examples

• The set of vectors {e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1)} (the standard basis) forms an orthonormal basis of R3.
Proof: A straightforward computation shows that the inner products of these vectors equals zero, <e1, e2> = <e1, e3> = <e2, e3> = 0 and that each of their magnitudes equals one, ||e1|| = ||e2|| = ||e3|| = 1. This means {e1, e2, e3} is an orthonormal set. All vectors (xyz) in R3 can be expressed as a sum of the basis vectors scaled
$(x,y,z) = xe_1 + ye_2 + ze_3, \,$
so {e1,e2,e3} spans R3 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 forms an orthonormal basis of R3.
• The set {fn : nZ} with fn(x) = exp(2πinx) forms an orthonormal basis of the complex space L2([0,1]). This is fundamental to the study of Fourier series.
• The set {eb : bB} with eb(c) = 1 if b = c and 0 otherwise forms an orthonormal basis of  2(B).
• Eigenfunctions of a Sturm–Liouville eigenproblem.
• An orthogonal matrix is a matrix whose column vectors form an orthonormal set.

## Basic formula

If B is an orthogonal basis of H, then every element x of H may be written as

$x=\sum_{b\in B}{\langle x,b\rangle\over\lVert b\rVert^2} b.$

When B is orthonormal, we have instead

$x=\sum_{b\in B}\langle x,b\rangle b$

and the norm of x can be given by

$\|x\|^2=\sum_{b\in B}|\langle x,b\rangle |^2.$

Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x, and the formula is usually known as Parseval's identity. See also Generalized Fourier series.

If B is an orthonormal basis of H, then H is isomorphic to  2(B) in the following sense: there exists a bijective linear map Φ : H ->  2(B) such that

$\langle\Phi(x),\Phi(y)\rangle=\langle x,y\rangle$

for all x and y in H.

## Incomplete orthogonal sets

Given a Hilbert space H and a set S of mutually orthogonal vectors in H, we can take the smallest closed linear subspace V of H containing S. Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.

## Existence

Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis.

## As a homogeneous space

The set of orthonormal bases for a space is a principal homogeneous space for the orthogonal group O(n), and is called the Stiefel manifold $V_n(\mathbf{R}^n)$ of orthonormal n-frames.

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds $V_k(\mathbf{R}^n)$ for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.

## References

1. ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4.
2. ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.
3. ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2.
4. ^ Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Basis (linear algebra) — Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference. In linear algebra, a basis is a set of linearly independent vectors that, in a linear… …   Wikipedia

• Orthonormal function system — An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. See basis (linear algebra), Fourier analysis, square integrable, Hilbert space for more. Categories: Mathematics stubsLinear algebraFunctional analysis …   Wikipedia

• orthonormal — adjective Date: 1932 1. of real valued functions orthogonal with the integral of the square of each function over a specified interval equal to one 2. being or composed of orthogonal elements of unit length < orthonormal basis of a vector space > …   New Collegiate Dictionary

• orthonormal — ˌ adjective Etymology: orth + normal 1. of real valued functions : orthogonal with the integral of the square of each function over a specified interval equal to one 2. : being or composed of orthogonal elements of unit length orthonormal basis… …   Useful english dictionary

• Orthonormal — Als orthonormal (genauer: zueinander orthonormal) werden in der Mathematik Vektoren bezeichnet, die zueinander orthogonal sind und alle die Norm (anschaulich: Länge) eins besitzen. Eine Basis eines Vektorraums aus orthonormalen Vektoren bildet… …   Deutsch Wikipedia

• Orthonormal frame — In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M… …   Wikipedia

• Basis function — In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The term is a degeneration of the term basis vector for a more general vector space; that is, each function in the function space… …   Wikipedia

• Standard basis — In mathematics, the standard basis (also called natural basis or canonical basis) of the n dimensional Euclidean space Rn is the basis obtained by taking the n basis vectors:{ e i : 1leq ileq n}where e i is the vector with a 1 in the ith… …   Wikipedia

• Change of basis — In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases. Contents 1 Expression of a basis 2 Change of basis for vectors 2.1 Tensor proof …   Wikipedia

• Schauder basis — In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.