# Orthogonal basis

In mathematics, particularly linear algebra, an **orthogonal basis** for an inner product space Template:Mvar is a basis for Template:Mvar whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

## As coordinates

Any orthogonal basis can be used to define a system of orthogonal coordinates Template:Mvar. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

## In functional analysis

In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.

## Extensions

The concept of an orthogonal (but not of an orthonormal) basis is applicable to a vector space Template:Mvar (over any field) equipped with a symmetric bilinear form Template:Langle·,·Template:Rangle, where *orthogonality* of two vectors **v** and **w** means Template:Langle**v**, **w**Template:Rangle = 0. For an orthogonal basis {**e**_{k}} :

where Template:Mvar is a quadratic form associated with Template:Langle·,·Template:Rangle: *q*(**v**) = Template:Langle**v**, **v**Template:Rangle (in an inner product space *q*(**v**) = | **v** |^{2}). Hence,

where Template:Mvar and Template:Mvar are components of **v** and **w** in {**e**_{k}} .

## References

- Template:Lang Algebra
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