# Orthogonal basis

In mathematics, particularly linear algebra, an **orthogonal basis** for an inner product space V is a basis for V 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 V. 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 V (over any field) equipped with a symmetric bilinear form ⟨·,·⟩, where *orthogonality* of two vectors **v** and **w** means ⟨**v**, **w**⟩ = 0. For an orthogonal basis {**e**_{k}} :

where q is a quadratic form associated with ⟨·,·⟩: *q*(**v**) = ⟨**v**, **v**⟩ (in an inner product space *q*(**v**) = | **v** |^{2}).

Hence for an orthogonal basis {**e**_{k}},

where v^{k} and w^{k} are components of **v** and **w** in the basis.

## References

- Lang, Serge (2004),
*Algebra*, Graduate Texts in Mathematics,**211**(Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4 - Milnor, J.; Husemoller, D. (1973).
*Symmetric Bilinear Forms*. Ergebnisse der Mathematik und ihrer Grenzgebiete.**73**. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.