# 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 {ek} :

${\displaystyle \langle \mathbf {e} _{j},\mathbf {e} _{k}\rangle =\left\{{\begin{array}{ll}q(\mathbf {e} _{k})&j=k\\0&j\neq k\end{array}}\right.\quad ,}$

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 {ek},

${\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle =\sum \limits _{k}q(\mathbf {e} _{k})v^{k}w^{k}\ ,}$

where vk and wk 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.