In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.
Any orthogonal basis can be used to define a system of orthogonal coordinates 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
Symmetric bilinear form
Hence for an orthogonal basis
The concept of orthogonality may be extended to a vector space (over any field) equipped with a quadratic form . Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form allows vectors and to be defined as being orthogonal with respect to when
- Basis (linear algebra) – Set of vectors used to define coordinates
- Orthonormal basis – Specific linear basis (mathematics)
- Orthonormal frame – Geometric structure that generalizes the Euclidean space
- Schauder basis
- Total set
- Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 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. Vol. 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.