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| In ], particularly ], an '''orthogonal basis''' for an ] |
In ], particularly ], an '''orthogonal basis''' for an ] {{mvar|V}} is a ] for {{mvar|V}} whose vectors are mutually ]. If the vectors of an orthogonal basis are ], the resulting basis is an ]. | ||
| In ], an orthogonal basis is any basis obtained from a orthonormal basis (or Hilbert basis) using multiplication by nonzero ]. | In ], an orthogonal basis is any basis obtained from a orthonormal basis (or Hilbert basis) using multiplication by nonzero ]. | ||
| Any orthogonal basis can be used to define a system of ]. | Any orthogonal basis can be used to define a system of ] {{mvar|V}}. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from ] orthogonal coordinates in ]s, as well as in ] and ] manifolds. | ||
| The concept of an orthogonal (but not of an orthonormal) basis is applicable to a ] {{mvar|V}} (over any ]) equipped with a ] {{math|{{langle}}·,·{{rangle}}}}, where '']'' of two vectors {{math|'''v'''}} and {{math|'''w'''}} means {{math|1={{langle}}'''v''', '''w'''{{rangle}} = 0}}. For an orthogonal basis {{math|{'''e'''<sub>''k''</sub>} }}: | |||
| A linear combination of the elements of a basis can be used to reach any point in the vector space, so of course the same property holds in the specific case of an orthogonal basis. | |||
| :<math>\langle\mathbf{e}_j,\mathbf{e}_k\rangle = | |||
| \left\{\begin{array}{ll}q(\mathbf{e}_k) & j = k \\ 0 & j \ne k | |||
| \end{array}\right.\quad,</math> | |||
| where {{mvar|q}} is a ] associated with {{math|{{langle}}·,·{{rangle}}}}: {{math|1=''q''('''v''') = {{langle}}'''v''', '''v'''{{rangle}}}} (in an inner product space {{math|1=''q''('''v''') = {{!}} '''v''' {{!}}<sup>2</sup>}}). Hence, | |||
| :<math>\langle\mathbf{v},\mathbf{w}\rangle = | |||
| \sum\limits_{k} q(\mathbf{e}_k) v^k w^k\ ,</math> | |||
| where {{mvar|v<sup>k</sup>}} and {{mvar|w<sup>k</sup>}} are components of {{math|'''v'''}} and {{math|'''w'''}} in {{math|{'''e'''<sub>''k''</sub>} }}. | |||
| ==References== | ==References== | ||
Revision as of 08:21, 10 June 2013
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.
In functional analysis, an orthogonal basis is any basis obtained from a orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.
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.
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} :
where q is a quadratic form associated with ⟨·,·⟩: q(v) = ⟨v, v⟩ (in an inner product space q(v) = | v |). Hence,
where v and w are components of v and w in {ek} .
References
- 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.
External links
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