Basis consisting of mutually orthogonal vectors
Inmathematics , particularlylinear algebra , anorthogonal basis for aninner product space V {\displaystyle V} basis forV {\displaystyle V} orthogonal . If the vectors of an orthogonal basis arenormalized , the resulting basis is anorthonormal basis
Any orthogonal basis can be used to define a system oforthogonal coordinates V . {\displaystyle V.} curvilinear orthogonal coordinates inEuclidean spaces , as well as inRiemannian andpseudo-Riemannian manifolds.
In functional analysis [ edit ] Infunctional analysis , an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzeroscalars .
Symmetric bilinear form [ edit ] The concept of an orthogonal basis is applicable to avector space V {\displaystyle V} field ) equipped with asymmetric bilinear form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , whereorthogonality v {\displaystyle v} w {\displaystyle w} ⟨ v , w ⟩ = 0 {\displaystyle \langle v,w\rangle =0} . For an orthogonal basis{ e k } {\displaystyle \left\{e_{k}\right\}} :⟨ e j , e k ⟩ = { q ( e k ) j = k 0 j ≠ k , {\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}} q {\displaystyle q} quadratic form associated with⟨ ⋅ , ⋅ ⟩ : {\displaystyle \langle \cdot ,\cdot \rangle :} q ( v ) = ⟨ v , v ⟩ {\displaystyle q(v)=\langle v,v\rangle } q ( v ) = ‖ v ‖ 2 {\displaystyle q(v)=\Vert v\Vert ^{2}} ).
Hence for an orthogonal basis{ e k } {\displaystyle \left\{e_{k}\right\}} ,⟨ v , w ⟩ = ∑ k q ( e k ) v k w k , {\displaystyle \langle v,w\rangle =\sum _{k}q(e_{k})v_{k}w_{k},} v k {\displaystyle v_{k}} w k {\displaystyle w_{k}} v {\displaystyle v} w {\displaystyle w}
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic formq ( v ) {\displaystyle q(v)} . Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form⟨ v , w △ = 1 2 ( q ( v + w ) − q ( v ) − q ( w ) ) {\displaystyle \langle v,w\triangle ={\tfrac {1}{2}}(q(v+w)-q(v)-q(w))} v {\displaystyle v} w {\displaystyle w} q {\displaystyle q} q ( v + w ) − q ( v ) − q ( w ) = 0 {\displaystyle q(v+w)-q(v)-q(w)=0} .
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 .
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