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In ], particularly ], an '''orthogonal basis''' for an ] {{mathmvar|''V''}} is a ] for {{mathmvar|''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 ].
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==
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,
