Editing Orthogonal basis

{{Short description|Basis for v whose vectors are mutually orthogonal}}
In [[mathematics]], particularly [[linear algebra]], an '''orthogonal basis''' for an [[inner product space]] <math>V</math> is a [[basis (linear algebra)|basis]] for <math>V</math> whose vectors are mutually [[orthogonal]].  If the vectors of an orthogonal basis are [[Normalize (linear algebra)|normalized]], the resulting basis is an ''[[orthonormal basis]]''.

== As coordinates ==
Any orthogonal basis can be used to define a system of [[orthogonal coordinates]] <math>V.</math> Orthogonal (not necessarily orthonormal) bases are important due to their appearance from [[Curvilinear coordinates|curvilinear]] orthogonal coordinates in [[Euclidean space]]s, as well as in [[Riemannian manifold|Riemannian]] and [[Pseudo-Riemannian manifold|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 [[Scalar (mathematics)|scalars]].

== Extensions ==

=== Symmetric bilinear form ===
The concept of an orthogonal basis is applicable to a [[vector space]] <math>V</math> (over any [[Field (mathematics)|field]]) equipped with a [[symmetric bilinear form]] {{tmath|1= \langle \cdot, \cdot \rangle }}, where ''[[orthogonality]]'' of two vectors <math>v</math> and <math>w</math> means {{tmath|1= \langle v, w \rangle = 0 }}. For an orthogonal basis {{tmath|1= \left\{e_k\right\} }}:
<math display=block>\langle e_j, e_k\rangle =
\begin{cases}
q(e_k) & j = k \\
0      & j \neq k,
\end{cases}</math>
where <math>q</math> is a [[quadratic form]] associated with <math>\langle \cdot, \cdot \rangle:</math> <math>q(v) = \langle v, v \rangle</math> (in an inner product space, {{tmath|1= q(v) = \Vert v \Vert^2 }}).

Hence for an orthogonal basis {{tmath|1= \left\{e_k\right\} }}, 
<math display=block>\langle v, w \rangle = \sum_k q(e_k) v_k w_k,</math>
where <math>v_k</math> and <math>w_k</math> are components of <math>v</math> and <math>w</math> in the basis.

=== Quadratic form ===
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form {{tmath|1= q(v) }}.  Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form <math>\langle v, w \rangle = \tfrac{1}{2}(q(v+w) - q(v) - q(w))</math> allows vectors <math>v</math> and <math>w</math> to be defined as being orthogonal with respect to <math>q</math> when {{tmath|1= q(v+w) - q(v) - q(w) = 0 }}.

== See also ==
* {{annotated link|Basis (linear algebra)}}
* {{annotated link|Orthonormal basis}}
* {{annotated link|Affine space#Affine coordinates|Orthonormal frame}}
* {{annotated link|Schauder basis}}
* {{annotated link|Total set}}

== References ==
{{reflist}}
* {{Lang Algebra | edition=3r2004 | pages=572–585 }}
* {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 | page=6}}

== External links ==
* {{MathWorld|title=Orthogonal Basis|urlname=OrthogonalBasis}}

{{linear algebra}}
{{Hilbert space}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Linear algebra]]
[[de:Orthogonalbasis]]