Editing Orthogonal basis (section)

== 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 }}.