Editing Bilinear form (section)

===Reflexive bilinear forms and orthogonal vectors===
{{block indent|left=1| '''Definition:''' A bilinear form {{math|''B'' : ''V'' × ''V'' → ''K''}} is called '''reflexive''' if {{math|1=''B''('''v''', '''w''') = 0}} implies {{math|1=''B''('''w''', '''v''') = 0}} for all '''v''', '''w''' in ''V''.}}
{{block indent|left=1| '''Definition:''' Let {{math|''B'' : ''V'' × ''V'' → ''K''}} be a reflexive bilinear form. '''v''', '''w''' in ''V'' are '''orthogonal with respect to ''B''''' if {{math|1=''B''('''v''', '''w''') = 0}}.}}

A bilinear form {{math|''B''}} is reflexive if and only if it is either symmetric or alternating.{{sfn|Grove|1997}}  In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the ''kernel'' or the ''radical'' of the bilinear form: the subspace of all vectors orthogonal with every other vector.  A vector {{math|'''v'''}}, with matrix representation {{math|''x''}}, is in the radical of a bilinear form with matrix representation {{math|''A''}}, if and only if {{math|1=''Ax'' = 0 ⇔ ''x''<sup>T</sup>''A'' = 0}}. The radical is always a subspace of {{math|''V''}}.  It is trivial if and only if the matrix {{math|''A''}} is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose {{mvar|W}} is a subspace.  Define the ''[[orthogonal complement]]''{{sfn|Adkins|Weintraub|1992|page=359}}
<math display="block"> W^{\perp} = \left\{\mathbf{v} \mid B(\mathbf{v}, \mathbf{w}) = 0 \text{ for all } \mathbf{w} \in W\right\} .</math>

For a non-degenerate form on a finite-dimensional space, the map {{math|''V/W'' → ''W''<sup>⊥</sup>}} is [[bijective]], and the dimension of {{math|''W''<sup>⊥</sup>}} is {{math|dim(''V'') − dim(''W'')}}.