View source for Bilinear form

This is a specialisation of a [[Bilinear map]] and a generalisation of [[Inner product]]

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==Definition==
Let {{M|(V,F)}} be a [[Vector space|vector space]] over a [[Field|field]] {{M|F}}, a mapping:
* <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow F</math> 
is a ''bilinear form''<ref name="Roman">Advanced Linear Algebra - Steven Roman - Third Edition - Springer - Graduate Texts in Mathematics</ref> if:
* It is [[Bilinear map|bilinear]], that is to say:
** It is [[Linear map|linear]] in each coordinate, which is to say:
*** <math>\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle\alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle]</math> and
*** <math>\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle]</math>
==Properties==
We say the bilinear form is <code>property</code> when it has any of the following properties:
{| class="wikitable" border="1"
|-
! Property
! Definition
! Comment
|-
! ''Symmetric''<ref name="Roman"/>
| <math>\forall x,y\in V[\langle x,y\rangle=\langle y,x\rangle]</math>
|
|-
! ''Skew-symmetric''<ref name="Roman"/> or ''Antisymmetric''
| <math>\forall x,y\in V[\langle x,y\rangle=-\langle y,x\rangle]</math>
| I use ''antisymmetric''
|-
! ''Alternate''<ref name="Roman"/> or ''Alternating''
| <math>\forall x\in V[\langle x,x\rangle=0]</math>
| I use ''alternating''. {{M|0}} denotes the additive identity of {{M|F}}
|}
==See also==
* [[Inner product]]
* [[Bilinear map]]
* [[Linear map]]

{{Todo|Unite with inner product over real or complex field. Page 260 in Roman and page 206 are what is needed}}

==References==
<references/>

{{Definition|Linear Algebra}}