Bilinear form

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

Definition

Let [ilmath](V,F)[/ilmath] be a vector space over a field [ilmath]F[/ilmath], a mapping:

• [math]\langle\cdot,\cdot\rangle:V\times V\rightarrow F[/math]

is a bilinear form if:

• It is bilinear, that is to say:
  • It is 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 property when it has any of the following properties:

Property	Definition	Comment
Symmetric	[math]\forall x,y\in V[\langle x,y\rangle=\langle y,x\rangle][/math]
Skew-symmetric or Antisymmetric	[math]\forall x,y\in V[\langle x,y\rangle=-\langle y,x\rangle][/math]	I use antisymmetric
Alternate or Alternating	[math]\forall x\in V[\langle x,x\rangle=0][/math]	I use alternating. [ilmath]0[/ilmath] denotes the additive identity of [ilmath]F[/ilmath]

See also

• Inner product
• Bilinear map
• Linear map

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TODO: Unite with inner product over real or complex field. Page 260 in Roman and page 206 are what is needed

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