# 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:

• $\langle\cdot,\cdot\rangle:V\times V\rightarrow F$

is a bilinear form[1] if:

• It is bilinear, that is to say:
• It is linear in each coordinate, which is to say:
• $\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle\alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle]$ and
• $\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle]$

## Properties

We say the bilinear form is property when it has any of the following properties:

Property Definition Comment
Symmetric[1] $\forall x,y\in V[\langle x,y\rangle=\langle y,x\rangle]$
Skew-symmetric[1] or Antisymmetric $\forall x,y\in V[\langle x,y\rangle=-\langle y,x\rangle]$ I use antisymmetric
Alternate[1] or Alternating $\forall x\in V[\langle x,x\rangle=0]$ I use alternating. [ilmath]0[/ilmath] denotes the additive identity of [ilmath]F[/ilmath]