Difference between revisions of "Bilinear map"

Revision as of 07:43, 9 June 2015

A bilinear map combines elements from 2 vector spaces to yield and element in a third (in contrast to a linear map which takes a point in a vector space to a point in a different vector space)

It is sometimes called a "Bilinear form"

Definition

Given the vector spaces [ilmath](U,F),(V,F)[/ilmath] and [ilmath](W,F)[/ilmath] - it is important they are over the same field - a bilinear map is a function:

[math]\tau:(U,F)\times(V,F)\rightarrow(W,F)[/math]
or
[math]\tau:U\times V\rightarrow W[/math] (in keeping with mathematicians are lazy)

Such that it is linear in both parts. Which is to say that the following "Axioms of a bilinear map" hold:

Axioms of a bilinear map

For a function [math]\tau:U\times V\rightarrow W[/math] and [math]u,v\in U[/math], [math]a,b\in V[/math] and [math]\lambda,\mu\in F[/math] we have:

1. [math]\tau(\lambda u+\mu v,a)=\lambda \tau(u,a)+\mu \tau(v,a)[/math]
2. [math]\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)[/math]

Common notations

If an author uses [math]T[/math] for linear maps they will probably use [math]\tau[/math] for bilinear maps.

If an author uses [math]L[/math] for linear maps they will probably use [math]B[/math] for bilinear maps.

TODO: Find the book I got this from!