Difference between revisions of "Bilinear map"
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==Relation to bilinear forms and inner products== | ==Relation to bilinear forms and inner products== | ||
A ''[[Bilinear form|bilinear form]]'' is a special case of a bilinear map where rather than mapping to a vector space {{M|W}} it maps to the field that the vector spaces {{M|U}} and {{M|V}} are over (which in this case was {{M|F}})<ref name="Roman"/>. An ''[[Inner product|inner product]]'' is a special case of that. See the pages: | A ''[[Bilinear form|bilinear form]]'' is a special case of a bilinear map where rather than mapping to a vector space {{M|W}} it maps to the field that the vector spaces {{M|U}} and {{M|V}} are over (which in this case was {{M|F}})<ref name="Roman"/>. An ''[[Inner product|inner product]]'' is a special case of that. See the pages: | ||
| − | * [[Bilinear form]] | + | * [[Bilinear form]] - a map of the form {{M|\langle\cdot,\cdot\rangle:V\times V\rightarrow F}} where {{M|V}} is a vector space over {{M|F}}<ref name="Roman"/> |
| − | * [[Inner product]] | + | * [[Inner product]] - a bilinear form that is either ''symmetric'', ''skew-symmetric'' or ''alternate'' (see the [[Bilinear form]] for meanings)<ref name="Roman"/> |
| − | + | ||
==Common notations== | ==Common notations== | ||
Revision as of 08:37, 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)
A bilinear form is a special case of a bilinear map, and an inner product is a special case of a bilinear form.
Contents
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[1] 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 variables. Which is to say that the following "Axioms of a bilinear map" hold:
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:
- [math]\tau(\lambda u+\mu v,a)=\lambda \tau(u,a)+\mu \tau(v,a)[/math]
- [math]\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)[/math]
Relation to bilinear forms and inner products
A bilinear form is a special case of a bilinear map where rather than mapping to a vector space [ilmath]W[/ilmath] it maps to the field that the vector spaces [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are over (which in this case was [ilmath]F[/ilmath])[1]. An inner product is a special case of that. See the pages:
- Bilinear form - a map of the form [ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow F[/ilmath] where [ilmath]V[/ilmath] is a vector space over [ilmath]F[/ilmath][1]
- Inner product - a bilinear form that is either symmetric, skew-symmetric or alternate (see the Bilinear form for meanings)[1]
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.
As always I recommend writing:
| Let [ilmath]\tau:U\times V\rightarrow W[/ilmath] be a bilinear map |
Or something explicit.
Examples of bilinear maps
- The Tensor product
- The Dot product - although this is an example of an inner product
