Difference between revisions of "Bilinear map"
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A bilinear map combines elements from 2 [[Vector space|vector spaces]] to yield and element in a third (in contrast to a [[Linear map|linear map]] which takes a point in a vector space to a point in a different vector space) | A bilinear map combines elements from 2 [[Vector space|vector spaces]] to yield and element in a third (in contrast to a [[Linear map|linear map]] which takes a point in a vector space to a point in a different vector space) | ||
| − | + | A ''[[Bilinear form|bilinear form]]'' is a special case of a bilinear map, and an ''[[Inner product|inner product]]'' is a special case of a bilinear form. | |
| + | __TOC__ | ||
==Definition== | ==Definition== | ||
| − | Given the [[Vector space|vector spaces]] {{M|(U,F),(V,F)}} and {{M|(W,F)}} - it is important they are over the same field - a bilinear map is a [[Function|function]]: | + | Given the [[Vector space|vector spaces]] {{M|(U,F),(V,F)}} and {{M|(W,F)}} - it is important they are over the same field - a ''bilinear map''<ref name="Roman">Advanced Linear Algebra - Steven Roman - Third Edition - Springer Graduate texts in Mathematics</ref> is a [[Function|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|mathematicians are lazy]]) | ||
| + | Such that it is [[Linear map|linear]] in both variables. Which is to say that the following "Axioms of a bilinear map" hold: | ||
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For a [[Function|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: | For a [[Function|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(\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> | # <math>\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)</math> | ||
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| + | ==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: | ||
| + | * [[Bilinear form]] | ||
| + | * [[Inner product]] | ||
| + | For more information | ||
==Common notations== | ==Common notations== | ||
| − | If an author uses <math>T</math> for [[Linear map|linear maps]] they will probably use <math>\tau</math> for bilinear maps. | + | * If an author uses <math>T</math> for [[Linear map|linear maps]] they will probably use <math>\tau</math> for bilinear maps. |
| + | * If an author uses <math>L</math> for [[Linear map|linear maps]] they will probably use <math>B</math> for bilinear maps. | ||
| + | As always I recommend writing: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | | Let {{M|\tau:U\times V\rightarrow W}} 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|inner product]]'' | ||
| + | ==See next== | ||
| + | * [[Bilinear form]] | ||
| + | ==See also== | ||
| + | * [[Bilinear form]] | ||
| + | * [[Inner product]] | ||
| + | * [[Linear map]] | ||
| + | * [[Tensor product]] | ||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
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Revision as of 08:34, 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:
For more information
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
