bilinear map
Let R be a ring, and let M, N and P be modules over R.
A function f:M×N→P
is said to be a bilinear map
if for each b∈M the function h:N→P
defined by h(y)=f(b,y) for all y∈N is linear (http://planetmath.org/LinearTransformation)
(that is, an R-module homomorphism),
and for each c∈N the function g:M→P
defined by g(x)=f(x,c) for all x∈M is linear.
Sometimes we may say that the function is R-bilinear,
.
A common case is a bilinear map V×V→V,
where V is a vector space over a field K;
the vector space with this operation
then forms an algebra over K.
If R is a commutative ring, then every R-bilinear map M×N→P
corresponds in a natural way to a linear map M⊗N→P,
where M⊗N is the tensor product of M and N (over R).
Title | bilinear map |
Canonical name | BilinearMap |
Date of creation | 2013-03-22 15:35:47 |
Last modified on | 2013-03-22 15:35:47 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 11 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 13C99 |
Synonym | bilinear function |
Synonym | bilinear operation |
Synonym | bilinear mapping |
Synonym | bilinear operator |
Synonym | bilinear pairing |
Synonym | pairing |
Related topic | Multilinear |
Related topic | BilinearForm |
Related topic | ScalarMap |
Defines | bilinear |