bilinear map
Let be a ring, and let , and be modules over . A function is said to be a bilinear map if for each the function defined by for all is linear (http://planetmath.org/LinearTransformation) (that is, an -module homomorphism), and for each the function defined by for all is linear. Sometimes we may say that the function is -bilinear, .
A common case is a bilinear map , where is a vector space over a field ; the vector space with this operation then forms an algebra over .
If is a commutative ring, then every -bilinear map corresponds in a natural way to a linear map , where is the tensor product of and (over ).
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 |