bilinear map

Let R be a ring, and let M, N and P be modules over R. A function f:M×NP is said to be a bilinear map if for each bM the function h:NP defined by h(y)=f(b,y) for all yN is linear ( (that is, an R-module homomorphismMathworldPlanetmath), and for each cN the function g:MP defined by g(x)=f(x,c) for all xM is linear. Sometimes we may say that the function is R-bilinear, .

A common case is a bilinear map V×VV, where V is a vector spaceMathworldPlanetmath over a field K; the vector space with this operationMathworldPlanetmath then forms an algebra over K.

If R is a commutative ring, then every R-bilinear map M×NP corresponds in a natural way to a linear map MNP, where MN is the tensor productPlanetmathPlanetmathPlanetmath 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