# bilinear map

Let $R$ be a ring, and let $M$, $N$ and $P$ be modules over $R$. A function $f\colon M\times N\to P$ is said to be a bilinear map if for each $b\in M$ the function $h\colon N\to P$ defined by $h(y)=f(b,y)$ for all $y\in N$ is linear (http://planetmath.org/LinearTransformation) (that is, an $R$-module homomorphism), and for each $c\in N$ the function $g\colon M\to P$ defined by $g(x)=f(x,c)$ for all $x\in M$ is linear. Sometimes we may say that the function is $R$-bilinear, .

A common case is a bilinear map $V\times V\to V$, 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 $M\times N\to P$ corresponds in a natural way to a linear map $M\otimes N\to P$, where $M\otimes N$ 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