# bilinear map

Let $R$ be a ring, and let $M$, $N$ and $P$ be modules over $R$.
A function $f:M\times N\to P$
is said to be a *bilinear map*
if for each $b\in M$ the function $h:N\to P$
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 $g:M\to P$
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 |