- Bilinear map
In

mathematics , a**bilinear map**is a function of two arguments that is linear in each. An example of such a map ismultiplication ofintegers .**Definition**Let "V", "W" and "X" be three

vector space s over the same base field "F". A bilinear map is a function:"B" : "V" × "W" → "X"such that for any "w" in "W" thelinear map from "V" to "X", and for any "v" in "V" theIn other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.

If "V" = "W" and we have "B"("v","w") = "B"("w","v") for all "v","w" in "V", then we say that "B" is "symmetric".

The case where "X" is "F", and we have a

, is particularly useful (see for examplebilinear form scalar product ,inner product andquadratic form ).The definition works without any changes if instead of vector spaces we use modules over a

commutative ring "R". It also can be easily generalized to "n"-ary functions, where the proper term is "multilinear".For the case of a non-commutative base ring "R" and a right module "M

_{R}" and a left module "_{R}N", we can define a bilinear map "B" : "M" × "N" → "T", where "T" is an abelian group, such that for any "n" in "N", "m" ↦ "B"("m", "n") is a group homomorphism, and for any "m" in "M", "n" ↦ "B"("m", "n") is a group homomorphism, and which also satisfies:"B"("mt", "n") = "B"("m", "tn")

for all "m" in "M", "n" in "N" and "t" in "R".

**Properties**A first immediate consequence of the definition is that $B(x,y)=o$whenever "x"=o or "y"=o. (This is seen by writing the

null vector "o" as 0·"o" and moving the scalar 0 "outside", in front of "B", by linearity.)The set "L(V,W;X)" of all bilinear maps is a

linear subspace of the space (viz. vector space , module) of all maps from "V"×"W" into "X".If "V","W","X" are

finite-dimensional , then so is "L(V,W;X)". For "X=F", i.e. bilinear forms, the dimension of this space is dim"V"×dim"W" (while the space "L(V×W;K)" of "linear" forms is of dimension dim"V"+dim"W"). To see this, choose a basis for "V" and "W"; then each bilinear map can be uniquely represented by the matrix $B(e\_i,f\_j)$, and vice versa. Now, if "X" is a space of higher dimension, we obviously have dim"L(V,W;X)"=dim"V"×dim"W"×dim"X".**Examples*** Matrix multiplication is a bilinear map M("m","n") × M("n","p") → M("m","p").

* If avector space "V" over thereal number s**R**carries an inner product, then the inner product is a bilinear map "V" × "V" →**R**.

* In general, for a vector space "V" over a field "F", abilinear form on "V" is the same as a bilinear map "V" × "V" → "F".

* If "V" is a vector space withdual space "V*", then the application operator, "b"("f", "v") = "f"("v") is a bilinear map from "V"* × "V" to the base field.

* Let "V" and "W" be vector spaces over the same base field "F". If "f" is a member of "V"* and "g" a member of "W"*, then "b"("v", "w") = "f"("v")"g"("w") defines a bilinear map "V" × "W" → "F".

* Thecross product in**R**^{3}is a bilinear map**R**^{3}×**R**^{3}→**R**^{3}.

* Let "B" : "V" × "W" → "X" be a bilinear map, and "L" : "U" → "W" be alinear operator , then ("v", "u") → "B"("v", "Lu") is a bilinear map on "V" × "U"

* The null map, defined by $B(v,w)\; =\; o$ for all ("v","w") in "V"×"W" is the only map from "V"×"W" to "X" which is bilinear and linear at the same time. Indeed, if ("v,w")∈"V"×"W", then if "B" is linear, $B(v,w)=\; B(v,o)+B(o,w)=o+o$ if "B" is bilinear.**ee also***

Tensor product

*Multilinear map

*Sesquilinear form

*Bilinear filtering **External links*** [

*http://www.umiacs.umd.edu/partnerships/lts/LTS_Report_Jan04.pdf Use of Bilinear maps in cryptography*] in [*http://wikileaks.org/wiki/On_the_take_and_loving_it NSA sponsored academic research*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Bilinear**— may refer to* Bilinear sampling, a method in computer graphics for choosing the color of a texture. * Bilinear form * Bilinear interpolation * Bilinear map, a type of mathematical function between vector spaces * Bilinear transform, a method of… … Wikipedia**Bilinear form**— In mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F , where F is the field of scalars. That is, a bilinear form is a function B : V × V → F which is linear in each argument separately::egin{array}{l} ext{1. }B(u + … Wikipedia**Bilinear transform**— The bilinear transform (also known as Tustin s method) is used in digital signal processing and discrete time control theory to transform continuous time system representations to discrete time and vice versa. The bilinear transform is a… … Wikipedia**Bilinear interpolation**— In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a regular grid. The key idea is to perform linear interpolation first in one direction, and then again in the other… … Wikipedia**Multilinear map**— In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function where and are vector spaces (or modules), with the following property: for each ,… … Wikipedia**Symmetric bilinear form**— A symmetric bilinear form is, as the name implies, a bilinear form on a vector space that is symmetric. They are of great importance in the study of orthogonal polarities and quadrics.They are also more briefly referred to as symmetric forms when … Wikipedia**SKY-MAP.ORG**— (or WIKISKY.ORG) is an interactive information management system which encompasses the entire outer space. The basic element of the system is a detailed map of the whole star sky that mirrors more than half a billion celestial objects. No… … Wikipedia**Abel–Jacobi map**— In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus.The name… … Wikipedia**Tensor product of modules**— In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (roughly speaking, multiplication ) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous… … Wikipedia**Tensor product**— In mathematics, the tensor product, denoted by otimes, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same:… … Wikipedia