- 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*]

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