Bilinear map

In mathematics, a bilinear map is a function of two arguments that is linear in each. An example of such a map is multiplication of integers.

Definition

Let "V", "W" and "X" be three vector spaces over the same base field "F". A bilinear map is a function:"B" : "V" × "W" → "X"such that for any "w" in "W" the
linear map from "V" to "X", and for any "v" in "V" the

In 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 bilinear form, is particularly useful (see for example scalar product, inner product and quadratic 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 "_RN", 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 a vector space "V" over the real numbers 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", a bilinear form on "V" is the same as a bilinear map "V" × "V" → "F".
* If "V" is a vector space with dual 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".
* The cross product in R³ is a bilinear map R³ × R³ → R³.
* Let "B" : "V" × "W" → "X" be a bilinear map, and "L" : "U" → "W" be a linear 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]