Bilinear map

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In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.

Contents

Definition

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

such that for any w in W the map

is a linear map from V to X, and for any v in V the map

is a linear map from W to X.

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 MR and a left module RN, we can define a bilinear map B : M × NT, where T is an abelian group, such that for any n in N, mB(m, n) is a group homomorphism, and for any m in M, nB(m, n) is a group homomorphism too, and which also satisfies

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 dimV×dimW (while the space L(V×W;K) of linear forms is of dimension dimV+dimW). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(ei,fj), and vice versa. Now, if X is a space of higher dimension, we obviously have dimL(V,W;X)=dimV×dimW×dimX.

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