Bilinear Map - Definition

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 map

v ↦ B(v, w)

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

w ↦ B(v, w)

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. Note that if we regard the product V × W as a vector space, then B is not a linear transformation of vector spaces (unless V = 0 or W = 0) because, for example B(2(v,w)) = B(2v,2w) = 2B(v,2w) = 4B(v,w).

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 over a field F, 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 too, and which also satisfies

B(mt, n) = B(m, tn)

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