Bilinear map

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

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, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when 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 the base field 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 generalizes to n-ary functions, where the proper term is multilinear.

For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map B : M × N → T with T an (R, S)-bimodule, and for which any n in N, m ↦ B(m, n) is an R-module homomorphism, and for any m in M, n ↦ B(m, n) is an S-module homomorphism. This satisfies

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