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.

A bilinear map can also be defined for modules. For that, see the article pairing.

Let ${\displaystyle V,W}$ and ${\displaystyle X}$ be three vector spaces over the same base field ${\displaystyle F}$. A bilinear map is a function $${\displaystyle B:V\times W\to X}$$ such that for all ${\displaystyle w\in W}$, the map ${\displaystyle B_{w}}$ $${\displaystyle v\mapsto B(v,w)}$$ is a linear map from ${\displaystyle V}$ to ${\displaystyle X,}$ and for all ${\displaystyle v\in V}$, the map ${\displaystyle B_{v}}$ $${\displaystyle w\mapsto B(v,w)}$$ is a linear map from ${\displaystyle W}$ to ${\displaystyle X.}$ In other words, when we hold the second entry of the bilinear map fixed while letting the first entry vary, yielding ${\displaystyle B_{w}}$, the result is a linear operator, and similarly for when we hold the first entry fixed.

Such a map ${\displaystyle B}$ satisfies the following properties.

If ${\displaystyle V=W}$ and we have B(v, w) = B(w, v) for all ${\displaystyle v,w\in V,}$ then we say that B is symmetric. If X is the base field F, then the map is called a bilinear form, which are well-studied (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

for all m in M, n in N, r in R and s in S, as well as B being additive in each argument.