Bilinear map

A bilinear map combines elements from 2 vector spaces to yield and element in a third (in contrast to a linear map which takes a point in a vector space to a point in a different vector space)

It is sometimes called a "Bilinear form"

Definition

Given the vector spaces $(U,F),(V,F)$ and $(W,F)$ - it is important they are over the same field - a bilinear map is a function:

$$\tau:(U,F)\times(V,F)\rightarrow(W,F)$$
or
$$\tau:U\times V\rightarrow W$$ (in keeping with mathematicians are lazy)

Such that it is linear in both parts. Which is to say that the following "Axioms of a bilinear map" hold:

Axioms of a bilinear map

For a function

$$\tau:U\times V\rightarrow W$$ and $$u,v\in U$$ , $$a,b\in V$$ and $$\lambda,\mu\in F$$ we have:

1. $$\tau(\lambda u+\mu v,a)=\lambda \tau(u,a)+\mu \tau(v,a)$$
2. $$\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)$$

Common notations

If an author uses

$$T$$ for linear maps they will probably use $$\tau$$ for bilinear maps.

If an author uses

$$L$$ for linear maps they will probably use $$B$$ for bilinear maps.