Bilinear map/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 variables. Which is to say that the following "Axioms of a bilinear map" hold:

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. $$\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)$$
3. Advanced Linear Algebra - Steven Roman - Third Edition - Springer Graduate texts in Mathematics