# Bilinear map/Definition

• $\tau:(U,F)\times(V,F)\rightarrow(W,F)$ or
• $\tau:U\times V\rightarrow W$ (in keeping with mathematicians are lazy)
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. $\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)$