Is every bilinear form a linear transformation?

Question

Let $B: \mathbb{R}×\mathbb{R}\to\mathbb{R}$ be a map defined by $B(a,b) = ab.$

This map $B$ is linear in both the coordinates but it is not linear as: $B((a,b)+(x,y)) = B(a+x,b+y) = (a+x)(b+y) \neq B(a,b) + B(x,y)$ So can we say that not every bilinear form is a linear transfornation?

Answer 1

A bilinear map $V\times W\to Z$ is linear if and only if it is the zero map, courtesy of the necessary condition $\Phi(x,y)=\Phi(0,y)+\Phi(x,0)=0+0=0$.