Bilinear Form

Bilinear form is a function that takes two vectors as inputs and returns a scalar.

What makes it bilinear is that it's linear for each input. If you keep one vector and vary the other, the function behaves linearly and vice versa.

A bilinear form $B$ on a vector space $V$ satisfies:

$$B(au + bv, w) = aB(u, w) + bB(v,w)$$

An example for this form would be dot product $B(u,v) = u \cdot v$.