Bilinear form

A bilinear form on vector spaces $V$ and $W$ over a field $\mathbb{F}$ is a map

such that for all $u_1,u_2\in V$, $v_1,v_2\in W$, and $a,b\in\mathbb{F}$,

A bilinear form is a scalar-valued function on a Cartesian product of vector spaces. Inner products (see inner product ) are important examples with additional positivity and symmetry properties.

Examples:

• On $\mathbb{R}^n$, the dot product $B(x,y)=\sum_{i=1}^n x_i y_i$ is a bilinear form.
• For a fixed matrix $A$, the rule $B(x,y)=x^\mathsf{T}Ay$ on $\mathbb{F}^n$ is a bilinear form.
• On $\mathbb{R}^{2}$, the form $B((x_1,x_2),(y_1,y_2))=x_1y_2-x_2y_1$ is a bilinear form.