Bilinear form - Imported Knowlpedia Content

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

B:V\times W\to \mathbb{F}

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

B(a u_1+b u_2,\, v)=a\,B(u_1,v)+b\,B(u_2,v),\qquad B(u,\, a v_1+b v_2)=a\,B(u,v_1)+b\,B(u,v_2).

Remarks

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.