34 - Bilinear form

Bilinear form

Let $V$ be a vector space over $\mathbf{F}$
A bilinear form $B$ on $V$ is a function $B:V\times V\to \mathbf{F}$ such that

1. $B({\alpha }_1{v}_1+{\alpha }_2{v}_2,v_3)={\alpha }_{1}B(v_1,v_3)+{\alpha }_{2}B(v_2,v_3)\text{ for all }{v}_1,v_2,v_3\in V\text{ and }{\alpha }_1,{\alpha }_2\in \mathbf{F}$
2. $B(v_1,{\alpha }_2{v}_2+{\alpha }_3{v}_3)={\alpha }_{2}B(v_1,v_2)+{\alpha }_{3}B(v_1,v_3)\text{ for all }{v}_1,v_2,v_3\in V\text{ and }{\alpha }_2,{\alpha }_3\in \mathbf{F}$

$(1)$ is linear in the first variable and $(2)$ is linear in the second variable

Dot Product

For $\mathbf{x}=(x_1,\cdots ,x_n)\in {\mathbb{F}}^n$ and $\mathbf{y}=(y_1,\cdots ,y_n)\in {\mathbb{F}}^n$
Then we define

$$B(\mathbf{x},\mathbf{y})=x_1{y}_1+\cdots +x_n{y}_n=\mathbf{x}{\mathbf{y}}^T$$