Bilinear Form

linear-algebra

Bilinear Form

Let $V$ be a vector space over a field $F$.

A bilinear form on $V$ is a map

$$B:V\times V\to F$$

such that for all $x,x_1,x_2,y,y_1,y_2\in V$ and all scalars $\alpha ,\beta \in F$:

• Linearity in the first argument $$B(\alpha x_1+\beta x_2,y)=\alpha B(x_1,y)+\beta B(x_2,y),$$
• Linearity in the second argument $$B(x,\alpha y_1+\beta y_2)=\alpha B(x,y_1)+\beta B(x,y_2).$$

Equivalently: for each fixed $y$, the map $x\mapsto B(x,y)$ is linear, and for each fixed $x$, the map $y\mapsto B(x,y)$ is linear.