Linear and multilinear algebra; matrix theory Definition

Bilinear form

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . A bilinear form on $V$ is a map $B \colon V \times V \to F$ such that, for all $u, v, w \in V$ and $a \in F$ :

$B(u + v, w) = B(u, w) + B(v, w)$ and $B(au, w) = a B(u, w)$ .
$B(u, v + w) = B(u, v) + B(u, w)$ and $B(u, aw) = a B(u, w)$ .

Remarks

The dot product on $\R^n$ is a bilinear form. For each $n \times n$ matrix $M$ over $F$ , the map $B(x, y) = x^T M y$ is a bilinear form on $F^n$ . The bilinear forms on $V$ form a vector space over $F$ ; if $\dim V = n$ , it has dimension $n^2$ . More generally, a bilinear map $V \times W \to F$ uses two different spaces.

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Vector space definition
Bilinear form definition

Tag	`0P799D6AMN`
URI	`m2://0P799D6AMN`
Kind	definition
Writing	W1 stated
MSC	15A63
Defines	Bilinear form
Uses	Vector space

Bilinear form

Definition

Remarks

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