Linear and multilinear algebra; matrix theory Definition

Nondegenerate bilinear form

Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$ . A bilinear form $B$ on $V$ is nondegenerate when, for each $u \in V$ , $B(u, v) = 0 \text{ for all } v \in V \implies u = 0.$ A bilinear form that is not nondegenerate is degenerate.

Remarks

$B$ is nondegenerate if and only if its matrix in a basis has a nonzero determinant. So, in finite dimensions, the condition in the first argument is equivalent to the same condition in the second argument. $B$ is nondegenerate if and only if $u \mapsto B(u, \cdot)$ is an isomorphism from $V$ to its dual space. Each inner product is nondegenerate; the form $B(x, y) = x_1 y_1$ on $\R^2$ is degenerate.

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Vector space definition
Bilinear form definition
Nondegenerate bilinear form definition

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Tag	`0KAN330PAH`
URI	`m2://0KAN330PAH`
Kind	definition
Writing	W0 stub
MSC	15A63
Defines	Nondegenerate bilinear form
Uses	Bilinear form

Nondegenerate bilinear form

Definition

Remarks

Reading path

Where this leads

Further reading

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