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.

Explanations

L1 human

A form is nondegenerate when it can detect every nonzero vector. For each nonzero $u$, some $v$ gives $B(u, v) \ne 0$. In matrix terms, the matrix of the form is invertible.