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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.\nA [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *nondegenerate* when, for each $u \\in V$,\n$$B(u, v) = 0 \\text{ for all } v \\in V \\implies u = 0.$$\nA bilinear form that is not nondegenerate is *degenerate*.\n\n## Remarks\n\n$B$ is nondegenerate if and only if its matrix in a basis has a nonzero determinant.\nSo, in finite dimensions, the condition in the first argument is equivalent to the same condition in the second argument.\n$B$ is nondegenerate if and only if $u \\mapsto B(u, \\cdot)$ is an isomorphism from $V$ to its dual space.\nEach inner product is nondegenerate; the form $B(x, y) = x_1 y_1$ on $\\R^2$ is degenerate.\n\n## Explanations\n\n### L1 human\n\nA form is nondegenerate when it can detect every nonzero vector.\nFor each nonzero $u$, some $v$ gives $B(u, v) \\ne 0$.\nIn matrix terms, the matrix of the form is invertible.\n",
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