DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space

Definition

Let $B$ be a bilinear form on an $n$ -dimensional vector space $V$ , and let $B = (v_{0}, \dots, v_{n - 1})$ be an ordered basis. The matrix of $B$ in $B$ is

$[B]_{B} : = (B (v_{i}, v_{j}))_{i, j < n} .$

For coordinate columns $x = [u]_{B}$ and $y = [v]_{B}$ , one has $B (u, v) = x^{T} [B]_{B} y$ .

The left radical and right radical are

${rad}_{L} (B) : = {u : B (u, v) = 0 for every v}, {rad}_{R} (B) : = {v : B (u, v) = 0 for every u} .$

The rank of $B$ is the rank of the associated map $B^{♭} : V \to V^{*}$ , $B^{♭} (u) (v) = B (u, v)$ . The form is nondegenerate when both radicals are zero. In finite dimension this is equivalent to $B^{♭}$ being an isomorphism, and also to $[B]_{B}$ being invertible.

The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space

Definition

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