LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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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.

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Diagonal criterion: if $⟨ v_{i}, w_{i} ⟩ \neq 0$ and $⟨ v_{i}, w_{j} ⟩ = 0$ for $i \neq j$ , then $v_{1}, \dots, v_{m}$ are linearly independent

Statement

Let $F$ be a field, let $V$ be an $F$ -vector space, and let $⟨ \cdot, \cdot ⟩ : V \times V \to F$ be a bilinear form. Suppose vectors $v_{1}, \dots, v_{m}, w_{1}, \dots, w_{m} \in V$ satisfy

$⟨ v_{i}, w_{i} ⟩ \neq 0 and ⟨ v_{i}, w_{j} ⟩ = 0 for i \neq j .$

Then $v_{1}, \dots, v_{m}$ are linearly independent.

Facts & Assumptions

Given: a field $F$ , a vector space $V$ over $F$ , a bilinear form $⟨ \cdot, \cdot ⟩$ on $V$ , and vectors $v_{1}, \dots, v_{m}, w_{1}, \dots, w_{m} \in V$ satisfying the displayed hypotheses.

[F1]

A bilinear form is linear in each variable separately (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

[F2]

A field has no zero divisors (Field).

Proof

technique · direct

1.1

Suppose $\sum_{i = 1}^{m} c_{i} v_{i} = 0$ . Pairing with $w_{j}$ and using linearity in the first variable from [F1] gives $\sum_{i = 1}^{m} c_{i} ⟨ v_{i}, w_{j} ⟩ = 0$ .

F1assume-contra

2.1

Every term with $i \neq j$ vanishes by hypothesis, so this reduces to $c_{j} ⟨ v_{j}, w_{j} ⟩ = 0$ . Since $⟨ v_{j}, w_{j} ⟩ \neq 0$ and a field has no zero divisors by [F2], we get $c_{j} = 0$ .

F2step 1.1

3.1

The index $j$ was arbitrary, so every coefficient is $0$ . Hence $v_{1}, \dots, v_{m}$ are linearly independent.

step 2.1discharge-contradiction∎

Remarks

The argument uses only bilinearity and the diagonal pattern. No positivity and no nondegeneracy is involved, which is why the criterion works over $F_{2}$ as well as over $R$ .

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §2.1.4 (standard reference, not scraped)

Diagonal criterion: if ⟨vi,wi⟩≠0⟨vi​,wi​⟩=0 and ⟨vi,wj⟩=0⟨vi​,wj​⟩=0 for i≠ji=j, then v1,…,vmv1​,…,vm​ are linearly independent