ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The form on $R^{2}$ with matrix $(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ is neither symmetric nor alternating

Example

On $R^{2}$ , let $B$ have matrix $A = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ in the standard basis. Then $B$ is neither symmetric nor alternating.

Facts & Assumptions

Given: The displayed matrix and standard basis $e_{1}, e_{2}$ .

[L1]

A matrix $A$ represents the form by $B (u, v) = u^{T} A v$ (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space).

[L2]

Symmetry requires $B (u, v) = B (v, u)$ , while alternation requires $B (v, v) = 0$ for every $v$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

Verification

technique · direct evaluation

1.1

By [L1], $B (e_{1}, e_{2}) = 1$ but $B (e_{2}, e_{1}) = 0$ , so symmetry fails by [L2].

L1L2algebra

1.2

Also $B (e_{1} + e_{2}, e_{1} + e_{2}) = 1$ , so alternation fails by [L2].

L1L2algebra

2.1

The same form therefore witnesses both failures.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Bilinear Forms, §1 (standard reference, not scraped)

The form on R2R2 with matrix (0100)(00​10​) is neither symmetric nor alternating