ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

A dual basis and transpose computed for a linear map on $F_{3}^{2}$

Example

Let $e_{1}, e_{2}$ be the standard basis of $F_{3}^{2}$ , with dual basis $ε_{1}, ε_{2}$ , and let

$[T] = (\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}) .$

Then $T^{*} (ε_{1}) = ε_{1} + 2 ε_{2}$ , $T^{*} (ε_{2}) = ε_{2}$ , and

$[T^{*}] = (\begin{array}{cc} 1 & 0 \\ 2 & 1 \end{array}) = [T]^{T} .$

Facts & Assumptions

Given: The displayed matrix in the standard basis over $F_{3}$ .

[L1]

Since $3$ is prime, $F_{3} = Z / 3 Z$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L2]

The coordinate functionals form the dual basis (The dual family of a finite basis is a basis of the dual space, with the same dimension).

[L3]

In dual bases, the matrix of $T^{*}$ is the transpose of that of $T$ (In dual bases, the matrix of $T^{*}$ is the transpose of the matrix of $T$ ).

Verification

technique · exhaustive coordinate evaluation

1.1

The matrix columns give $T (e_{1}) = e_{1}$ and $T (e_{2}) = 2 e_{1} + e_{2}$ . Therefore $T^{*} (ε_{1})$ takes values $1, 2$ on $e_{1}, e_{2}$ , while $T^{*} (ε_{2})$ takes values $0, 1$ .

L1L2algebra

2.1

By the dual-basis expansion [L2], these four values give $T^{*} (ε_{1}) = ε_{1} + 2 ε_{2}$ and $T^{*} (ε_{2}) = ε_{2}$ .

step 1.1L2

3.1

The coordinate columns in step 2.1 give $[T^{*}] = (\begin{array}{cc} 1 & 0 \\ 2 & 1 \end{array})$ , which independently agrees with [L3].

step 2.1L3∎

Depends on

Used by

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A dual basis and transpose computed for a linear map on F32F32​