ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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.

Changing both domain and codomain bases of a map $F^{2} \to F^{3}$ uses both sides of the formula

Example

Let $T : F^{2} \to F^{3}$ be $T (x, y) = (x + 2 y, 3 x - y, x + y)$ . Use the standard bases $B, C$ and the new bases

$B^{'} = (e_{0} + e_{1}, e_{1}), C^{'} = (f_{0} + f_{2}, f_{1}, f_{2}) .$

Then

$[T]_{B^{'}}^{C^{'}} = (\begin{array}{cc} 3 & 2 \\ 2 & - 1 \\ - 1 & - 1 \end{array}) .$

Facts & Assumptions

Given: The displayed map and four ordered bases.

[L1]

The two-sided formula is $[T]_{B^{'}}^{C^{'}} = P_{C^{'} \leftarrow C} [T]_{B}^{C} P_{B \leftarrow B^{'}}$ ( $[T]_{B^{'}}^{C^{'}} = P_{C^{'} \leftarrow C} [T]_{B}^{C} P_{B \leftarrow B^{'}}$ ).

Verification

technique · direct

1.1

Here $[T]_{B}^{C} = (\begin{array}{cc} 1 & 2 \\ 3 & - 1 \\ 1 & 1 \end{array})$ , $P_{B \leftarrow B^{'}} = (\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array})$ , and $P_{C^{'} \leftarrow C} = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ - 1 & 0 & 1 \end{array})$ . Multiplying in the order of [L1] gives $(\begin{array}{cc} 3 & 2 \\ 2 & - 1 \\ - 1 & - 1 \end{array})$ .

givenL1

2.1

Independently, $T (e_{0} + e_{1}) = (3, 2, 2) = 3 (f_{0} + f_{2}) + 2 f_{1} - f_{2}$ and $T (e_{1}) = (2, - 1, 1) = 2 (f_{0} + f_{2}) - f_{1} - f_{2}$ , producing the same two coordinate columns.

step 1.1L1∎

Depends on

Used by

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Dependency tree · two levels

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Changing both domain and codomain bases of a map F2→F3F2→F3 uses both sides of the formula