CounterexampleConstruction: AI-adaptedVerification: 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.

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Row-equivalent matrices can have different column spaces even though their column ranks agree

Statement refuted

The false claim is: row-equivalent matrices have the same column space.

Facts & Assumptions

Given: Over any field, let $A = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ and $B = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array})$ .

[L1]

Row operations preserve all column relations and column rank, but do not assert equality of the literal column spaces (Elementary row operations preserve every linear relation among the columns and hence preserve column rank).

[L2]

The column space is the span of the columns and column rank is its dimension (Row space, column space, nullspace, row rank, column rank and matrix rank).

Counterexample

technique · direct

1.1

Swapping the two rows of $A$ gives $B$ , so the matrices are row equivalent.

L1algebra

2.1

Yet $Col (A) = span {(1, 0)^{T}}$ and $Col (B) = span {(0, 1)^{T}}$ , which are distinct one-dimensional subspaces. Their column ranks are both $1$ , exactly as [L1] predicts.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, Remark 3.15 (standard reference, not scraped)