Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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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=(1000)A=\begin{pmatrix}1&0\\0&0\end{pmatrix} and B=(0010)B=\begin{pmatrix}0&0\\1&0\end{pmatrix}.

[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 AA gives BB, so the matrices are row equivalent.

L1algebra
2.1

Yet Col⁡(A)=span⁡{(1,0)T}\operatorname{Col}(A)=\operatorname{span}\{(1,0)^{\mathsf T}\} and Col⁡(B)=span⁡{(0,1)T}\operatorname{Col}(B)=\operatorname{span}\{(0,1)^{\mathsf T}\}, which are distinct one-dimensional subspaces. Their column ranks are both 11, exactly as [L1] predicts.

step 1.1L2algebra∎

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: 35 results over 14 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