Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

The system Ax=b is consistent if and only if rank⁡(A)=rank⁡([A∣b])

Statement

For every finite system Ax=b over a field, Ax=b is consistent⟺rank⁡(A)=rank⁡([A∣b]).

Facts & Assumptions

Given: A coefficient matrix A and augmented matrix [A∣b].

[L1]

Consistency is equivalent to absence of a pivot in the augmented column of the RREF (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

Matrix rank equals the number of pivots (Row rank equals column rank, and both equal the number of pivots).

Proof

technique · direct
1.1

Row reduce the augmented matrix. Its coefficient block is an echelon form row equivalent to A, while the whole reduced matrix is an echelon form of [A∣b]. Thus [L2] identifies their respective pivot counts with the two ranks. Those counts differ exactly when the augmented column supplies one additional pivot.

L1L2
2.1

By [L1], absence of that additional augmented pivot is equivalent to consistency. Step 1.1 therefore proves both directions.

step 1.1L1∎

Depends on

Used by

Dependency tree · two levels

9 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