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.

For an m×n matrix A, rank⁡(A)+dim⁡N(A)=n

Statement

For A∈Mm×n(F), rank⁡(A)+dim⁡FN(A)=n.

Facts & Assumptions

Given: The linear map LA:Fn→Fm, x↦Ax.

[L1]

Matrix rank equals the rank of LA (The rank of a matrix equals the rank of the linear map x↦Ax).

[L2]

Rank–nullity gives dim⁡V=rank⁡T+dim⁡ker⁡T for a linear map with finite-dimensional domain (Rank-nullity: dim⁡FV=nullity⁡T+rank⁡T).

[L3]

N(A)={x:Ax=0} (Row space, column space, nullspace, row rank, column rank and matrix rank).

Proof

technique · direct
1.1L2

Apply rank–nullity to LA:Fn→Fm to obtain n=rank⁡LA+dim⁡ker⁡LA.

2.1step 1.1L1L3∎

By [L1] the first term is rank⁡A, and by [L3] the kernel is N(A). Substitution proves the formula, including n=0.

Depends on

Used by

Dependency tree · two levels

17 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