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 $m \times n$ matrix $A$ , $rank (A) + \dim N (A) = n$

Statement

For $A \in M_{m \times n} (F)$ , $rank (A) + \dim_{F} N (A) = n .$

Facts & Assumptions

Given: The linear map $L_{A} : F^{n} \to F^{m}$ , $x \mapsto A x$ .

[L1]

Matrix rank equals the rank of $L_{A}$ (The rank of a matrix equals the rank of the linear map $x \mapsto A x$ ).

[L2]

Rank–nullity gives $\dim V = rank T + \dim \ker T$ for a linear map with finite-dimensional domain (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[L3]

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

Proof

technique · direct

1.1

Apply rank–nullity to $L_{A} : F^{n} \to F^{m}$ to obtain $n = rank L_{A} + \dim \ker L_{A}$ .

2.1

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

step 1.1L1L3∎

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

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §2.9 (standard reference, not scraped)

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