LemmaStatement: 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.

Elementary row operations preserve every linear relation among the columns and hence preserve column rank

Statement

If $B$ is row equivalent to $A$ , then for every coefficient column $c$ , $A c = 0 ⟺ B c = 0.$ Thus row operations preserve all linear relations among corresponding columns, preserve which column subfamilies are linearly independent, and preserve column rank.

Facts & Assumptions

Given: Row-equivalent matrices $A, B \in M_{m \times n} (F)$ and $c \in F^{n}$ .

[L1]

Row operations on an augmented matrix preserve its solution set (Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system).

[L2]

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).

[L3]

A family is linearly independent exactly when its only zero linear combination has all coefficients zero (Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent).

Proof

technique · direct

1.1L2L3

The equation $A c = 0$ says exactly that the columns of $A$ , with coefficients $c_{j}$ , have zero linear combination.

2.1step 1.1L1

Apply the same row operations to $[A ∣ 0]$ . Its last column stays zero, so [L1] gives $A c = 0$ if and only if $B c = 0$ .

3.1step 2.1L2L3∎

Restricting $c$ to any chosen column subfamily shows that it is independent for $A$ exactly when it is independent for $B$ . Hence the largest independent column families and their dimension agree, although the literal column spaces may differ.

Elementary row operations preserve every linear relation among the columns and hence preserve column rank

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources