LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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 $c$ , $Ac=0\quad\Longleftrightarrow\quad Bc=0.$ 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 $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent).

Proof

technique · direct

1.1

The equation $Ac=0$ says exactly that the columns of $A$ , with coefficients $c_j$ , have zero linear combination.

L2L3

2.1

Apply the same row operations to $[A\mid0]$ . Its last column stays zero, so [L1] gives $Ac=0$ if and only if $Bc=0$ .

step 1.1L1

3.1

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.

step 2.1L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 19 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

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §3.5 (standard reference, not scraped)
J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.4 (standard reference, not scraped)