DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 and row equivalence for finite matrices over a field

Definition

Let $F$ be a field and let $A \in M_{m \times n} (F)$ . An elementary row operation on $A$ is one of the following operations, where row indices lie in $m$ :

interchange two distinct rows $R_{p}$ and $R_{q}$ ;
replace $R_{p}$ by $c R_{p}$ for some nonzero $c \in F$ ;
replace $R_{p}$ by $R_{p} + c R_{q}$ for distinct $p, q$ and some $c \in F$ .

Matrices $A, B \in M_{m \times n} (F)$ are row equivalent, written $A \sim B$ , when a finite sequence of elementary row operations transforms $A$ into $B$ . The empty sequence is allowed. Thus the definition also covers matrices with no rows or no columns.

Elementary row operations and row equivalence for finite matrices over a field

Definition

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