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.

Finite rectangular matrices over a commutative ring, their entries, rows and columns

Definition

Let $R$ be a commutative ring and let $m, n \in N$ . An $m \times n$ matrix over $R$ is a function $A : m \times n ⟶ R .$ Its value at $(i, j)$ is written $a_{i j}$ . The set of all such matrices is $M_{m \times n} (R)$ , and $M_{n} (R) : = M_{n \times n} (R)$ .

For $i < m$ , row $i$ is the function $j \mapsto a_{i j}$ on $n$ ; for $j < n$ , column $j$ is the function $i \mapsto a_{i j}$ on $m$ . This includes zero-sized shapes: a matrix with empty index set is the unique function from that empty set.

Depends on

Dependency tree · 2 levels

Sources

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Notation 4.16