DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $m,n\in\mathbb N$ . An $m\times n$ matrix over $R$ is a function $A:m\times n\longrightarrow R.$ Its value at $(i,j)$ is written $a_{ij}$ . 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 $j\mapsto a_{ij}$ on $n$ ; for $j<n$ , column $j$ is the function $i\mapsto a_{ij}$ on $m$ . This includes zero-sized shapes: a matrix with empty index set is the unique function from that empty set.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 17 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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Notation 4.16 (standard reference, not scraped)