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.

The transpose $A^{T}$ of a matrix

Definition

For $A = (a_{i j}) \in M_{m \times n} (F)$ , the transpose of $A$ is the matrix $A^{T} \in M_{n \times m} (F)$ defined by

$(A^{T})_{j i} : = a_{i j} .$

Thus transposition interchanges rows and columns. The transpose of a zero-sized matrix is the unique matrix of the transposed zero-sized shape.

Depends on

The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Dependency tree · 2 levels

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3C, Exercises 14–15