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.

Invertible linear maps, linear isomorphisms, and inverse linear maps

Definition

Let $T : V \to W$ be linear. It is invertible when there is a linear map $S : W \to V$ such that

$S \circ T = {id}_{V}, T \circ S = {id}_{W} .$

Such an $S$ is the inverse linear map of $T$ , denoted $T^{- 1}$ . The two inverse equations make $T$ bijective and determine $T^{- 1}$ uniquely. An invertible linear map is also called a linear isomorphism, and $V$ and $W$ are linearly isomorphic, written $V ≅ W$ , when such a map exists.

Invertible linear maps, linear isomorphisms, and inverse linear maps

Definition

Depends on

Dependency tree · 2 levels

Sources