DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Invariant subspaces, restrictions, and induced quotient operators

Definition

Let $T : V \to V$ be an endomorphism. A linear subspace $W \leq V$ is $T$ -invariant when $T (W) \subseteq W$ . For such a $W$ , the restriction of $T$ to $W$ is $T ∣_{W} : W ⟶ W, w ⟼ T (w) .$ The operator induced by $T$ on the quotient is the map $\overset{ˉ}{T} : V / W ⟶ V / W, \overset{ˉ}{T} (v + W) : = T (v) + W,$ where $V / W$ is the quotient of The quotient vector space $V / W$ and its canonical projection. Its well-definedness, linearity, and relation to the canonical projection are proved in Invariance makes the induced quotient operator well defined and linear, with $π T = \overset{ˉ}{T} π$ ↗.

Depends on

Dependency tree · 2 levels

Sources