DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge 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

Used by

Cyclic subspaces, cyclic vectors, and vector annihilators Definition
Complete invariant flags are equivalent to upper-triangular matrices Proposition
For invariant W, χ_T=χ_T|_Wχ_bar T Proposition
Invariance makes the induced quotient operator well defined and linear, with π T=bar Tπ Proposition
Polynomial evaluation commutes with restriction and invariant quotients Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 9 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

K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Section 6.4 (standard reference, not scraped)
Cornell Math 4330, Quotient Spaces, Exercise QuoSpace 7 (standard reference, not scraped)