DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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.

Polynomial evaluation at an endomorphism: $p (T) = \sum_{k} a_{k} T^{k}$

Definition

Let $T : V \to V$ be an endomorphism and let $p (x) = \sum_{k \geq 0} a_{k} x^{k} \in F [x]$ . Define

$p (T) : = \sum_{k \geq 0} a_{k} T^{k} \in L (V, V),$

where $T^{0} : = I_{V}$ and $T^{k + 1} : = T \circ T^{k}$ . The sum is finite because the coefficient sequence of $p$ has finite support. In particular, the zero polynomial evaluates to the zero endomorphism and the constant polynomial $1$ evaluates to $I_{V}$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 10 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

M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)
H. Pinkham, Linear Algebra, §12.3 (standard reference, not scraped)

Polynomial evaluation at an endomorphism: p(T)=∑kakTkp(T)=∑k​ak​Tk