PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The vector annihilator is the unique monic generator of ${Ann}_{T} (v)$ and divides the minimal polynomial

Statement

For an endomorphism $T$ of a finite-dimensional vector space and $v \in V$ , ${Ann}_{T} (v)$ is a nonzero ideal of $F [x]$ . It has a unique monic generator $m_{T, v}$ , and $p (T) v = 0 ⟺ m_{T, v} ∣ p .$ Moreover $m_{T, v} ∣ μ_{T}$ . In particular $m_{T, 0} = 1$ .

Facts & Assumptions

Given: A finite-dimensional endomorphism $T$ and a vector $v$ .

[L1]

The annihilator ideal is ${Ann}_{T} (v) = {p : p (T) v = 0}$ (Cyclic subspaces, cyclic vectors, and vector annihilators).

[L2]

The minimal polynomial satisfies $μ_{T} (T) = 0$ (The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ).

[L3]

Every ideal of $F [x]$ is principal (For every field $F$ , $F [x]$ is a principal ideal domain).

Proof

technique · direct

1.1

Linearity of polynomial evaluation shows that [L1] is closed under addition and under multiplication by arbitrary polynomials, so it is an ideal; it is nonzero because [L2] puts $μ_{T}$ in it.

L1L2algebra

2.1

By [L3] the ideal is $(a)$ for a nonzero polynomial $a$ ; scaling by the inverse of its leading coefficient gives a monic generator $m_{T, v}$ , and two monic generators of one ideal are associates and hence equal.

step 1.1L3algebra

3.1

Membership in $(m_{T, v})$ is exactly divisibility by $m_{T, v}$ , proving the displayed equivalence. Since $μ_{T}$ belongs to the ideal, $m_{T, v} ∣ μ_{T}$ ; if $v = 0$ , the ideal contains $1$ and its monic generator is $1$ .

step 1.1step 2.1L1L2∎

Depends on

Used by

Cited to discharge well-definedness by Cyclic subspaces, cyclic vectors, and vector annihilators.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 7 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 7.1 (standard reference, not scraped)

The vector annihilator is the unique monic generator of Ann⁡T(v)AnnT​(v) and divides the minimal polynomial