T-Annihilator of a Vector

The polynomial applied to $T$ that annihilates a specific vector, or all vectors.

Definition

• With $V$ as a $\mathbb{F}$ vector space
• Let $\alpha \in V$
• With $T\in \mathcal{L}(V)$
• Then $M(T)=\{f\in P(\mathbb{F})|f(T)=0\}$ is the general $T$-annihilator
• Then $M({\alpha }_{j}T)=\{f\in P(\mathbb{F})|f(T)\alpha =0\}$ is the $T$-annihilator of $\alpha$.

Intuition

• A Polynomial $p\in M(T)$ will set $p(T)v=0,\forall v\in V$
• A Polynomial $p\in M(a_{j}T)$ will set $p(T)a=0$

Properties

• This is a Polynomial Ideal.

Concepts

• Minimal Polynomial
• A Vector exists with T-Annihilator Equal to the Minimal Polynomial