I've been racking my brain on the following problem with not much progress, and after encountering the sketch of its solution, I'm still unable to really grasp what's going on.
Let $V$ be a vector space of dimension $0< n \in \mathbb{N}$ over a field $\mathbb{F}$, and let $A \in M_n(\mathbb{F})$.
Given that every $v \in V$ has a polynomial $f_v \in \mathbb{F}[x]$ such that $\langle f_v \rangle = \{p \in \mathbb{F}[x] \mid p(A)v=0\}$, prove that for all $v_1, v_2 \in V$ there is a $v_3 \in V$ such that $f_{v_3}=lcm(f_{v_1}, f_{v_2})$.
The solution uses the following facts (each of which I understand on its own), but I don't understand how they are used to form a solution:
- There is a one-to-one correspondence between $(V,T)$ (where $T$ is the linear transformation corresponding to $T$) and $(V, \cdot)$ as an $F[x]$-module, and therefore, by the fundamental theorem for finitely generated $R$-modules over a PID $R$ ($\mathbb{F}[x]$ in this case), there is an isomorphism: $$V \cong \mathbb{F}[x]/\langle a_1(x)\rangle\oplus\dots\oplus\mathbb{F}[x]/\langle a_k(x)\rangle$$ where $a_i(x)$ are the invariant factors.
- Moving to the elementary factors, one can view the two vectors (if I understood correctly) as a vector of primary elements.
- The required vector is somehow defined using the primary factors for these two vectors.
Since I'm unable to see how these facts combine and lead to the solution, this is about as much of the sketch that I could figure out.
If someone is able to shed light on what the solution actually is and what I am missing to complete the full picture, I would very much appreciate it.