T-annihilators and whether it kills the set Question : $V=F[t]v$ is T-cyclic, $g(t)$ is a polynomial. Show that if $g(T)v=0$, then $g(T)=0$.
Any ideas how to solve this?
plus, does this statement implies T-annihilators of v are exactly same as Ann(T)? I don't really get the differences between two concepts.

First of all, it's important to keep in mind the module action here: $t\cdot v:=T(v)$ (which induces $p\cdot v=p(T)\big(v\big)$), where $T:V\to V$ is a linear transformation and $p\in F[t]$ is a polynomial.
Now, $v$ being a cyclic vector (for the above defined module structure) means indeed that $V=F[t]v=\{p\cdot v:p\in F[t]\}$, that is, $v$ itself generates the whole space as $F[t]$-module.
Nevertheless it follows that the set $\{v,Tv,T^2v,T^3v,\dots\}$ is a generator system for the $F$-vector space $V$.
So, if $g(T)v=0$ then for all $n\in\Bbb N$, we have
$$g(T)T^nv\ =\ T^n g(T)v\ =\ 0$$
so $g(T)$ is a linear transformation of $V$ which is identically zero on a generating system, hence has to be constant $0$.
