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I'm stuck on a part of a question: if $T : V \rightarrow V$ is a linear endomorphism of a $\mathbb{C}$-vector space $V$, then the associated $\mathbb{C}[t]$- module is cyclic (that is $V =\frac{k[t]}{<g>}$ for some monic polynomial $g \in k[t]$) if and only if there is a vector $v$ such that $V = span\{T^{i}(v)\}$.

I really can't even see where to start with this! Any hints greatly appreciated.

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I think you want to translate the condition of being cyclic into the condition that there exists $v\in V$ for which $$\varphi:k[t]\rightarrow V$$ $$p(t)\mapsto p(T)v$$ is a surjective map of $k[t]$-modules. In this case, the kernel has a monic generator.

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