Prove that if $T^2$ has a cyclic vector over finite dimensional vector space $V$, then $T$ has a cyclic vector over $V$.

Question

Prove that if $T^2$ has a cyclic vector over finite dimensional vector space $V$, then $T$ has a cyclic vector over $V$.

Answer 1

Hint:

$$T^2v=T(Tv)$$

where $v\in V$ is a cyclic vector.

Answer 2

Hint: $span\{ v, T^2v, T^4v, \dots \} \subseteq span\{ v, Tv, T^2v, T^3v, \dots \} $