$T$ is a linear transformation such that $T^{2}v \neq 0$ and $T^{3}v = 0$. $S = \{ v, Tv, T^{2}v \}$. Which are necessarily true?
- $T(S) \subseteq S$
- $\forall s \in S, Ts \neq s$
- $|S| = 3$
Attempt
- $T(T^{2}v) = 0$ which need not be in $S$? So false?
- $Tv \neq v$ or we will not have the assumptions. But $Tv = T(Tv)$ possible?
- $Tv = T^{2}v$ possible?
Additional Comment
Feel free to ignore my attempts, and not feel compelled to answer them. Also, please tell me if there is anything missing in or wrong with the question.