TITLE: Linear transformation such that $T^{2}v \neq 0$ and $T^{3}v = 0$.
QUESTION [2 upvotes]: $T$ 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.

REPLY [0 votes]: Based on the comments and the fact that $v \neq 0$, we have 2,3 are true.
On the other hand, $T(T^2v) = 0$ cannot be in $S$. So 1 is false.
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