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$T$ 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?

  1. $T(S) \subseteq S$
  2. $\forall s \in S, Ts \neq s$
  3. $|S| = 3$

Attempt

  1. $T(T^{2}v) = 0$ which need not be in $S$? So false?
  2. $Tv \neq v$ or we will not have the assumptions. But $Tv = T(Tv)$ possible?
  3. $Tv = T^{2}v$ 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.

  • 1
    It is actually impossible to have $Tv=T^2v$; the manner of finding a contradiction is the same as when you assume $v=Tv$, so I suggest you take a moment and try to find it.2012-11-07
  • 0
    Can't believe I missed that! Applying $T$ again on both sides gives $T(Tv) = T(T^{2}v)$ which is a contradiction. Thanks!2012-11-07
  • 0
    Is this question from a GRE mathematics subject test?2012-11-07

1 Answers 1

0

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.

  • 0
    Be careful. :) They are not all true.2012-11-07
  • 0
    @peoplepower - Ah thanks. And for your hint too. I was stuck with this question for some time.2012-11-07