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I am trying to solve following exercise problem from a book.

Let $V$ be a nonzero finite dimensional vector space over a field $F$. Let $T: V\to V$ be a linear transformation such that $T^n=0$ but $T^{n-1}\neq 0$ for some natural number $n>1$. I have to prove that there exists a nonzero vector $v\in V$ such that $\{v, T(v), T^2(v), \ldots, T^{n-1}(v)\}$ is linearly independent over filed $F$.

I try to prove by method of contradiction. Suppose that $T^n=0$ but $T^{n-1}\neq 0$ holds but there doesn't exist nonzero vector $v\in V$ such that $A =\{v, T(v), T^2(v), \ldots, T^{n-1}(v)\}$ is linearly independent over filed $F$. So, vectors in $A$ must be linearly dependent. So, the linear combination $a_1v +a_2T(v)+a_3T^2(v)+ \ldots +T^{n-1}(v)=0$ for some nonzero $a_i\in F$.

After, this step I am not able to proceed. Is there any other approach to solve this problem. Thanks a lot for your help.

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  • $\begingroup$ Can you use the Jordan form? $\endgroup$
    – copper.hat
    Feb 23 '18 at 3:40
  • $\begingroup$ @copper.hat Dear sir I have not been introduced jordan form? $\endgroup$
    – mathscrazy
    Feb 23 '18 at 4:01
  • $\begingroup$ Follow @user24142's hint below to show that $v,Tv,...,T^{n-1}v$ are linearly independent directly. $\endgroup$
    – copper.hat
    Feb 23 '18 at 4:05
  • $\begingroup$ @copper.hat I am trying sir. Thanks again $\endgroup$
    – mathscrazy
    Feb 23 '18 at 4:07
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Start with the assumption that $v$ is such that $T^n v = 0$ and $T^{n-1} v \neq 0$. Now, make you assumption, using $v$. What happens if we apply $T$ to both sides of the equation? What if we apply it again? Is there a number of applications that lets you prove your result?

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  • $\begingroup$ I am trying and will let you know. Thanks a lot for the hint. $\endgroup$
    – mathscrazy
    Feb 23 '18 at 4:08

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