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.
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?