$V$ is a vector space over the complex numbers with finite dimension and consider $T\in L(V)$ and $v \in V $\ {$0$} such that $T^{n-1}v \neq 0 $ but $T^nv=0 \in V$

Prove that T has a unique eigenvalue and find the matrix $A =[T]_\beta$

where $\beta=$ $v,Tv,T^2v,...,T^{n-1}v$}

I just know that $\beta$ is a set independent, how can I prove what I asked for?

Do I have to use the cyclic vector? How can I find the matrix A?

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