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This is the problem that I am stuck on.

Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose $\cal{B}_v=$$\{v,T(v),T^2(v),T^3(v),...,T^{n-1}(v)\}$ is a basis for $V$. Also further assume that the only $T$-invariant subspaces are $\{0\}$ and $V$ . Find the minimal polynomial of $T$ and $T^t$.

Attempt: I have managed to show that $T$ has a irreducible minimal polynomial. I think $T$ and $T^t$ has the same polynomial but don't know how to show it. Someone told me to use the primary decomposition theorem. But I don't know how to use that. Is there a way to get around the primary decomposition and prove the result?

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Hint

We express $T^n(v)$ in the basis $\mathcal B_v$ by

$$T^n(v)=-a_0v-a_1T(v)-\cdots- a_{n-1}T^{n-1}v$$ and we write the matrix of $T$ relative to the basis $\mathcal B_v$ so we find the companion matrix $C(p)$. We prove by calculating $\det(tI_n-C(p))$ that the characteristic polynomial $\chi_T$ is equal to $a_0+a_1t+\cdots+a_{n-1}t^{n-1}+t^n$ and it's equal to the minimal polynomial $\mu_T$.

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