We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. There exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with Jordan blocks of sizes $m_k$ associated to eigenvalues $\lambda_k$. We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Justify that $f$ is cyclic.