Show that if $M$ is in $\mathcal{M}_n(\mathbb{C})$, then the following propositions are equivalent: i. there exists $x_0$ in $\mathbb{C}^n$ such that $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ is a basis of $\mathbb{C}^n$; ii. $M$ is similar to the matrix $C(a_0, \ldots, a_{n-1})$ defined by $$C(a_0, \ldots, a_{n-1}) = \left(\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_0 \\ 1 & \ddots & & \vdots & a_1 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & a_{n-1} \end{array}\right)$$ where $(a_0, \ldots, a_{n-1})$ are complex numbers.
Show that if $M$ is in $\mathcal{M}_n(\mathbb{C})$, then the following propositions are equivalent:
i. there exists $x_0$ in $\mathbb{C}^n$ such that $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ is a basis of $\mathbb{C}^n$;
ii. $M$ is similar to the matrix $C(a_0, \ldots, a_{n-1})$ defined by
$$C(a_0, \ldots, a_{n-1}) = \left(\begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_0 \\ 1 & \ddots & & \vdots & a_1 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & a_{n-1} \end{array}\right)$$
where $(a_0, \ldots, a_{n-1})$ are complex numbers.