This is a follow-up homework question(s) to a conceptual question I had asked earlier with regard to the cyclic basis of the vector space $V$ denoted by $\mathcal{B}= \{v,Av,…,A^{k-1}v\}$, where $A \in M_n(F)$, and $v \in F^n$. Also, there is a linear transformation $T: V \rightarrow V$ which is basically matrix multiplication by $A$, i.e. $T(v)= Av$. The question is as follows:
We are asked to find the explicit basis $\mathcal{B}$ for the module $V = F[x]v$, and the minimal polynomial $f= \min_{T}(x)$ for $T$ given specific choices for $A$ and $v$. We are also asked to find the coordinate matrix $[T]_{\mathcal{B}}$.
$1$. $A = \begin{bmatrix}12&2&3&-7\\ -5&2&-2&4\\6&1&5&-5\\13&3&4&-7\end{bmatrix}$, $F= \mathbb{Q}$, $v = \begin{bmatrix} 0\\ -1\\ -1\\ \ -1 \end{bmatrix}.$
$2$. $A= \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix}$, $0< \theta < 2\pi$, $F= \mathbb{R}$, $v= e_1$.
$3$. $A= \begin{bmatrix} 2&0&0\\ 0&3&0\\ 0&0&4 \end{bmatrix}$, $F= \mathbb{Q}$, $v= e_1+ e_2$.
$4$. $A= \begin{bmatrix} 1&1\\ 1&0 \end{bmatrix}$, $F= \mathbb{Z}_2$, $v= e_1$.
Well, for all of these, I computed the bases via matrix multiplication by $A, A^2,…, A^{k-1}$ with $v$ first. Is that the right approach? Does finding the minimal polynomial $f$ of the transformation $T$ require finding the characteristic polynomial of $A$ (which I already found), but what next? Finally, is the coordinate matrix $[T]_\mathcal{B}$ the companion matrix $C(f)$ of $f$?