Say we have $F:=\Bbb F_3(\alpha)$ where α is a root of $f(x)=x^3-x-1$ then $ F:=\{a+b\alpha +c\alpha^2|a,b,c \in \Bbb F_3\}$ So a basis for F as a vector space over $\Bbb F_3$ is $\beta:=\{1,\alpha,\alpha^2\}$.
Now say we have a map $T:F \rightarrow F$ s.t. $T(x)=\alpha x $ $\forall x \in F$
Then would I be correct in saying that the matrix M of T with respect to the basis β would be found in the following way ?
say $\beta:=\{1 , \alpha, \alpha^2\}=\{v_1,v_2,v_3\}$
$T(1)=\alpha=0v_1+1v_2+0,v_3$
$T(\alpha)=\alpha^2=0v_1+0v_2+1v_3$
$T(\alpha^2)=\alpha^3=\alpha+1=1v_1+1v_2+0v_3$
So
$T(v_1)=\begin {pmatrix} 0\\ 1 \\ 0 \end{pmatrix}$
$T(v_2)=\begin {pmatrix} 0\\ 0 \\ 1 \end{pmatrix}$
$T(v_3)=\begin {pmatrix} 1\\ 1 \\ 0 \end{pmatrix}$
So $M=[T(v_1)_B,T(v_2)_B,T(v_3)_B]= \begin {pmatrix} 0 & 0 &1\\ 1 &0 &1 \\ 0 &1&0 \end{pmatrix} $
I found this method on https://yutsumura.com/matrix-of-linear-transformation-with-respect-to-a-basis-consisting-of-eigenvectors/. Is it correct ?