Determinant of a polynomial

I just realized my mistakes:
first error: $\Phi(f)=4f-3$ is an affine transformation, not a linear one. (Thanks @Bernard)

second one:
I have to view polynomials $f\in V$ as vectors, e.g.:

Let $g(X):=4X^2-8X+2$
and $\Phi:V \to V : (aX^2+bX+c)\mapsto -2bX^2-(c+3b)X+4a$

$g$ can be represented as $\vec{v}_g:=(4, -8, 2)^T$.
$\Phi$ can be represented as $A_\Phi:=\begin{bmatrix}0 & -2 & 0\\0 & -3 & -1\\4 & 0 & 0\end{bmatrix}$.
The calculation of $\Phi(g)$ can be written as $A_\Phi * \vec{v}_g$ which would yield the result $(16, -22, 16)^T$ corresponding to the correct 'output' polynomial of $\Phi(g):x\mapsto 16X^2-22X+16$.

Calculating the determinant and characteristic polynomial etc. can be done with the matrix $A_\Phi$ and is a piece of cake ;)