We know that polynomials are a vector space, as they are non-empty, have the elements $1$, $0_V$, an additive inverse and define an operation $\times : \mathbb{K} \times V \to V$ and $+ : V \times V \to V$.
The basis of the polynomial vector space $\mathbb{R}[X]$ is the set $\{1,X,X^2,X^3,…,X^n\}$ (assuming that all polynomials have the maximum degree of $n$.
One could define a isomorphic function $\Phi:V \to V : f \mapsto 4f-3$.
As this function is a linear function, one could assume that one could calculate the Eigenvalues, Eigenvectors etc. of $\Phi$ — just as we do it for 'regular' vector space transformations $\mathbb{R}^n \to \mathbb{R}^m$.
However, while starting to calculate, I got stuck on one question: How do I calculate the characteristic polynomial of $\Phi$?
I of course know the formula $CP_\Phi(λ)=\text{det}(\Phi-λ)$, but it does not get me very far in this case, as I do not know how to calculate the determinant of a polynomial (remember, that $\Phi$ is a polynomial in this case).
Should I transform $\Phi$ into a representation matrix $A_\phi$ and solve $\text{det}(A_\Phi-λ)$ instead? What would be the elements of this matrix? Should I take the polynomial basis $\langle 1,X,X^2,…\rangle$?
I have the feeling that I am near the goal, but that I have missed something obvious or trivial.
Disclaimer: English is not my native language, do please forgive any 'wrong' technical term.
EDIT №1:
See my answer for explanation of my errors.
Best Answer
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 ;)