Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) \in V$. For which values of $a$, $f_a$ is diagonalizable?
What I did:
Let be $p(x) = bx^3 + cx^2+dx+e$ a generic polynomial.
So $f_a(p(x)) = b(1-ax)^3 + c(1-ax)^2+d(1-ax)+e=…=(-a^3b)x^3+a^2(3b+c)x^2-a(3b+2c+d)x+(b+c+d+e)$
So the transformation matrix is: \begin{bmatrix} -a^3 & 0 & 0 & 0 \\ 3a^2 & a^2 & 0 & 0 \\ -3a & -2a & -a & 0 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix}
Finally I found the characteristic polynomial that seems to be $p(\lambda)=(a+\lambda)^2(a-\lambda)(a^3+\lambda)$ Can someone tell me if my procedure is correct? Now i have to discuss the eigenvalues found?