Let U be the R-vector space consisting of all polynomials of degree at most n with
coefficients that are real.
The Derivative Map
F : U [tex]\rightarrow[/tex] U
f(x) [tex]\rightarrow[/tex] f'(x)
Is the derivative function F diagonalizable?
The Attempt at a Solution
My instinct says yes but I'm not too sure why. Am I right in saying that eigenvalues are invariant after a linear transform. I'm pretty sure that before the function is applied you can have a matrix that has n+1 distinct eignenvalues but not sure if this is the right lines or if it is how to set about writing it down in a proof format.
Any help would be great thanks !