Hello, we have recently started studying linear transformation in university and I couldn't understand them well (we have done three lectures about them)
And I have to do the following question:
Let f:R[X] -> R[X] be the linear transformation sending a polynomial P(X) to f(P(X))= P(X+1) - P(X).
a) Let f4: R4[X] -> R[X] be the linear transformation induced by restriction of f to the R-vector space of polynomial of degree at most 4. Determine the kerne and the image of f4. (4 is a subscript)
b)Answer part (a) again with f4 repaced by fn (n any non-negative integer).
Deduce that for any P in R[X], there exists Q in R[X] such that f(Q)=P
c) Let Q in R[X] and let S in R[X] be a polynomial such that f(S)=Q. Show that any other solutions of the equation f(P)=Q can be written P=S+S' with S' in ker(f).
I don't know even how to start this problem and how to continue it.
I know the definitions of image, kernel etc but I find it difficult to put the theory into a practical problem. Any help of how to face such types of problems and maybe any general guidance for linear transformations would be appreciated!!!
Thanks in advance!