Hi all, have no idea how to go about doing this!

In this question let β = {1, x, x^{2}, x^{3}} be the standard basis for_{[x]. }Let Φ:_{3}[x]_{3}[x] be the linear transformation defined by :

Φ : p(x) (x^{2})(d^{2}p/dx^{2})+2(x+1)dp/dx

a) Find matrix [Φ]^{β}_{β of }Φ relative to the basis β

b) Consider the different basis γ = {1, 1+x, 1+3x+3x^{2}, 1+6x+15x^{2}+15x} for_{3}[x]. Find the change-of-basis matrices [id]^{β}_{γ and }[id]^{γ}_{β. c) use your answer to part b to calculate }[Φ]^{γ}_{γ and observe that it's diagonal. d) use answer to part c to calculate }Φ Φ ... Φ(q) (10 times), where q = 15 +15x +15x^{2}+ 15^{3}. (I think it may be best to use coordinate representation of q in the basis γ here. )