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5. (10 points) (a) (5 points) Let P (R) denote the vector space of polynomials of degree at most n, where addition is the usual addition of functions, and scalar multiplication is the usual way we multiply a function by a real number. Consider the map T: P3(R) + P(R), given by T(px)) = 3p" (2) - 2p (). a Find a basis B for P3(R) and a basis B' for P(R) (and explain why these are bases). Compute the (B.B')-matrix for T.

(b) (5 points) Prove that if p is a non-constant polynomial with degree at most 3, then p can not be a solution to the differential equation 3p" (2) - 2p'(x) = 0. (The reason for "non-constant" above is that of course any constant function will satisfy this because both its derivative and its second derivative will be 0).