This is a problem regarding duality in advanced linear algebra. (P_n(R))' is the dual for P_n(R), which is a n degree polynomial that takes in a real number.

Problem 2. Let a E R and let n be a non-negative integer. Define Di(p) := p() (a) for all p E Pn(IR) for i = 0, 1,...,n. (The notation p() refers to the ith derivative of p. By convention, p) is p itself.) Each D; is an element of (Pn (IR) )'. (a) Show that Do, D1, . .., Dn is a basis of (Pn (IR) )'. (b) Let b E R. Consider T E (Pn (IR) )' defined by T(p) = p(b). In view of the preceding part, T is uniquely expressible as a linear combination of Do, D1, ..., Dn. Find this linear combination. Note that a and b are constants, and of course your answer to part (b) will depend on a and b