Let P2(R) denote the vector space of real polynomial functions of degree less than or equal to two and let B := [p0,p1,p2] denote the natural ordered basis for P2(R) (so pi(x) = xi). Define g ∈ P2(R) by g(x) = 2x2 − 3x + 1. Write g as a linear combination of the elements of B. Compute the coordinate vector gB of g with respect to B.

Define h1,h2,h3 ∈ P2(R) by h1(x) = 2, h2(x) = 3x−2 and h3(x) = 2x2 −3x+1. Define C := [h1, h2, h3]. Write each element of B as a linear combination of the elements of C. Explain why the calculations you have performed prove that C is a basis for P2(R).

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