1. Let n e N and let V be the set of polynomials in the indeterminate x with coefficients in R of degree at most n. That is, V = {a,x" + an-1ª"-1+..+ a1x + ao : ao,a1,..., ɑn E R}. (a) Show that V is a vector space over R under addition of polynomials and scalar multipli- cation defined by A (ana" + ... + a1x + ao) = (Aan)æ" + ..+ (Aa1)x + (Aao) for AeR. (b) Calculate the dimension of V by finding a basis and justify your answer. (c) We define a scalar product 3 on V by p(x) = a„x" + · ·.+a1x+ ao; B(p(x), q(x)) = E a;bi where i=0 q(x) bnx" + · ..+ bịx + bo. Prove that 3 is a positive definite scalar product.

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