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Question 1. Let (Fun(R,R), +, -) be the real vector space of all functions from R to R. Recall that the addition of two functions f, g E Fuu(R, R) and the multiplication of a function f E Fun(R, R) with a real scalar A E R are explained by (f +g)(t) 2 t) +g(t) and (Ath) = At) for all t 6 R. a) Pick any real number q E R. Let V(q,0) = {f E Fuu(R, R) : ag) 2 0} be the set of all functions f : R > R with the property that their value at q is equal to zero, i.e. f (q) = 0. Explain why V(q, U) is a subspace of Fun(R, R). b) Pick any real number q E R. Let V(q, 71') = {f E Fun(R,R) : f(q) = 7r} be the set of all functions f : R > R with the property that their value at q is equal to 11', i.e. q) = 11'. Is V(q, 7r) a subspace of Fun(R, R)? Explain your answer. c) Finally, consider the subspace V = Span({f1, f2, f3}) of the vector space (Fun(R,R), +, ) spanned by the following three functions f1(t) = cos(t)2, f2(t) = sin(t)2, f3(t) = 4 for t E R. Show that B = {f1, f2} is a basis of the vector space V and compute the Bcoordinates of the vector f1 + f2 + f 3 E V