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Recall from Assignment 4 the linear transformation gon2(IR) > 8 A v> (Tr(B"'A))n where (a) Apply the rank-nullity theorem to (,0 to determine rank(go). You may use the value of nullity(cp) given in the solutions to Assignment 4. (b) Let A e M2013), and let a.n = Tr(B"A) for each n 2 0. Show that an+2 = 2an+1 4a,. for all n 2 0. You may use the fact that 82 = 2B U. (c) Part (b) shows that Image(p) is contained in the space V = {(an) E 8 | an\" = 2an+1 4an for all n 2 0}. Using the fact that dim(V) = 2, deduce that Image(tp) = V. (d) In light of part (c), show that if an, a1, a2, . . . are real numbers such that an\" = 2an+1 40,,I for all n 2 0, then there is a matrix A e M2(R) such that Tr(B"'A) = an for all n 2 0