Problem 5. Dene the hyperbolic cosine and hyperbolic sine functions by ex '28 33, sinh(x) = em 28 m (i) Show that for any function of the form f (cc) = a1 cosh(a:) + a2 sinh(sc), with (11, a2 6 R, we can nd numbers ()1, b2 6 IR such that f(3:) = 1518'B + E926\". (Hint: Express b1 and b2 in terms of a1 and (12.) (ii) Show that for any function of the form cosh(x) = g(.:c) = 016'\" + age'3, with c1, (:2 6 IR, we can nd numbers d1, d2 6 IR such that 9(33) = d1 cosh(:c) + d2 sinh(a:). (Hint: Express (1 and 12 in terms of cl and 02) Problem 6. Let V be the space of all real symmetric 2 x 2 matrices, V = {A E M2(IR) | AT: A}. (i) Show that V is a subspace of M2 (R) (ii) How many matrices in V have only zeros and ones as entries? Explain. Problem 7. Show that the set 1 1 0 O S = 1 , 2 , 0 , 1 O 3 0 3 spans all of R3 or nd an element of R3 not belonging to span(S). Problem 8. Explain why the subset a U: b 2a+b of R3 is or is not a subspace of R3. a,bE]R