1. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication op- erations on u = (ui, uz) and v= (u. 12): u+ v = (1 + 1.42 + 02). ku = (0, kuz) (a) Compute u + vand ku for u =(-1,2), v = (3, 4), and k = 3. (b) In words, explain why V is closed under addition and scalar multiplication (C) Since addition on V is the standard addition operation on R?, certain vector space axioms hold for V because they are known to hold for R? Which axioms are they? (d) Show that Axioms 7, 8 and 9 hold. (C) Show that Axiom 10 fails and hence that V is not a vector space under the given operations u 2. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication op- erations on u = (us. uz) and v= (v1.12): u+v= (u + v1+1,42 +0+1), ku = (kuj, kus) (a) Compute u + vand ku for u = (0,4), v = (1, -3), and k = 2 (b) Show that (0,0) +0. (C) Show that (-1, -1) = 0. (d) Show that Axiom 5 holds by producing an ordered pair -u such that u + (-u) = 0 for u = (uu, uz). (e) Find two vector space axioms that fail to hold. In Exercises 3-12, determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. 11. The set of all pairs of real numbers of the form (1, x) with the operations (1, y) + (1. y) = (1, y + y) and k(1, y) = (1, ky)