4. [6 marks] Let V = {(a, b) : a, be R}. Define addition and scalar multiplication in V as follows: (a, b) + (c, d) = (ad + be, db), c(a, b) = (cab-1, be ) respectively for a, b, c, and de R. V is a vector space. Verify each of the following axioms associated with vector spaces. i) For each f E V, there exists 0 6 V such that f + 0 = f. [Hint: For any vector space Or = 0 for 0 ( F and , 0 E V] First, determine if the zero vector exists: Let (a, b) E V 0 = 0 . (a, b) = (0ab-', bo) = (0, 1) c V. Now determine whether f + 0 = f. (a, b) + (0, 1) = (a(1) + 6(0), d(1)) = (a, b). ii) For each f E V, there exists -f E V such that f + (-f) =0. Hint: For any vector space -1x = -x for all c ( V] -(a, b) = -1(a, b) = (-ab-2,b-1) EV (a, b) + (-ab-2,b-1) = (ab-1 + b(-ab-2), bb-1) = (0, 1) =0