Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2):

u + v = (u1 + v1, u2 + v2), ku = (0, ku2)

(a) Compute u + v and 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 R2, certain vector space axioms hold for V because they are known to hold for R2. Which axioms are they?

(d) Show that Axioms 7, 8, and 9 hold.

(e) Show that Axiom 10 fails and hence that V is not a vector space under the given operations

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