Let V be the set of all ordered triples of real numbers, and consider the following

Chapter 4, Problem 1

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Let V be the set of all ordered triples of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, u2, u3) and v = (v1, v2, v3): u + v = (u1 + v1, u2 + v2, u3 + v3), ku = (ku1, 0, 0) (a) Compute u + v and ku for u = (3, 2, 4), v = (1, 5, 2), and k = 1. (b) In words, explain why V is closed under addition and scalar multiplication. (c) Since the addition operation on V is the standard addition operation on R3, certain vector space axioms hold for V because they are known to hold for R3. Which axioms in Definition 1 of Section 4.1 are they? (d) Show that Axioms 7, 8, and 9 hold. (e) Show that Axiom 10 fails for the given operations.