Chapter 4.1, Problem 29E

Exercises 25–29 show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and −u + u = 0 for all u.

29.  Prove that (−1)u = −u. [Hint: Show that u + (−1)u = 0. Use some axioms and the results of Exercises 26 and 27.]