Chapter 4.1, Problem 25E

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.

25. Complete the following proof that the zero vector is unique. Suppose that w in V has the property that u + w = w + u = u for all u in V. In particular, 0 + w = 0. But 0 + w = w, by Axiom _____.  Hence w = 0 + w = 0.