Chapter 4.1, Problem 26E

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.

26. Complete the following proof that −u is the unique vector in V such that u + (−u) = 0. Suppose that w satisfies u + w = 0. Adding −u to both sides, we have

$\begin{array}{cccccccc}\left(-u\right)& +& \left[u+w\right]& =& \left(-u\right)& +& 0& \\ \left[\left(-u\right)+u\right]& +& w& =& \left(-u\right)& +& 0& byAxiom\_\_\_\_\_\_\left(a\right)\\ 0& +& w& =& \left(-u\right)& +& 0& byAxiom\_\_\_\_\_\_\left(b\right)\\ & & w& =& -u& & & byAxiom\_\_\_\_\_\_\left(c\right)\end{array}$