Page 349 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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7. Finally, . _________
Prove part (d) of Theorem 5.1.1.
30.
The following is a proof that the cancellation law for addition holds in a vector space. Justify
31. each step by filling in the blank line with the word hypothesis or by specifying the number of
one of the vector space axioms given in this section.
Hypothesis: Let u, v, and w be vectors in a vector space V and suppose that .
Conclusion: Then .
Proof:
1. First, and are vectors in V. _________
2. Then . _________
3. The left side of the equality in step (2) is .
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_________
4. The right side of the equality in step (2) is
_________
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From the equality in step (2), it follows from steps (3) and (4) that
Do you think it is possible for a vector space to have two different zero vectors? That is, is it
32. possible to have two different vectors and such that these vectors both satisfy Axiom 4?
Explain your reasoning.
Do you think that it is possible for a vector u in a vector space to have two different
33. negatives? That is, is it possible to have two different vectors and , both of
which satisfy Axiom 5? Explain your reasoning.
The set of ten axioms of a vector space is not an independent set because Axiom 2 can be
34. deduced from other axioms in the set. Using the expression
and Axiom 7 as a starting point, prove that .
Hint You can use Theorem 5.1.1 since the proof of each part of that theorem does not use
Axiom 2.
