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Complete the unfinished details of Example 4.
23.
Complete the unfinished details of Example 6.
24.
We showed in Example 6 that every plane in that passes through the origin is a vector
25. space under the standard operations on . Is the same true for planes that do not pass through
the origin? Explain your reasoning.
It was shown in Exercise 14 above that the set of polynomials of degree 1 or less is a vector
26. space under the operations stated in that exercise. Is the set of polynomials whose degree is
exactly 1 a vector space under those operations? Explain your reasoning.
Consider the set whose only element is the moon. Is this set a vector space under the
27. operations moon + moon = moon and k(moon)=moon for every real number k? Exaplain your
reasoning.
Do you think that it is possible to have a vector space with exactly two distinct vectors in it?
28. Explain your reasoning.
The following is a proof of part (b) of Theorem 5.1.1. Justify each step by filling in the blank
29. line with the word hypothesis or by specifying the number of one of the vector space axioms
given in this section.
Hypothesis: Let u be any vector in a vector space V, the zero vector in V, and k a scalar.
Conclusion: Then .
Proof:
1. First, . _________
2. _________
3. Since is in V, is in V. _________
4. Therefore, . _________
5. _________
6. _________
