[Apostol's Calculus] Exercises 1.5

Notice: These proofs are my own. I am not a professional mathematician. If you notice errors or inconsistencies in these solutions, let me know! I appreciate all feedback.

The key to solving these exercises is to analyze each of the given sets and determine whether or not they violate any of the 10 axioms of real linear vectors spaces V_n:

1. Closure under addition
2. Closure under multiplication
3. Commutativity of addition
4. Associativity of Addition
5. Existence of 0 element
6. Existence of negatives
7. associativity of scalar multiplication
8. Distributive law for addition in V
9. Distributive law for addition of numbers
10. Existence of Identity Element

For these solutions, I will simply state "Yes" if the example given is a real linear space, but I will provide a formal proof if it is not.

Answers:

1. Yes
2. Yes
3. Yes
4. Yes
5. No (see proof)
6. Yes
7. Yes
8. Yes
9. Yes
10. Yes
11. No (see proof)
12. Yes
13. Yes
14. No (see proof)
15. Yes
16. Yes
17. Yes
18. Yes
19. Yes
20. Yes
21. Yes
22. Yes
23. No (see proof)
24. Yes
25. No (see proof)
26. Yes
27. Yes
28. Yes
31. (a) No
    (b) No
    (c) No
    (d) No