Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied.
1. True, for a set with two operations to be a vector space all 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set cannot be a vector space.
2. False, for a set with two operations to be a vector space a majority of 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set can still be a vector space.
3. False, for a set with two operations to be a vector space at least one of the 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set can still be a vector space.
4. False, for a set with two operations to be a vector space at least 9 of the 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set can still be a vector space.
5. False, for a set with two operations to be a vector space at least 8 of the 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set can still be a vector space.
(b) The set of all first-degree polynomials with the standard operations is a vector space.
1. True. The set is a vector space because all 10 axioms are satisfied.
2. False. The set is not a vector space because it is not closed under addition.
3. False. The set is not a vector space because it does not have the commutative property of addition.
4. False. The set is not a vector space because it does not have the associative property of addition.
5. False. The set is not a vector space because a scalar identity does not exist.
(c) The set of all pairs of real numbers of the form (0, y), with the standard operations on R^2, is a vector space.
1. True. The set is a vector space because all 10 axioms are satisfied.
2. False. The set is not a vector space because it is not closed under addition.
3. False. The set is not a vector space because an additive inverse does not exist.
4. False. The set is not a vector space because it is not closed under scalar multiplication.
5. False. The set is not a vector space because a scalar identity does not exist.