Question

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. C[O, 3], the set of all continuous functions defined on the interval [0, 3], with the standard operations a. The set is a vector space. b. The set is not a vector space because it is not closed under addition, c. The set is not a vector space because an additive inverse does not exist. d. The set is not a vector space because it is not closed under scalar multiplication. e. The set is not a vector space because the associative property of scalar multiplication is not satisfied.

Answer 1

Answer: a) The set is a vector space.

Explanation:

a. The set C[0, 3] is a vector space with the standard operations of addition and scalar multiplication.

b. This statement is false. The set C[0, 3] is closed under addition, meaning that if f(x) and g(x) are both continuous functions on [0, 3], then their sum f(x) + g(x) is also a continuous function on [0, 3].

c. This statement is also false. For every continuous function f(x) defined on [0, 3], there exists an additive inverse function -f(x), such that f(x) + (-f(x)) = 0 for all x in [0, 3].

d. This statement is false. The set C[0, 3] is closed under scalar multiplication, meaning that if f(x) is a continuous function on [0, 3] and c is a scalar, then the product cf(x) is also a continuous function on [0, 3].

e. This statement is false. Scalar multiplication of continuous functions on [0, 3] is associative, meaning that (c_{1} c_{2))f(x) =c_{1} ( c_{2) f(x)) for all x in [0, 3] and all scalars c_{1} and c_{2)

Therefore, the correct answer is (a) The set is a vector space.