Question

Is w a subspace of the vector space? If not, state why. (Select all that apply.) w is the set of all nonnegative functions in c(-[infinity], [infinity]).

1) Yes, because w is closed under addition.
2) Yes, because w is closed under scalar multiplication.
3) No, because w is not closed under addition.
4) No, because w is not closed under scalar multiplication.
5) Cannot be determined.

Answer 1

The set of nonnegative functions in C(-∞, ∞) is not a subspace of the vector space of functions as it is not closed under scalar multiplication.

Step-by-step explanation:

The set W of all nonnegative functions in C(-∞, ∞) is not a subspace of the vector space of all functions because it is not closed under scalar multiplication.

A subspace must satisfy certain conditions, one of which is closure under scalar multiplication. If we take any nonnegative function f(x) ≥ 0 and multiply it by a negative scalar, the resulting function will not be nonnegative, which violates the requirement for a subspace. Thus, the correct answer is 'No, because W is not closed under scalar multiplication.'

Answer 2

The set W of all nonnegative functions in C(-∞, ∞) is not a subspace of the vector space because it is not closed under scalar multiplication (Option 4).

Step-by-step explanation:

The question is whether the set W, consisting of all nonnegative functions in C(-∞, ∞), is a subspace of the vector space. To determine whether W is a subspace, it must satisfy three conditions:

• The zero vector must be in W.
• W must be closed under vector addition.
• W must be closed under scalar multiplication.

The set W fails to be a subspace because it is not closed under scalar multiplication. If we multiply a nonnegative function by a negative scalar, the result is not a nonnegative function, hence it is not in W.

Therefore, the correct choice would be No, because W is not closed under scalar multiplication, which corresponds to Option 4.