Question

Is w a subspace of the vector space? If not, state why.

1) Yes, because w is the set of all nonnegative functions in c(-[infinity], [infinity]).
2) No, because w is not closed under addition.
3) No, because w is not closed under scalar multiplication.
4) No, because w does not contain the zero vector.

Answer 1

No, w is not a subspace of the vector space because it is not closed under addition.

Step-by-step explanation:

No, w is not a subspace of the vector space. In order for a set to be a subspace, it must satisfy three properties: closure under addition, closure under scalar multiplication, and containing the zero vector. Option 2 states that w is not closed under addition, which means that there exist two nonnegative functions in w whose sum is not a nonnegative function. Therefore, w fails to meet the closure under addition property and cannot be a subspace.