Question

Is w a subspace of v? If not, state why. Assume that v has the standard operations.

1) Yes, because w is a subset of v.
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, because w does not contain the zero vector.

Step-by-step explanation:

For w to be a subspace of v, it must fulfill three conditions:

1. It must be a subset of v.
2. It must be closed under vector addition.
3. It must be closed under scalar multiplication.

In this case, option 4 states that w does not contain the zero vector. Since the zero vector is a necessary element for a subspace, we can conclude that w is not a subspace of v.