Question

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

1) Yes, because w satisfies all the properties of a subspace.
2) No, because w does not satisfy the closure property under addition.
3) No, because w does not contain the zero vector.
4) No, because w does not contain the additive inverse of each vector.

Answer 1

No, because w does not contain the zero vector.

Step-by-step explanation:

In order for w to be a subspace of v, it must satisfy three properties:

1. It must contain the zero vector.
2. It must be closed under addition; if a and b are vectors in w, then a + b is also in w.
3. It must be closed under scalar multiplication; if a is a vector in w and c is a scalar, then ca is also in w.

Based on the given options, the correct answer is:

No, because w does not contain the zero vector.

Answer 2

To determine if w is a subspace of v, it must contain the zero vector, be closed under addition, and be closed under scalar multiplication. The answer depends on which, if any, of these properties w violates.

Step-by-step explanation:

To determine whether w is a subspace of v, we need to check if it satisfies the three necessary properties:

• The zero vector of v must be in w.
• w must be closed under addition.
• w must be closed under scalar multiplication.

1) If w satisfies all these properties, then w is a subspace of v.

2) If w is not closed under addition, then the sum of two vectors in w could yield a vector outside of w, denying it subspace status.

3) If w does not contain the zero vector, it cannot be a subspace because the zero vector represents the identity element for vector addition.

4) If w does not contain the additive inverse of each vector, it cannot fulfill the condition that for any vector in w, its inverse under addition must also be in w.

Therefore, the answer to the original question depends on which of these properties w violates, if any.