Question

W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R³

whose third component is -4 . W is a subspace of R³ . W is not a subspace of R³
because it is not closed under addition. W is not a subspace of R³
because it is not closed under scalar multiplication.

Answer 1

W is not a subspace of R³ because it is not closed under addition and scalar multiplication.

Step-by-step explanation:

W is not a subspace of R³ because it is not closed under addition and scalar multiplication.

To be a subspace, a set must satisfy three conditions:

1. The zero vector must belong to the set.
2. The set must be closed under addition.
3. The set must be closed under scalar multiplication.

In this case, the set W consists of all vectors in R³ whose third component is -4. While the zero vector belongs to W (since its third component is also -4), W is not closed under addition or scalar multiplication.

For example, if we take vectors v1 = (1, 2, -4) and v2 = (3, 4, -4) in W, their sum v1 + v2 = (4, 6, -8) is not in W because its third component is not -4.

The same goes for scalar multiplication. If we take vector v = (1, 2, -4) in W and multiply it by a scalar c = 2, c * v = (2, 4, -8) is not in W because its third component is not -4.