Question

Which of the following sets is a subspace of R⁵?

1) Set A
2) Set B
3) Set C
4) Set D

Answer 1

Set B is the only subspace of R⁵ among the given sets.

Step-by-step explanation:

To determine if a set is a subspace of R⁵, it needs to satisfy three conditions:

1. The set needs to be closed under vector addition.
2. The set needs to be closed under scalar multiplication.
3. The set needs to contain the zero vector (0, 0, 0, 0, 0).

Looking at the given sets:

1. Set A = {2, 4, 6}, does not satisfy the third condition as it does not contain the zero vector, so it is not a subspace.
2. Set B = {14, 16, 18}, satisfies all three conditions and is a subspace of R⁵.
3. Set C = {1, 3, 5, 7}, does not satisfy the second condition as it is not closed under scalar multiplication, so it is not a subspace.
4. Set D = {0, 12}, does not satisfy the first condition as it is not closed under vector addition, so it is not a subspace.

Therefore, Set B is the only subspace of R⁵ among the given sets.