Question

Which of the following subsets of 2 are subspaces of 2?

A) Option A
B) Option B
C) Option C
D) Option D

Answer 1

To determine which subsets of ℒ^2 are subspaces, we need to check for closure under addition and scalar multiplication, and contain the zero vector. Option C (metal alloys) and Option D (elements) are subspaces of ℒ^2.

Step-by-step explanation:

To determine which of the following subsets of ℒ^2 are subspaces of ℒ^2, we need to identify if they satisfy the three properties of a subspace: closure under addition, closure under scalar multiplication, and contain the zero vector.

1. Option A: The set of all compounds containing 2 elements is not closed under addition or scalar multiplication, so it is not a subspace.
2. Option B: The set of all heterogeneous mixtures is not closed under addition or scalar multiplication, so it is not a subspace.
3. Option C: The set of all metal alloys is closed under addition and scalar multiplication, and contains the zero vector, so it is a subspace.
4. Option D: The set of all elements is closed under addition and scalar multiplication, and contains the zero vector, so it is a subspace.

Therefore, Options C and D are subsets of ℒ^2 that are subspaces of ℒ^2.