Question

Are the following subsets also subspaces? If yes, prove it. If no, give an explicit counterexample to show that it is not.

(a) W = {A ∈ M₂ₓ₂ : A = A} ⊆ M₂ₓ₂

Answer 1

Subset W = {A ∈ M₂ₓ₂ : A = A} is a subspace of M₂ₓ₂ because it passes all three conditions for subspaces: containing the zero matrix, being closed under addition, and being closed under scalar multiplication.

Step-by-step explanation:

To determine whether the subsets W = {A ∈ M₂ₓ₂ : A = A} are also subspaces of M₂ₓ₂, we must verify three conditions:

1. The zero matrix is in W.
2. W is closed under addition.
3. W is closed under scalar multiplication.

For (a), the subset W consists of all matrices that are equal to themselves, which is essentially all possible 2x2 matrices since every matrix satisfies the condition A = A. This means that W is simply the entire set M₂ₓ₂.

Let's confirm each condition required for W to be a subspace:

• The zero matrix 0 is in W because 0 = 0.
• If A and B are in W (A = A and B = B), then A + B = A + B, which means that A + B is also in W. Thus, W is closed under addition.
• If A is in W and c is a scalar, then cA = cA, which means that cA is also in W. Thus, W is closed under scalar multiplication.

Since all three conditions are met, W is indeed a subspace of M₂ₓ₂.