Question

Is the set H of all matrices of the form a b 0 d a subspace of M22?

Answer 1

Yes, the set H of all matrices of the form a b 0 d a is a subspace of M22.

Step-by-step explanation:

Yes, the set H of all matrices of the form a b 0 d a is a subspace of M22.

To show that H is a subspace, we need to check three conditions:

1. Zero vector: The zero vector, denoted as 0, must belong to H. In this case, the zero vector is the matrix [0 0 0 0 0], which is of the form a b 0 d a. Thus, the zero vector is in H.
2. Closed under addition: If two matrices A and B are in H, then their sum A + B must also be in H. Let's assume that A = [a b 0 d a] and B = [x y 0 z x]. The sum A + B is [a + x b + y 0 d + z a + x], which is in the form a b 0 d a. Therefore, H is closed under addition.
3. Closed under scalar multiplication: If a matrix A is in H and k is a scalar, then the product kA must also be in H. Let's consider A = [a b 0 d a]. The product kA is [ka kb 0 kd ka], which is of the form a b 0 d a. Hence, H is closed under scalar multiplication.

Since H satisfies all three conditions, it is a subspace of M22.