Let B = {1, x, e−x, xe−x} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix A for Dx relative to the basis B.

To find the matrix A for the differential operator Dx relative to the basis B, we need to apply the differential operator to each basis vector and express the result as a linear combination of the basis vectors.

1. Apply Dx to the first basis vector 1. The derivative of a constant is 0, so Dx(1) = 0. This can be expressed as 01 + 0x + 0e^(-x) + 0xe^(-x) relative to the basis B. So, the first column of the matrix A is [0, 0, 0, 0]^T.

2. Apply Dx to the second basis vector x. The derivative of x is 1, so Dx(x) = 1. This can be expressed as 11 + 0x + 0e^(-x) + 0xe^(-x) relative to the basis B. So, the second column of the matrix A is [1, 0, 0, 0]^T.

3. Apply Dx to the third basis vector e^(-x). The derivative of e^(-x) is -e^(-x), so Dx(e^(-x)) = -e^(-x). This can be expressed as 01 + 0x + (-1)e^(-x) + 0xe^(-x) relative to the basis B. So, the third column of the matrix A is [0, 0, -1, 0]^T.

4. Apply Dx to the fourth basis vector xe^(-x). By the product rule, the derivative of xe^(-x) is e^(-x) - xe^(-x), so Dx(xe^(-x)) = e^(-x) - xe^(-x). This can be expressed as 01 + 0x + 1*e^(-x) + (-1)*xe^(-x) relative to the basis B. So, the fourth column of the matrix A is [0, 0, 1, -1]^T.

Therefore, the matrix A for the differential operator Dx relative to the basis B is:

A = [0, 1, 0, 0; 0, 0, 0, 0; 0, 0, -1, 1; 0, 0, 0, -1]