##### Solution to Practice Midterm #8
Cullen_Im
1. Start by finding T(x + 1) = 1 + 2x + x^2 since (x + 1) is the first basis vector in beta. Solve 1 + 2x + x^2 with respect to the bases in gamma by setting 1 + 2x + x^2 = a(2) + b(x^2 + 1) + c(1 - x) to find that a = 1, b = 1, c = -2. Therefore, the first column vector in the linear transformation matrix from beta to gamma is (1 1 -2). 2. find T(x - 2) = 1 - 4x + x^2 since (x - 2) is the second basis vector in beta. Solve 1 - 4x + x^2 with respect to the bases in gamma by setting 1 - 4x + x^2 = a(2) + b(x^2 + 1) + c(1 - x) to find that a = -2, b = 1, c = 4. Therefore, the second column vector in the linear transformation matrix from beta to gamma is (-2 1 4). 3. find T(x^2 + x) = 2 + x^2 since (x^2 + x) is the third and final basis vector in beta. Solve 2 + x^2 with respect to the bases in gamma by setting 2 + x^2 = a(2) + b(x^2 + 1) + c(1 - x) to find that a = 1/2, b = 1, c = 0. Therefore, the third column vector in the linear transformation matrix from beta to gamma is (1/2 1 0). 4. The final linear transformation matrix from beta to gamma has row 1: 1, -2, 1/2; row 2: 1, 1, 1; row 3: -2, 4, 0