To find the matrix M(T) of the linear transformation T: P₃(R) → P₂(R) with respect to the given bases, we need to calculate the images of the basis vectors of P₃(R) under the transformation T and express them as linear combinations of the basis vectors of P₂(R). Given: - T(p(t)) = p'(t) - 2p''(t) - Basis for P₃(R): {1, t, t², t³} - Basis for P₂(R): {1, t + 1, t² + 1} Step 1: Calculate T(1), T(t), T(t²), and T(t³). T(1) = 0 - 2(0) = 0 T(t) = 1 - 2(0) = 1 T(t²) = 2t - 2(2) = 2t - 4 T(t³) = 3t² - 2(6t) = 3t² - 12t Step 2: Express T(1), T(t), T(t²), and T(t³) as linear combinations of the basis vectors {1, t + 1, t² + 1} of P₂(R). T(1) = 0 = 0 · 1 + 0 · (t + 1) + 0 · (t² + 1) T(t) = 1 = 1 · 1 + 0 · (t + 1) + 0 · (t² + 1) T(t²) = 2t - 4 = -4 · 1 + 2 · (t + 1) + 0 · (t² + 1) T(t³) = 3t² - 12t = -12 · 1 + 0 · (t + 1) + 3 · (t² + 1) Step 3: Construct the matrix M(T) by arranging the coefficients from the linear combinations as columns. M(T) = [ [ 0, 1, -4, -12 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 3 ] ] Therefore, the matrix M(T) of the linear transformation T: P₃(R) → P₂(R) with respect to the bases {1, t, t², t³} for P₃(R) and {1, t + 1, t² + 1} for P₂(R) is: M(T) = [ [ 0, 1, -4, -12 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 3 ] ]