Find the matrix MBB(T) of the linear transformation T(f(t)) = f(2t + 5) from P₂ to P2 with respect to the standard basis {1, t, t²} for P₂. MBB(T) = 000 01 000

Find the matrix A' for T relative to the basis B'. T: R² A' = R², T(x, y) = (x + y, 3y), B' = {(-4, 1), (1, -1)}

The vectors p1(t) = 1 + t², p2(t) = −t + 3t², and p3(t) = 1+t+t2 form a basis B 2(t), p3(t)} for P₂(R). Let p(t) = 1 + 2t + 4t², find its B-coordinates [p(t)B. =

7. (a) Consider the ordered basis ß = [1,1 +x, 1 + x + x²] for V = P₂ and F = R. Suppose is in P2 and you know these coordinates: [] = (1, 2, 3). Find the vector 7. (b) Let A be a 2 x 2 matrix and a be a real number. Prove that det (a A)=a² det(A)