(10%) Define the linear transformation T by T(x) = Ax. Find (a) ker(T), (b) nullity(T), (c) range(T), and (d) rank(T). [3 6 -1 15 = 4 -2 3 8 10 -14 4-4 20 12 -3 3. (10%) Let B = {1, x,x², x³ } be a basis for P3, and T:= P3 → P4 be the linear transformation represented by T(x) = f t dt. a. Find the matrix A for T with respect to B and the standard basis for P4. Use A to integrate p(x) = 8 - 4x + 3x³. b. 4. (10%) Let B = {(1,3), (-2,-2)} and B' = {(-12,0), (-4,4)} be bases for R², and let A = =14] be the matrix for T: R² R² relative to B. → a. Find the transition matrix P from B' to B. b. " Use the matrices P and A to find [] and [T()] where [v]g' = [-1 2]. I C. Find P1 and A' (the matrix for T relative to B'). d. Find [T()]B'. 5. (15%) Please find the characteristic equation and the eigenvalues (and corresponding eigenvectors) of the matrix. 0 -3 2 -8 2 -4 0 2 6. (15%) Find (if possible) a nonsingular matrix P such that P-1AP is diagonal. Verify that P-¹AP is a diagonal matrix with the eigenvalues on the main diagonal. [5 3 A = 0 0 lo 2 2 7. (20%) Find a matrix P such that PT AP orthogonally diagonalizes A. Verify that PT AP gives the correct diagonal form. [9 30 01 900 A = 093 0 3 91 3 0 Lo