Math 225 Practice Final Problems Show all work! Correct answers without supporting work will not be given credit. Calculators and notes are not permitted. (1) Let A = 6 0 0 0 2 7 0 0 0 0 9 0 0 0 1 5 . a) What are the eigenvalues of A? For each eigenvalue find a basis for the -eigenspace. b) Is A diagonalizable. Why? (2) Let A be the matrix in problem 1. Find the general solution to x'(t) = Ax(t). (20 points) (3) Let A be a 33 rank 1 matrix. Is A diagonalizable? Why? (4) a) Find the general solution to (D2+4)(D-3)2y=0. b) Find the general solution to (D2+4)(D-3)2y=e2x. (5) Let x = (1,1 ,2), and let y = (2,2,2). Find a pair of orthogonal vectors that span the same space as x and y. (6) Which of the following matrices are similar to one another? Carefully justify your answer. a) 100 (140) 829 b) 7 0 0 (240) 819 c) 9 6 5 d) 3 0 0 (0 48) (1 3 0) 0 0 1 0 1 5 e) 3 0 0 (0 3 0) 0 2 5 . (7) Let A = 5 0 1 5 . a) The vector 0 is an eigenvector for A corresponding to the eigenvalue 5. Find a 1 generalized eigenvector. b) What is the general solution to x(t) = Ax(t)? (8) Find the general solution to the system x(t) = Ax(t) where A is the matrix -1 0 0 1 5 0 1 6 -2 . (9) Let A be the matrix in problem (8). Find a matrix S such that S-1AS is diagonal. (10) Let A = 4 -3 , and let b = e2t 2 -1 et . Find a particular solution to x(t) = Ax(t) + b . (11) Let T: V V be a linear transformation such that T(V) = T2(V). Show that 0 is the only vector that lies in both the null space of T, and the range space of T. (12) Let B={x1,. . . ,xn} be a basis for the vector space V, and let W be a subspace of V. Does W necessarily have a basis that consists of vectors in B? Carefully explain your