MATH 218 Final, Fall 2005

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2005-2006 Name:.................................. ID:................................ Section 1 Section 2 Section 3 Mrs. Z. Mouzeihem Ms. M. Houri Ms. D. Audi Time: 120 min I- (35 points) Let A =    1 2 1 1 3 7 2 2 2 4 2 2    (a) Find a basis for the row space of A. (b) Find a basis for the column space of A. 1 (c) Find a basis for the null space of A. (d) Find a basis for the null space of AT. (e) If W ⊥ is the column space of A deduce a basis for W. 2 II- (25 points) Let T : P2 → P2 be a linear transformation such that: T(p(x)) = p(2x + 3) a) Find the matrix A = [T]B with respect to the standard basis B = {1, x, x2}. b) Find the eigenvalues for the matrix A = [T]B c) Write the formula for [T(p(x))]B without justification. 3 d) Deduce the values of λ such that T(p(x)) = λp(x) has nonzero solutions. III- (20 points) Let S = {u1, u2, u3} be a basis for R3, with u1 = (2, 2, 0), u2 = (2, 0, 0) and u3 = (0, 0, 3) a) Find an orthonormal basis S1 = {q1, q2, q3} for R3. b) Write u = (1, 1, 1) in terms of the basis S1. 4 IV- (20 points) Let T : R3 → R4 be defined by: T(x, y, z) = (x + 2y, x + 3z, z − x, 0) a) Show that T is a linear transformation. b) Is T one-to-one? Justify c) Find the rank of T. d) Find a basis for the range of T. 5 V- (25 points) Let A be a 3 × 3 matrix with eigenvalues λ1 = −2, λ2 = 0 and λ3 = 4, and let v1, v2 and v3 be the corresponding eigenvectors respectively, where v1 = (0, 3, −2), v2 = (−3, −3, 0) and v3 = (−2, 0, 3). Let E1 = span{v1} be the eigenspace corresponding to λ1 = −2, E2 = span{v2} the eigenspace corresponding to λ2 = 0 and E3 = span{v3} the eigenspace corresponding to λ3 = 4. a) Write x = (−10, −3, 4) as a linear combination of the vectors {v1, v2, v3}, and deduce Ax. b) Find the null space of A (Hint: Use the fact that λ = 0 is an eigenvalue for A.) c) Find a matrix P that diagonalizes A and determine P −1AP 6 VI- (15 points) Short Proofs 1) Show that if A is an orthogonal matrix then det(A) = ±1. 2) State and prove the theorem of Pythagoras for an inner product space V . 3) Let T : V → W be a linear transformation, show that Ker(T) is a subspace of V. 7 VII- (30 points) Show your work. a) Let A, B and C be n × n matrices, if B = A−1and C = BT find C−1. b) Let T : R2 → R2 be a linear operator with T(1, 1) = (2, 1) and T(3, 2) = (0, 3), find T(5, 4). c) Let A =    1 0 3 2 5 2 1 0 8    and C =    2 0 0 0 0 2 0 2 0    . If B is a 3 × 3 matrix and AB = C find det(B). d) Let A = � 4 1 d −4 � , find the value of d that makes A orthogonally diagonalizable. 8 e) Find the null space of a 5 × 4 matrix with linearly independent columns. f) Let {u, v, w} be a basis for a vector space V. Show that {w, u + w, u + v + w} are linearly independent. VII- (30 points) Answer TRUE or FALSE without proof (5 points for each right answer and 2 points penalty for each wrong answer). a- Let A be an n × n matrix, if λ = 0 is an eigenvalue for A then A is invertible. b- Let T : Pn → Pn be a linear transformation, if range of T is Pn then T is one-to one. c- Let A, B and C be n × n matrices, if AB = AC then B = C. d- Let A be an m × n matrix and b be an m × 1 vector, if b is not in the column space of A then AX = b has no solution. e- Let T : V → R be a linear transformation with V = span{u}. If T(u) = 0 then T ≡ 0. f- Cramer’s rule can be used to solve any system of linear equations. GOOD LUCK!! 9