MATH 218 Final, Spring 2004

= LIBRARY OF KrRot American University of Beirut MATH 218 Elementary Linear Algebra Spring 2004 Fina] Exam Date: Wednesday, June 9, 2004 - 11:30 am to 1:30 pm Instructor: Dr. Mohamed Kobeissi Name: ID #: Section: This is NOT an open-book exam. Your exam should have 1] pages including this one, and there are 6 questions totaling 100 points. You can continue each exercise on the reverse side of the paper if needed. Question 1. Good luck Exercise 1 Let A= {| -—-1 k 1 a. (7 points) For what value of k the matrix A is invertible? b. (3 points) For k = 2, solve the system AX = | 0 Exercise 2 Letz = (1,-1,1,1),y = (-1,-1,1, —1), < = (0,1,1.0), andlet W =span{z, y, z} be a subspace of R’. W a. (5 points) Show that fis an orthogonal set of vectors relative to the Euclidean inner product. b. (3 points) What is the dimension of IW’? Justify. c. (8 points) Find a basis for 4+, the orthogonal complement of W’. d. (4 points) Find a vector t € R? such that the {z, y, z,t} form a basis of R‘. Justify. Exercise 3 (/0 points) Let WV be the subspace of R® spanned by {(1, 1, 2). (1.1, L}, and let 1 b=] 1 0 Find projy-), the orthogonal projection of b on WW’. 1 Exercise 4 Consider the matrix A = | 0 2 Ow oo moO Nh a. (4 points) Find the characteristic equation of A. b. (8 points) Find the eigenvalues of A, and a basis for each eigenspace of A. c. (J0 points) Find an orthogonal matrix P that diagonalize A, and show the relation between A, P. P~} and D. d. (3 points) What is the rank of A? Justify. Exercise 5 In an inner product space V, prove the following: a. (5 points) < u,v >= [jut vl]? — Zlfu- vll?. b. (5 points) \lu + ul] < ||ul] + |lel].- (hint: you may use the Cauchy-Schwarz inequality, | < u,v > | < |luli.|le|) Exercise 6 (25 points, 5 points each) Prove (concisely) or disprove (by a counter example), the following statements. a. Let A bean x n matrix. i) If A is invertible, then the orthogonal complement of the row space of A is reduced to 0. ii) If 0 is an eigenvalue of A‘, then A is invertible. iii) If \ is an eigenvalue of .4 with corresponding eigenvector 2’, then 3 is an eigenvalue of A3 with the same corresponding eigenvector. b. The Nullity of a 3 x 4 matrix A can be equal to zero. c. If a set of 3 vectors in R® have the property that each 2 of them are linearly independent, then this set is linearly independent.