MATH 218 Final, Spring 2003

Signature: Student Number: Dr. J. Nikiel : Mathematics 218, Sections 1 and 2 Final Examination, June 7, 2003, 15:00—17:00 Important Instructions: 1. This exam consists of two sets of problems. The ten “TRUE/FALSE answer” questions are numbered A,B,...,J; each of them is worth 4 pts. The eight “workout” questions are numbered 1,2,...,8; each of them is worth 8 pts. The maximum score is 104 points. 2. Do not separate these pages. 3. The pink booklets are merely a source of scrap paper. I will not read what is in them. If you have any comments/requests write them here: No calculators. Good Luck! Part I. Write your answer TRUE or FALSE under each of the following problems/statements: Problem A. If det(A) = 2, then A is an orthogonal matrix. Your answer: Problem B. If A and B are square matrices of the same size, then det(A + B) = det(A) + det(B). Your answer: Problem C. If A and B are symmetric matrices of the same size, then aA + GB is a symmetric matrix for each choice of scalars a and f. Your answer: Me UNIVER! LIBRARY of BEIRUT Problem D. If 7@ is an eigenvector of a square matrix A, then the same ¥ is an eigenvector of A’. Your answer: Problem FE. Let d, = (1,1,1,1), #2 = (2,2,2,0), v3 = (3,3,0,0) and d = (4,0, 0,0). Then S = {%1, 00,03, %4} is a basis for R+. Your answer: Problem F. If A is a matrix, then each vector U belonging to the null-space of A must be orthogonal to each vector t# belonging to the row space of A. Your answer: Problem G. If } = 1 is an eigenvalue of a symmetric matrix A, then \ = —1 is an eigenvalue of A, too. Your answer: Problem H. If A and B are square matrices of the same size, then (AB)? = A? B?. Your answer: Problem I. If a square matrix A has a row of 1’s or a column of 1’s, then det(A) # 0. Your answer: Problem J. If {u1,w2,..-,t%} is an orthogonal set of vectors in an inner product space V such that dim(V) =n, then n < k. Your answer: Part II. You must provide all essential details of your solution to each of the following problems 1,...,8 on the page that contains the problem (continue on the reverse side of that page when needed). Problem 1. 12 0 0 2 ; 0321), {-1 Use Cramer’s Rule to solve the system 210 2] 7=| 9 Evaluate deter- 1101 1 minants in the denominators, but do not evaluate determinants in the nominators. fiwEnicaw UNIVER: % LIBRARY OF BeIROT me. Problem 2. Find a matrix P that diagonalizes the following matrix A. Then find PAP. 111 A={0 2 2 0 0 3 Be iste TNTVER: LIBLARY OF GEIRUT Problem 3. Find the least squares solution of the linear system 1 oi 1 2 2)|-#=1 3 1 0 0 ‘S UNIVERSIT® i Libr, RY a tt Problem 4. Find an orthonormal basis for the row space of the following matrix 1 2 3 0 Nr rR rome Noor i NIVERSIT LIBRARY a OF beRTT Let A be a matrix of size 6 x 5. Suppose that the null-space of A is spanned by the vectors Problem 5. U, = (1,1,1,0,0), v = (0,0,1,1,1) and a = (1,1,2,1,1). Use this information to find the rank of A. Problem 6. Let A be a square matrix of size n x n. Write four conditions equivalent to “A is singular”. oe vENSIT Libwary OF BEIRUT Problem 7. Let V be an inner product space and W be a subspace of V. Let T’: V > W denote the orthogonal projection of V onto W. Prove that ker(T) = W+. Nese | | Libiary OF BeIROT Problem 8. Let Vi and V2 be subspaces of R’. Let V = V; V2. Suppose that dim(V,) = 4 and dim(V2) = 5. Prove that V must be infinite.