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May2005 - THE FACULTY OF ARTS AND SCIENCE University of...

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Unformatted text preview: THE FACULTY OF ARTS AND SCIENCE University of Toronto ' FINAL EXAMINATION, April-May 2005 MAT 223H1S - Linear Algebra I f Examiners: S. Cohen E. Pujals M. Saprykina R. Stanczak Last Name: Student # First Name: (a) TIME ALLOWED: 3hours (b) NO AIDS ALLOWED (0) TOTAL MARKS: 100 (d) WRITE SOLUTIONS IN THE SPACE PROVIDED (e) GIVE ANSWERS FOR THE MULTIPLE CHOICE QUESTIONS ON THE SECOND PAGE, ONLY THOSE ANSWERS WILL COUNT (f) DO NOT REMOVE ANY PAGES. THERE ARE 14 PAGES (9) MARKING SCHEME: (1) Multiple Choice Questions : see page 3 (2) Written Questions: indicated by each question MARKER'S REPORT NAME: Student No Give answers for multiple choice questions below. ONLY THOSE ANSWERS WILL COUNT Right way of giving the answer : Q PART I . MULTIPLE CHOICE QUESTIONS MARKING SCHEME: 4 marks for each correct answer 1. Consider the system of linear equations, where k is a real number. X1+ X2 + X3 = 4 X3 =2 (k2—4)X3 = k—2 Which of the following statements are true? A. The system cannot have a unique solution for any value of k . B. The system has always infinitely many solutions. C. There is a value of k for which the system has no solutions. A and B A and C B and C A only B only all @@®@®® 2. Which of the following is equivalent to the statement : “ A is an invertible nxn matrix ” A. A +AT is an invertible nxn matrix B. A2 isaninvertible nxn matrix. C. A is aproduct of nm elementary matrices. D. dim[(rowA)i]=n. (D all Q) A, B and C CD B, C and D (D B andC @CandD ©AandB cont’d . . . ., .) 3. Which of the following sets are subspaces of P2 ? A- W = {p(x)€P2 l 1900 = p(-'X)} 3- W = {p(x)€P2 I 19(1) = 0} C- W = {p(x)eP2 | 13(0) = 1} all A and B A and C B and C B only Aonly @@®@®® 4. Find the dimension of the following subspace of R5 ®1 (232 ($3 @4 ®5 @0 W: { (al 32,33 34 ,as ) I an = a3 and a3=a4 +3212} —1 2 0 5. Let A = [ l — 2 1 . An orthonormal basis for the column space is O 0 1 l O 0 1 —2 O 1 1 0 ®{O,l,0} ®{—1,2,1} @{T—1,0} O 0 1 0 0 l 2 O 1 1 1 1 1 11 l —l 0 ©{—1——1,——1} @{—1,§1} ®{——l,i1} 2 O 6 2 0 2 2 0 21 a 3 6. Let W={ b |a,be]R} and let X= 2 .Find the distance between X and the b 6 vector in W that is closest to X. ® JE ® 8 @@@@ gonfd.“ 7. Suppose A is a 3x3 matrix with detA= 2. Then det(-2A‘3ade) = ? @—4 (22 ®_1 2 @4 69—8 ©—1 8. Find the least square solution to the system X1 —X3=1 X2+X3=1 X1+X2 =- X1~X2+X3=—1 cont’d . .. l —2 2 9. Given is that A‘1 = 0 1 0 .If X satisfies the matrix equation XA'1=AT O —1 1 then the third row of the matrix X is (D [2 l -3] (23 [0 1 1] [2 l 0] [-3 0 -2] [—2 l 2] [-2 l 5] @@®@ 10. Suppose A = l, 2, 3 are eigenvalues of an 3x3 matrix A . Which of the following is true ? A. detA=6. B. The eigenvalues of the matrix A2 + 6A'1 are 7, 7, 11. C. A is diagonalizable. (D A and B (Z) A and C ® B and C @ B only (5) all © C only cont’d ... 11. Suppose V is a vector space and dimV = 4. Which of the statements are true? A. Every spanning (generating) set for V is a part of a basis for V. B. Any five vectors in V are linearly dependent. C. Any five vectors in V span V. A and B A and C B and C B only all C only @@@@®® abc 12. Suppose d e f =~1.Find x1 from the system g h i axl +be +cx3 =2a—3c dx, +ex2 +fic3 =2d—3f gxl +hx2 +ix3 =2g—3i @@®@@@ | cont’d . . . PART II. OPEN ENDED QUESTIONS 13. Let A bean nxn invertible matrix such that A"I =AT. (a) [6 marks] Prove that MAX” = “X” for all X in R". (b) [5 marks] Show that the only eigenvalues of A are 1, ~ 1. gont'd . . . 9 13.(c) [6 marks] If A is symmetric prove that A"1 = AT if and only if A2 = I. confd." 10 14. (a)[7 marks] Suppose S = {1, f, g } is a linearly independent set of functions in the vector space 9 of all functions f: R —) IR with usual addition and multiplication by scalar. Let h e 5’ be NOT in span(S). Prove that the set S; = {1, 1+f, f +g, g +h} is linearly independent. cont’d ... ll 14.(b) Suppose that the linear transformation T: R"—~> R” is induced by an mxn matrix A ’ where (m—1)<n and rankA = m —l. (i)[6 marks] Can T be ONTO ? Can T be ONE-TO-ONE ? Justify your answers. (ii)[5 marks] Give an example of such a matrix A in case n = 3. .cont’d . ..A 1 1 . The characteristic polynomials of both 1 matrices are the same cA(x) = cB(x) = (x —1)2(x —4). (a) [7 marks] Find an invertible matrix P and the diagonal matrix D such that P‘ ‘AP = D. .cont’d 13 15.(b) [5 marks] Is B diagonalizable ? Justify your answer. (0) [5 marks] Your friend tells you “B is not similar to A”. Prove or disprove his statement. 14 ...
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