Blog

Semester 1, 2017 The University of Sydney School of Mathematics and Statistics MATH1002 Linear Algebra June 2017 Lecturers: B. Armstrong, D. Badziahin, A. Casella, T.-Y. Chang, J. Ching, R. Haraway, V. Nandakumar, A. Thomas Time Allowed: Writing – one and a half hours; Reading – 10 minutes Family
Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . SID: . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . . This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination; there are 20 questions; the questions are of equal value; all questions may be attempted. Answers to the Multiple Choice questions must be entered on the Multiple Choice Answer Sheet. The Extended Answer Section is worth 50% of the total examination; there are 3 questions; the questions are of equal value; all questions may be attempted; working must be shown. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. Marker’s use only Page 1 of 24 Semester 1, 2017 Page 2 of 24 Multiple Choice Section In each question, choose at most one option. Your answers must be entered on the Multiple Choice Answer Sheet. 1. The cosine of the angle between the vectors i+ 2j+ 2k and 3i+ 4k is equal to (a) 14 225 (b) 11 15 (c) 11√ 15 (d) 15 11 (e) 225 11 2. If u = i+ 2j+ 2k and v = 3i 2j+ k then u · v is equal to (a) 6i+ 7j 4k (b) 5 (c) 6i 7j+ 4k (d) 9 (e) 9 3. The unit vector in the direction of 2i j 2k is (a) 1 6 (2i j 2k) (b) 1 9 (2i j 2k) (c) 1 3 (2i j 2k) (d) i+ j+ k (e) i j k 4. If a = i+ j+ k and b = i j+ k then a× b is equal to (a) the vector 0 (b) 2i 2j (c) the scalar 0 (d) 2i+ 2j (e) i j 2k 5. Let v = 2i + 6j k and w = 3i + αj + 3 2 k. Find the value of α so that v and w are parallel. (a) 9 (b) 9 (c) 2 3 (d) 3 2 (e) 0 Semester 1, 2017 Page 3 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 4 of 24 6. Which one of the following is true for all linearly independent vectors v and w (a) v and w are orthogonal. (b) v and w are parallel. (c) If av + bw = cv + dw, where a, b, c, d ∈ R, then a = c and b = d. (d) The vectors v +w and v w are linearly dependent. (e) None of the above. 7. The system of equations 2x 2y 2z = 4 x 5z = 0 5x 4y 9z = 8 has the general solution (a) x = 5, y = 2, z = 1. (b) x = 0, y = 1, z = 0. (c) x = 5t, y = 4t 2, z = t where t ∈ R. (d) x = 1 + 2t, y = 2 + 9t, z = t where t ∈ R. (e) x = 5t, y = 4t+ 2, z = t where t ∈ R. 8. Let A = P D1 P 1, and B = P D2 P 1 where D1 = [ 1 0 0 2 ] , D2 = [ 3 0 0 1 ] , and P = [ 1 1 1 0 ] . Then (AB)5 is (a) [ 25 35 25 0 35 ] . (b) [ 35 25 35 0 25 ] . (c) [ 35 0 0 25 ] . (d) [ 25 1 0 35 ] . (e) [ 25 35 + 25 0 35 ] . 9. The determinant of the matrix 2 2 03 1 2 1 2 1 is equal to (a) 4 (b) 4 (c) 16 (d) 16 (e) 0 10. Let B be a 4× 4 matrix and suppose that det(B) = 2 . Then det ( 1√ 2 B ) is (a) 1/2 (b) 1/ √ 2 (c) 1 (d) √ 2 (e) 2 Semester 1, 2017 Page 5 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 6 of 24 11. Which one of the following augmented matrices is in row echelon form (a) 1 0 20 0 0 0 0 1 (b) 1 3 0 11 6 1 4 0 0 0 2 (c) [ 0 0 1 2 1 1 0 1 ] (d) [ 1 3 2 0 0 1 ] (e) None of the above. 12. Which one of the following is true for all n× n matrices A and B (a) (A+B)2 = A2 + 2AB +B2 (b) if A2 = B2 then A = ±B (c) if AB = 0 then A = 0 or B = 0 (d) if A is invertible then AB is invertible (e) if A is invertible and AB is invertible then B is invertible 13. If A = 1 1 01 1 1 0 1 1 , which one of the following is true (a) A is not invertible. (b) A is invertible and A 1 = 0 1 11 1 1 1 1 0 . (c) A is invertible and A 1 = 1 1 0 1 1 1 0 1 1 . (d) A is invertible and A 1 = 0 1 11 1 1 1 1 0 . (e) None of the above. 14. Which one of the following statements is true, given that A is a matrix of size 3× 3, B is a matrix of size 3× 2, and C is a matrix of size 2× 3 (a) A2 +BC is a 3× 3 matrix. (b) ACB is defined. (c) 2A+ CB is defined. (d) (BC)2 is a 2× 2 matrix. (e) B(A BC) is defined. Semester 1, 2017 Page 7 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 8 of 24 15. Consider the three planes with equations P1 : x + 2y z = 1, P2 : 2x 4y + 2z = 2, P3 : 3x 6y + 3z = 6. Which of the following is true (a) None of the planes are parallel to each other. (b) P1 and P2 are parallel to each other but not parallel to P3. (c) P1 and P3 are parallel to each other but not parallel to P2. (d) P2 and P3 are parallel to each other but not parallel to P1. (e) All of the planes are parallel to each other. 16. Consider the following system of equations: x + y + z w = 0 y + 2z w = 0 2x + 6z 4w = 0 Which one of the following statements about this system is true (a) There is a unique solution. (b) The general solution is expressed using exactly 1 parameter. (c) The general solution is expressed using exactly 2 parameters. (d) The general solution is expressed using 3 or more parameters. (e) There is no solution. 17. Which one of the following sequences of row operations, when applied to the matrix[ a b c d e f ] , produces the matrix [ d a e b f c 3a 3b 3c ] (a) First R1 := R1 R2, then R2 := 3R2, then R1 R2. (b) First R1 R2, then R1 := 3R1, then R1 := R1 R2. (c) First R2 := R2 R1, then R1 R2, then R1 := 3R1. (d) First R1 := 3R1, then R1 R2, then R1 := R1 R2. (e) First R1 R2, then R1 := R1 R2, then R2 := 3R2. Semester 1, 2017 Page 9 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 10 of 24 18. Suppose v and w are two non-zero vectors lying in this page: v w Which of the following is true (a) (v ×w) · v is a non-zero scalar. (b) v and v ×w are parallel. (c) (v ×w)× v is perpendicular to both v and w. (d) (w × v)× (v ×w) is parallel to v but not w. (e) v ×w points upwards, towards the ceiling. 19. The two lines given by the respective parametric scalar equations x = 3 + t y = 5 + 2t z = 5 t t ∈ R and x = 3 2sy = 2 4sz = 1 + 2s s ∈ R (a) do not intersect. (b) intersect at the point (7, 3, 9). (c) intersect at the point ( 2, 15, 0). (d) intersect at the point ( 3, 2, 1). (e) coincide. 20. Suppose a 3 × 3 matrix A has 3 distinct eigenvalues λ1, λ2 and λ3. Which one of the following is NOT necessarily true (a) The characteristic polynomial of A has 3 distinct roots. (b) det(A) = λ1λ2λ3. (c) A is invertible. (d) There is a 3× 3 invertible matrix P so that PAP 1 = λ2 0 00 λ3 0 0 0 λ1 . (e) If B is any 3× 3 invertible matrix then BAB 1 has eigenvalues λ1, λ2 and λ3. Semester 1, 2017 Page 11 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 12 of 24 This blank page may be used for rough working; it will not be marked. Be sure to enter your answers on the Multiple Choice Answer Sheet. Semester 1, 2017 Page 13 of 24 This blank page may be used for rough working; it will not be marked. End of Multiple Choice Section Make sure that your answers are entered on the Multiple Choice Answer Sheet The Extended Answer Section begins on the next page Semester 1, 2017 Page 14 of 24 Extended Answer Section There are three questions in this section, each with a number of parts. Write your answers in the space provided. If you need more space there are extra pages at the end of the examination paper. 1. (a) Let v = 2i j+ 3k and w = 4i 3j+ k. (i) Find the area of the parallelogram inscribed by the vectors v and w. (ii) Is the parallelogram inscribed by the vectors v and w a rectangle Either prove that it is, or prove that it isn’t. Semester 1, 2017 Page 15 of 24 (b) Let P be the plane with Cartesian equation 3x 4y+z = 5. What are the Cartesian equations of the line L that is perpendicular to the plane P and passes through the point Q = (2, 1, 7) Semester 1, 2017 Page 16 of 24 (c) Let A = [ a b c d ] and suppose that for all vectors v = [ v1 v2 ] and w = [ w1 w2 ] , (Av) · (Aw) = v ·w. Prove that if B = [ a c b d ] then BA = I. [Hint: consider particular vectors v and w.] Semester 1, 2017 Page 17 of 24 2. (a) A quadratic polynomial P (x) = ax2 + bx + c satisfies P (1) = 1, P (2) = 2 and P (4) = 3. Find a, b and c. Semester 1, 2017 Page 18 of 24 (b) Find the value(s) of the parameter a such that the following system of linear equa- tions is inconsistent. x+ 2y + z = 1 2x+ 4y + az = 2 x+ 2ay + 2z = 1 Semester 1, 2017 Page 19 of 24 (c) You are given that the matrix M = 0 2 18 15 6 21 36 14 satisfies the equation M3 = M2 M I. Compute M25. Semester 1, 2017 Page 20 of 24 3. (a) Let A = [ 0 4 1 0 ] . Find a diagonal matrix D so that A = PDP 1 for some invertible matrix P . You do not need to find the matrices P or P 1. (b) You are given that the matrix B = 1 0 32 1 5 1 1 0 has λ = 2 as one of its eigenvalues. Find the ( 2)-eigenspace of B. Semester 1, 2017 Page 21 of 24 (c) Let C and D be n× n matrices, with D invertible. Prove that if λ is an eigenvalue of C, then λ2 is an eigenvalue of D 2C2D2. Semester 1, 2017 Page 22 of 24 There are no more questions. More space is available on the next page. Semester 1, 2017 Page 23 of 24 This blank page may be used if you need more space for your answers. Semester 1, 2017 Page 24 of 24 This blank page may be used if you need more space for your answers. End of Extended Answer Section This is the last page of the question paper. B Semester 1, 2017 Multiple Choice Answer Sheet 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Write your SID here → Code your SID into the columns below each digit, by filling in the appropriate oval. Answers → a b c d e a b c d e Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 The University of Sydney School of Mathematics and Statistics MATH1002 Linear Algebra Family
Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . Other Names: . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . . . . . . Indicate your answer to each question by filling in the appropriate oval. This is the first and last page of this answer sheet Correct Responses to MC Component of MATH1002 Linear Algebra : Semester 1, 2017 Q1 → b Q2 → b Q3 → c Q4 → d Q5 → b Q6 → c Q7 → c Q8 → a Q9 → b Q10 → a Q11 → d Q12 → e Q13 → d Q14 → a Q15 → e Q16 → c Q17 → e Q18 → e Q19 → a Q20 → c