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8009A Semester 2 2010

The University of Sydney School of Mathematics and Statistics

MATH1014 Introduction to Linear Algebra November 2010

Lecturers: S Britton, J Henderson, A Molev

Time Allowed: One and a half hours Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SID: . . . . . . . . . . . . .

Seat Number:

.............

Marker’s use only

This examination has two sections: Multiple Choice and Extended Answer.

The Multiple Choice Section is worth 50% of the total examination; there are 25 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.

Approved non-programmable, non-graphics calculators may be used. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM.

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8009A Semester 2 2010

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Multiple Choice Section In each question, choose at most one option. Your answers must be entered on the Multiple Choice Answer Sheet. 

   4 2 In questions 1, 2, 3 and 4, a =  0  and b = −1. −3 −2 1. Find 2 a − 5 b.     −2 6 (b)  5  (a)  −5  −22 −2

  −2 (c)  5  4

 18 (d)  −5  −16

 14 (e) −5 −2

(c) 2

(d) 13

(e) 14

3. Find a unit vector in the direction of a.       1 1 4/5 (a)  0  (b)  0  (c)  0  −1 −3/4 −3/5

√  4/ 7 (d)  0√  −3/ 7

 4/25 (e)  0  −3/25

4. Find the cross product a × b.     −3 −5 (b)  2  (a)  2  −4 −4

  −3 (d)  2  6

2. Find the dot product a · b. (a) 0

(b) 1



 3 (c) −14 4





    4 k    5. For which value of k is k perpendicular to 1? 5 2 (a) 1

(b) −7

(c) −2

(d) 0







 −5 (e) −14 6

(e) −1

6. Vectors a, b, c, d are vectors in R3 . Select the option which is NOT a meaningful expression. (a) a + ( c × d)

(c) ( a · b) + ( c · d)

(e) ( a · b) × ( c × d)

(b) ( a × b) · c

(d) ( a · b)( c × d)

8009A Semester 2 2010

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8009A Semester 2 2010

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7. A line in R2 has Cartesian equation 3x + 4y = 24. Which of the following is a vector equation of the line?             3 4 x 8 3 x = +t ,t ∈ R (b) = +t ,t ∈ R (a) y 4 −3 y 0 4             x 4 4 x 4 0 (c) ,t ∈ R +t = ,t ∈ R (d) +t = −3 3 y 3 6 y       x 6 4 (e) = +t ,t ∈ R y 8 3  5 8. Find the Cartesian equation of the plane through the point (1, 3, 1) with normal −2 . 1 

(a) x + 3y + z = 0

(b) 5x − 2y + z = 2

(c) 5x − 2y + z = 0

(d) 6x + y + 2z = 11

(e) x + 3y + z = 10

9. The vector [0, 1, 1, 1, 0, 1] is the parity check code vector for which one of the following code vectors? (a) [0, 1, 1, 1, 0] (b) [0, 1, 1, 1, 0, 0] (c) [1, 0, 0, 0, 1, 0]

(d) [0, 1, 1, 1, 1]

(e) [0, 1, 1, 1, 0, 1, 0] 10. The augmented matrix for a system of linear equations in x, y and z reduced to   1 −2 1 5  0 1 3 0 . 0 0 1 −2 What is the solution to the system of equations? (a) x = 5, y = 0, z = −2.

(b) x = 8, y = 3, z = 2.

(c) x = 19, y = 6, z = −2.

(d) x = −9, y = −6, z = 2.

(e) x = −5, y = −6, z = −2.

11. Evaluate 1 × 2 + 3 × 4 + 5 × 6 + 7 × 8 + 9 × 10 in Z11. (a) 1

(b) 3

(c) 5

(d) 7

(e) 9

8009A Semester 2 2010

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8009A Semester 2 2010

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 1 4 2 Questions 12, 13 and 14 refer to a Leslie matrix, L = 0.9 0 0, for an insect 0 0.6 0 population with 3 age classes. 

  100 12. If the initial insect population vector is x0 = 200 , what is the population vector x1 ? 150           100 1200 100 1200 450 (b)  90  (c) 180 (d)  180  (e)  90  (a) 800  300 120 90 90 120 13. How many daughters do female insects in the second age group produce on average? (a) 0.6

(b) 0.9

(c) 1

(d) 2

(e) 4

14. What percentage of insects in the first age group survive to the second age group? (a) 0%

(b) 10%

15. Find the eigenvalues of (a) 2 and 1.



(c) 60%

(d) 90%

(e) 100%

(c) 2 and −5.

(d) −2 and 5.

(e) −2 and −1.

 2 3 . 4 1

(b) 2 and 5.

16. Suppose that A is a 2 × 4 matrix and B is a 4 × 5 matrix. Which of the following is true? (a) AB is a 2 × 5 matrix. (b) AB is a 2 × 4 matrix. (c) AB is a 4 × 4 matrix. (d) AB is a 5 × 2 matrix. (e) AB is not defined.

 1 0 2 17. The matrix  1 −1 1  has an eigenvalue of 3. 2 0 1 Which one of the following is a corresponding eigenvector?         3 0 1 3 (b) 3 (c) 2 (d)  0  (a) 1  3 0 1 −3 

  2 (e) 1 2

8009A Semester 2 2010

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8009A Semester 2 2010

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 2 0 0 18. What are the eigenvalues of the matrix  3 −1 0? 1 4 5 

(a) 2, 3, 1.

(b) 2, −1, 5.

(d) 2, 0, 0.

(e) −2, −1, −5.

 2 −3 0 0 2. 19. Calculate the determinant of  1 −1 4 1

(c) −2, 1, −5.



(a) −25

(b) −19

(c) −13

(d) −7

(e) −5



 1 2 0 5 2 3 is the augmented matrix of a system of linear equations. 20. The matrix  0 0 1 1 0 0 0 1 0 Which one of the following statements is true? (a) The system has infinitely many solutions, with one parameter. (b) The system has infinitely many solutions, with two parameters. (c) The system has five unknowns. (d) The system is inconsistent. (e) The system has a unique solution. 21. The augmented matrix for the system of equations

 1 2 1 reduces to 0 1 1/5 0 0 0

x + 2y + z = k 2x − y + z = 2k 3x + y + 2z = k 2  k 0 . 2 k − 3k

Which one of the following statements is true? (a) If k = 0 the system is inconsistent.

(b) The system is consistent for any value of k . (c) There are no values of k for which the system is consistent. (d) The system is consistent if k = 0 or if k = 3. (e) The system is consistent only if k = 0.

8009A Semester 2 2010

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22. Suppose A, B , C are n × n matrices, and that B and C are invertible. If CA(BC)−1 = B then A must be equal to: (a) C −1 B 2 C

(c) B 2

(b) I

(d) C −1 BCB

23. If A is a 3 × 3 matrix and det(A) = 7, what is det(2A)? (a) 7

(b) 14

(c) 21

(d) 56

(e) CBC −1 B −1

(e) 63

24. Alex is 4 years younger than Bob. Two years ago, Bob was twice as old as Alex. Let a = Alex’s present age and b = Bob’s present age. Which system of equations describes this situation? (a)

b−a = 4 b − 2a = 2

(d) a − b = 4 2b − a = 2

(b) a − b = 4 2a − b = 2 (e)

(c)

b−a = 4 2b − a = 2

b−a = 4 2a − b = 2

25. Suppose A is an invertible 3 × 3 matrix. Which one of the following statements about the system of linear equations A x = b is true? (a) There is no solution. The system is inconsistent for all b ∈ R3 . (b) There is a unique solution for any b ∈ R3 .

(c) There are infinitely many solutions for any b ∈ R3 .

(d) The system is always consistent, but there may be a unique solution, or infinitely many, depending on b. (e) The system may or may not be consistent, depending on b.

8009A Semester 2 2010

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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

8009A Semester 2 2010

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Extended Answer Section There are three questions in this section. Write your answers in the spaces provided below the questions.

MARKS

1. (a) Let A = (1, 5, 0),

B = (4, 3, 5), C = (7, −2, 6), D = (1, 2, −4) . −→ −−→ (i) Find the vector AB and the vector CD. −−→ −→ (ii) Find the scalar k such that CD = k AB. −→ −−→ (iii) Do AB and CD point in the same direction or opposite directions? Explain your answer.

1 1

1

8009A Semester 2 2010

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MARKS

(b) (i) (ii)

Write down the normal vector to the plane with equation 3x + 2y − 2z = 0. Show that the plane with equation 3x + 2y − 2z = 0 is perpendicular to the plane with equation 2x + 4y + 7z = 3.

1

2

8009A Semester 2 2010

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MARKS

(c) Line ℓ1 passes through the point (3, 1, 1) and has vector equation       5 3 x y  = 1 + t  1  (where t ∈ R). 1 −2 z Line ℓ2 also passes through the point (3, 1, 1) and has parametric equations x= 3+s y = 1 + 2s z = 1+s

(where s ∈ R).

Find the Cartesian (general) equation of the plane containing both ℓ1 and ℓ2 .

3

8009A Semester 2 2010

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MARKS

(d) The GTIN-13 code vector v = [1, 1, 0, 4, 2, 3, 0, 8, 0, 1, 2, 0, 5] has a single error in the second component (reading from left to right). Find the correct v given that the check vector is c = [1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1], and operations are performed in Z10 .

3

8009A Semester 2 2010

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MARKS

 1 0 0 2. (a) Let A =  2 1 −1  6 0 1 

and

B=



2 0 7 −1 3 1



.

(i)

Find 2B.

1

(ii)

Find BA.

2

(iii) Find A−1 .

3

  1 (iv ) Using your answer to part (iii ) (or otherwise) solve A x = 1. 1

1

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8009A Semester 2 2010

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(b) Write down the augmented matrix for the following system of equations, and reduce the matrix to row echelon form. Hence solve the system of equations. x − 3y + 4z = 2 3x + y − 8z = 6 x − 13y + 24z = 2

3

8009A Semester 2 2010

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MARKS 3

(c) Suppose A is a 3 × 3 matrix, and there exists a vector b in R such that A x = b is an inconsistent system of equations. How many solutions are there to A x = 0? Justify your answer.

2

8009A Semester 2 2010

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3. (a) A Markov process with 3 states is used to model the changing use of oil, gas and electricity to heat houses in a certain country. State 1 corresponds to the use of oil. State 2 corresponds to the use of gas. State 3 corresponds to the use of electricity. The transition matrix for this Markov process, based on yearly observations, is   0.8 0.1 0.1 P = 0.2 0.9 0  . 0 0 0.9 (i)

(ii)

According to this model, if a house is currently heated using gas, what is the probability that it will be heated using oil in two years’ time?

2

Find a steady state vector.

3

(iii) If this Markov process continues indefinitely, what percentage of houses will eventually be using oil for heating?

1

8009A Semester 2 2010

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8009A Semester 2 2010

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 2 10 0 (b) The Leslie matrix for a population with three age classes is L =  0.8 0 0 . 0 0.5 0 Find the eigenvalues of L. 

3

8009A Semester 2 2010

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MARKS

(c) The trace of a matrix A, written tr(A), is the sum of the elements on its main diagonal.   1 2 (For example, if A = , then tr(A) = 1 + 4 = 5.) 0 4 Prove that a 2 × 2 matrix A has two distinct, real, eigenvalues if and only if [tr(A)]2 > 4|A|.

End of Extended Answer Section This is the last page of the question paper.

3

8009B Semester 2 2010

Multiple Choice Answer Sheet Write your SID here −→

The University of Sydney School of Mathematics and Statistics

Code your SID into the columns below each digit, by filling in the appropriate oval.

MATH1014 Introduction to Linear Algebra

0 1 2 3 4 5 6 7 8 9

✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

 ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁

✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂ ✄ ✁✂

 ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁  ✁

0 1 2 3 4 5 6 7 8 9

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . . . . . .

Indicate your answer to each question by filling in the appropriate oval.

Answers −→

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13

a ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁

b c ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁

d ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁

e ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁

Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25

This is the first and last page of this answer sheet

a ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁ ✄  ✂ ✁

b c ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁

d e ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁ ✄ ✄  ✂ ✁✂ ✁

Correct Responses to MC Component of MATH1014 Introduction to Linear Algebra 8009: Semester 2 2010 Q1 −→ c Q2 −→ e Q3 −→ c

Q4 −→ a Q5 −→ c Q6 −→ e

Q7 −→ d Q8 −→ c Q9 −→ a

Q10 −→ c

Q11 −→ b Q12 −→ b Q13 −→ e

Q14 −→ d Q15 −→ d Q16 −→ a Q17 −→ e

Q18 −→ b Q19 −→ d Q20 −→ a

Q21 −→ d Q22 −→ a Q23 −→ d Q24 −→ e

Q25 −→ b...