Final 2017 - Math 1600 exam PDF

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####### MATH 1600B FINAL EXAM 2017Section 1 Section 2P. Gupta A. Dhillon Circle your section number aboveFirst Name(print!) :Last Name(print!) :Student number :####### INSTRUCTIONS: This booklet has 15 pages and 14 questions. Last page is forrough work. Answer questions in the space provided. Circle...

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MATH 1600B

FINAL EXAM

2017

Section 1

Section 2

P. Gupta

A. Dhillon Circle your section number above

First Name (print!)

:

Last Name (print!)

:

Student number

:

INSTRUCTIONS: 1. This booklet has 15 pages and 14 questions. Last page is for rough work. 2. Answer questions in the space provided. 3. Circle the correct answer for multiple choice questions. 4. In Part B, circle the answer either T (for true) or F (for false). If you circle T you must give a reason. If you circle F you must provide a example showing that the statement is false. 5. In Part C, show all work. 6. Calculators, electronic aids and notes cannot be used. 7. Time allowed : 3 hours. 8. Having a phone on your person during the exam is considered cheating. It is irrelevant if you use it or not. You will receive a mark of 0/100 if you are caught with a phone.

Name :

2017 Grading table Questions

Points

1-7

28

8

30

9

6

10

6

11

6

12

6

13

6

14

6

15

6

Total

100

Score

Page 2 of 15

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2017 Part A. Mulptiple Choice (28 points)

Circle your answer for each question below. Each question in Part A is worth 2 points. No partial credits will be awarded. 1. (4 points) Suppose that

Then u · v =

  u = 1 4 −1

  and v = 2 1 −1 .

A. 0. B. 8.

C. 6. D. 5. E. 7.

2. (4 points) Let A be an invertible n × n matrix with entries in R. Which of the following statements is false? A. det(A) 6= 0.

B. The linear system Ax = b has infinitely many solutions. C. rank(A) = n. D. nullity(A) = 0. E. The rows of A are linearly independent.

3. (4 points) Consider the matrix  1 1 2 A = 1 4 2  . 2 0 2 

What is det(A)? A. 3 B. 6 C. 0 D. −6 E. −3

Page 3 of 15

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2017

4. (4 points) Which of the following polynomials in the variable λ is the characteristic polynomial of   −4 15 1  0 2 0 ? −6 13 3 A. −(λ + 1)(1 − λ)2

B. −(λ − 2)2 (λ + 3) C. (2 − λ)(4 + λ)2

D. −(2 − λ)(3 − λ)(4 + λ) E. −(1 − λ)(λ + 2)(3 − λ)

5. (4 points) Suppose A and B are n × n matrices with A ∼ B. Which of the following statements is false? A. rank(A) = rank(B). B. A and B have the same eigenvalues. C. row(A) = row(B). D. det(A) = det(B). E. A and B have the same characteristic polynomials.

6. (4 points) Which of the following matrices is not an orthogonal matrix?   0 −1 A. 1 0 √   1/2 3/2 √ B. − 3/2 1/2 √   √ 1/ √2 −1/√ 2 C. 1/ 2 1/ 2 √   1 − 2 D. √ 2 1   −1 0 E. 0 1

Page 4 of 15

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2017

7. (4 points) Consider W = span



   1 2 −1 0 , 2 4 3 1 .

Which of the following sets is a basis for W ⊥ ? n   o −2 1 0 0 , 1 −1 −1 5 A. B.

n

  o 1 2 −1 0 , 0 0 −5 1

C.

n

D.

n

E.

n

  o 1 2 −1 0 , 1 −1 −1 5   o 1 0 1 2 , −3 1 −1 5

  o 1 2 −1 0 , 1 0 5 1

Page 5 of 15

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2017 Part B. True or False (30 points, 5 points each)

8. (a) Let A be a 5 × 4 matrix with nullity(A) = 2. Then rank(A) = 2.

T or F

(b) Let A be an invertible n × n matrix. Let B be an n × n matrix chosen so that det(AB) = 0. Then B is not invertible. T or F

(c) The function T : R2 → R2 given by T (x, y) = (x + y, x + 2y) is a linear transformation. T or F

Page 6 of 15

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2017

(d) Let A be a 2 × 2 matrix. Let v and w be two eigenvectors of A. Then, v + w is an eigenvector of A. T or F

(e) Every non-zero 2 × 2 matrix A is invertible.

T or F

(f) Let A be a 3 × 3 matrix with rank(A) = 2. If b ∈ R3 , then Ax = b is always consistent. Here   x1  x = x2 . x3 T or F

Page 7 of 15

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2017 Part C. Long answer (42 points)

9. (6 points) Let S be a collection of vectors in Rn . (a) Define what it means for S to be a subspace of Rn .

(b) Give an example of a collection of vectors in R2 that do not form a subspace of R2 . Make sure you explain why your example is not a subspace....