9781133110873 08 ans · a2 answer key section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21....

6
Chapter 8 Section 8.1 (page 398) 1. 3. 5. 1 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 8 31. 33. 35. 37. 2, 3 39. 41. 43. 45. 47. 49. 51. 53. 55. Proof 57. (a) (b) (c) , where is an integer. 59. False. See the Remark, page 392. 61. Proof Section 8.2 (page 403) 1. 3. 5. 4 7. 9. 11. 5 13. 15. 17. 19. 21. 23. 25. 27. 29. (a) (b) (c) (d) 31. 33. Not invertible 35. 37. 39. Proof 41. (a) and (b) Proofs 43. (a) (b) (c) where is an integer k 1i n 1 i 1 i n 4k n 4k 1 n 4k 2 n 4k 3 , 1i 2000 1, 1i 2010 1 1i 5 i 1i 4 1, 1i 3 i, 1i 2 1, 1i 1 1i i, 5 3 10 3 i A 1 i 0 0 0 i 0 0 0 i A 1 1 3 i 2 i 3i 6 3 25 4 25 i 2 5 1 5 i 2 11i 3 4i 1, 2, 3 ± 2 i 2 3 , 1 ± 3 i 1 10 9 10 i 1 2 5 2 i 13 10 9 10 i 7 11 62 11 i 1 2i zw 3 i 10 w z 5 2 10 wz 3 i 10 65 5 1 2 3 1 2 1 2 3 4 5 z and z = 4 Real axis Imaginary axis 4 8 Real axis 4 8 4 4 8 8 z = 8i z = 8i Imaginary axis 2 4 6 2 4 z = 6 3i z = 6 + 3i Real axis Imaginary axis 2 4 8i 6 3i k i n 1 i 1 i n 4k n 4k 1 n 4k 2 n 4k 3 i 5 i i 4 1 i 3 i i 2 1 i 2010 1 i 1 i 5 25i 15 10i 15 30i 5 3i 2 2i 4 4i 2i 6 2 2i 4 4i 2 6i 2 1 2i 1 3i 4i 3 2 , ± 5i 1, 1 ± i ± 2, ± 2 1 2 ± 3 2 i 2 2i a 2 b 2 2abi 20 10i 4 Imaginary axis u = 3 + i u = 3 i 2u = 6 2i 2 2 4 4 6 Real axis 1 2 3 4 1 2 3 4 Real axis Imaginary axis u = 2 + i v = 2 + i u + v = 4 + 2i 4 2 2 2 4 Real axis u = 6 v = 2i u v = 6 + 2i Imaginary axis 4 2i 6 2i Real axis Imaginary axis 2 4 2 4 6 8 v = 5 i u v = 2i u = 5 + i 2 4 4 6 8 10 Imaginary axis 2 4 6 8 10 u = 2 + 6i v = 3 3i u + v = 5 + 3i Real axis 2i 5 3i Real axis axis 1 2 3 4 5 1 2 3 4 5 z = 1 + 5i Imaginary Real axis axis 1 2 3 4 5 1 2 3 4 5 z = 5 + 5i Imaginary 2 2 4 axis Real axis z = 6 2i 2 4 6 Imaginary x 3 x 6 4 6 Answer Key A1

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Page 1: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

Chapter 8

Section 8.1 (page 398)1. 3. 5. 1 7. 9.

11. 13.

15.

17. 19.

21. 23.

25.

27. 29. 8 31.33. 35. 37. 2, 3 39.

41. 43. 45.

47. 49.

51. 53. 55. Proof

57. (a) (b)

(c) , where is an integer.

59. False. See the Remark, page 392. 61. Proof

Section 8.2 (page 403)1. 3.

5. 4

7. 9. 11. 513.

15. 17. 19.

21. 23. 25.

27.29. (a) (b) (c) (d)

31. 33. Not invertible

35. 37.

39. Proof 41. (a) and (b) Proofs43. (a)

(b)

(c) where is an integerk�1�i�n � �1

�i�1

i

n � 4k n � 4k � 1n � 4k � 2n � 4k � 3

,

�1�i�2000 � 1, �1�i�2010 � �1�1�i�5 � �i�1�i�4 � 1,

�1�i�3 � i,�1�i�2 � �1,�1�i�1 � 1�i � �i,

�53 �

103 iA�1 � �

�i00

0�i

0

00

� i�A�1 � �

13� i

�2 � i�3i

6�325 �

425i2

5 �15i2 � 11i3 � 4i

1, 2, �3 ± �2 i

�23, 1 ± �3i1

10 �9

10i�12 �

52i

13

10�

9

10i

7

11�

6�2

11 i1 � 2i

zw � �3 � i � �10wz � �5�2 � �10wz � �3 � i � �10

�65�5

1

2

3

−1

−2

1 2 3 4 5

z and z = 4Realaxis

Imaginaryaxis

4

8

Realaxis

4 8−4

−4

−8

−8

z = 8i

z = −8i

Imaginaryaxis

2 4 6

2

4

z = 6 − 3i

z = 6 + 3i

Realaxis

Imaginaryaxis

−2

−4

8i6 � 3i

kin � �1i�1�i

n � 4kn � 4k � 1n � 4k � 2n � 4k � 3

i5 � ii4 � 1i3 � �ii2 � �1

i2010 � �1i1 � i

��525i

�15 � 10i15 � 30i��5 � 3i

��2 � 2i4 � 4i

2i6��2 � 2i

4 � 4i 2

�6i�� 2�1 � 2i

1 � 3i�4i��

32, ±5i1, 1 ± i

±2, ±2�12 ± 3

2i�2 � 2i�a2 � b2� � 2abi20 � 10i

4

Imaginary axis

−u = −3 + i

u = 3 − i

2u = 6 − 2i−2

−2

−4

−4 6

Realaxis

1

2

3

4

1 2 3 4

Realaxis

Imaginaryaxis

u = 2 + iv = 2 + i

u + v = 4 + 2i

4

2

−22 4

Realaxisu = 6

v = −2i

u − v = 6 + 2i

Imaginaryaxis

4 � 2i6 � 2i

Realaxis

Imaginaryaxis

−2

−4

2

4

6

8v = 5 − i

u − v = 2i

u = 5 + i

24

4

68

10

Imaginaryaxis

−2−4

6 8 10

u = 2 + 6i

v = 3 − 3i

u + v = 5 + 3i

Realaxis

2i5 � 3i

Realaxis

axis

1

2

3

4

5

1 2 3 4 5

z = 1 + 5i

Imaginary

Realaxis

axis

1

2

3

4

5

−1−2−3−4−5

z = −5 + 5i

Imaginary

2

−2

−4

axis

Realaxis

z = 6 − 2i

2 4 6

Imaginary

x � 3x � 6�4��6

Answer Key A1

9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A1

Page 2: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

A2 Answer Key

Section 8.3 (page 411)

1. 3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

15�2

8�

15�2

8i

123 θ = 4

r = 3.75

1 2 3−2−2−3

Realaxis

Imaginaryaxis

π

3

4�

3�3

4i

1

2

θ = 53

Realaxis

Imaginaryaxis

−2

−2 1 2

32r =

π

2i

1

θ = π2

−1−1

r = 2Realaxis

Imaginaryaxis

1

�5 cos��2.0344� � i sin��2.0344��

1 43

1

34

z = −1 − 2i

−1−3

−4

Realaxis

Imaginaryaxis

2�3�cos �

6� i sin

6

1 2 4

12

4z = 3 + 3 i

−1−4−2

−4

Realaxis

Imaginaryaxis

7�cos 0 � i sin 0�

3

6

z = 7

Realaxis

3 6−3−3

−6

Imaginaryaxis

6�cos �

2� i sin

2

2

4

Realaxis

Imaginaryaxis

z = 6i

−4

−4

−2 2 4

4�cos��2�

3 � i sin��2�

3 �

123

Realaxis

1 2 3−2−2−3

−3

z = −2 − 2 3 i

Imaginaryaxis

�8�cos��3�

4 � i sin��3�

4 �

1

2

1 2

Realaxis

z = −2 − 2i

Imaginaryaxis

6�cos � � i sin ���8�cos���

4 � i sin���

4 �

9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A2

Page 3: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

23.

25.

7

27. 29.

31.

33. 35. 37.

39. 41. 43. 256

45.

Square roots:

47.

Square roots:

49.

Square roots:

51.

Square roots:

53. (a)

(b)

(c)

55. (a)

(b)

(c)

�3 � i�1 � �3 i��3 � i1 � �3 i

1r = 2

θθ

θ

θ

==

=

=

5

11

4

36

6

3

12

4

3

−1−1

1

Realaxis

Imaginaryaxis

π π

π

π

2�cos 11�

6� i sin

11�

6

2�cos 4�

3� i sin

4�

3

2�cos 5�

6� i sin

5�

6

2�cos �

3� i sin

3 �2�3 � 2i2�3 � 2i

θ

=

= 7

6

6

1

2

−2−2

r = 4Realaxis

Imaginaryaxis

π

π

4�cos 7�

6� i sin

7�

6 4�cos

6� i sin

6

�2�cos7�

6� i sin

7�

6 � ��62

��22

i

�2�cos�

6� i sin

6 ��62

��22

i

1 � �3i � 2�cos�

3� i sin

3

81�4�cos15�

8� i sin

15�

8 � 1.554 � 0.644i

81�4�cos7�

8� i sin

7�

8 � �1.554 � 0.644i

2�2i � 2�2�cos7�

4� i sin

7�

4

�3�cos7�

4� i sin

7�

4 ��62

��62

i

�3�cos3�

4� i sin

3�

4 � ��62

��62

i

�3i � 3�cos3�

2� i sin

3�

2

�2�cos5�

4� i sin

5�

4 � �1 � i

�2�cos�

4� i sin

4 � 1 � i

2i � 2�cos�

2� i sin

2 �

81

2�

81�3

2 i�8

�32i�44�cos �

6� i sin

6

12 �cos��

6 � i sin���

6 �

0.25�cos 0 � i sin 0�12�cos �

2� i sin

2

24 θ = 0

r = 7Realaxis

2 4 6−2−4−4−6

Imaginaryaxis

�4i

1 2 3

123

θ = 32

r = 4

−2−2−1−3

−3

Realaxis

Imaginaryaxis

π

Answer Key A3

9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A3

Page 4: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

A4 Answer Key

57. (a)

(b)

(c)

59. (a)

(b)

(c)

61. (a)

(b)

(c) 2

63. (a)

(b)

(c) 1

65.

67.

1

1

θ

θ

θ

=

=

=

π

π

3

3

2

3

1

−1

Realaxis

Imaginaryaxis

r = 1

cos 5�

3� i sin

5�

3

cos � � i sin �

cos �

3� i sin

3

θ

θ

θ

θ

=

=

=

=

13

5

9

8

8

8

8

4

2

3

1

Realaxis

Imaginaryaxis

r = 42

2−2−2

π

π

π

π

4�cos13�

8� i sin

13�

8 4�cos

9�

8� i sin

9�

8 4�cos

5�

8� i sin

5�

8 4�cos

8� i sin

8 �i�1i

1

1

θ

θ

θ

θ

= 0

=

=

= π 32

2

1

4

2

3

r = 1

−1

Realaxis

Imaginaryaxis

π

π

cos 3�

2� i sin

3�

2

cos � � i sin �

cos �

2� i sin

2

cos 0 � i sin 0

�1 � �3 i�1 � �3 i

θ

θ

= 0

=

=

4

2

3

3

1

3

2

r = 2

−1−1 1

Realaxis

Imaginaryaxis

π

π

2�cos 4�

3� i sin

4�

3

2�cos 2�

3� i sin

2�

3 2�cos 0 � i sin 0�3.83 � 3.21i�4.70 � 1.71i0.868 � 4.92i

2

4

6 θ

θθ

1

32

=

==

4

1610

9

99

2 4 6−2

−4

−6

r = 5Realaxis

Imaginaryaxis

π

ππ

5�cos 16�

9� i sin

16�

9

5�cos 10�

9� i sin

10�

9

5�cos 4�

9� i sin

4�

9

5�2

2�

5�2

2 i

�5�2

2�

5�2

2 i

2

4

θ

1

2

=

=

3

7

4

4

2 4 6−2−6

−4

−6

r = 5Realaxis

Imaginaryaxis

π

π

5�cos 7�

4� i sin

7�

4

5�cos 3�

4� i sin

3�

4

9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A4

Page 5: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

69.

71.

73. 75. Proof77. (a) (b)79. (a)

(b) Counterclockwise rotation of clockwise rotation of

81. True

Section 8.4 (page 418)1. 3.5. 7.9. Linearly dependent 11. Linearly independent

13. is not a basis for 15. is a basis for 17. (a)

(b)

19. (a)

(b)

21.23. (a)

(b)The expressions are equal.

25. (a)(b)The expressions are equal.

27. 29. 31. 33. 35.37. 39. 41. Not a complex inner product43. A complex inner product 45. 47.49. 51. (a) and (b) Proofs53. and can be any complex numbers.55.

57.

59–63. Proofs 65.

67. 69.

71. where

73. 75. (a), (b), and (d) are subspaces.

77. False.

Section 8.5 (page 428)

1.

3.

5.

7. is not unitary because it is singular.9. is not unitary because it is not a square matrix.

11.

So, is not unitary.

13.

So, is unitary.

15.

So, is unitary.A

AA* � �10

01� � I2

A

AA* � I2 � �10

01�

A

AA* � � 4�4i

4i4� � I2

AA

�1 � i

01i

2 � i10

�2i

1 � i�i

2�4i

�i2 � i

�10�

�0

5 � i��2i

5 � i64

�2i43�

��ii

2�3i�

u � v � u1v1 � u2v2 � . . . � unvn

�0

0�t � R�ker�T� � ��0, t, �ti�,

�2 � i

1 � 2i

�1 � 5i�, �2

1��1 � i2i�, �0

0�

T�u1

u2� � �2 � i

10

�i��u1

u2�

� �v, 2u� � �v, u � u�

�u, v� � �u, v� � �v, u� � �v, u�

� k�u, v� � �u, w� �u, kv � w� � �u, kv� � �u, w�

z2z3 � 0, z1

�4 � i23 � 3i�6

�2�15�3�17�73�42�2

4 � i4 � i

1 � 4i1 � 4i

�4 � 3i

��12 �

12i��0, 0, 1 � i�

�2 � i��1, 1, 0� �

��i, 2 � i, �1� � ��2 � 2i��1, 0, 0� �

�1 � 3i��i, i, 0� � i�i, i, i� ��i, 2 � i, �1� � ��2 � 2i��i, 0, 0� �

�1, 2, 0� � ��1, 0, 0� � 2�1, 1, 0� � 0�0, 0, 1 � i��1, 2, 0� � i�i, 0, 0� � 2i�i, i, 0� � 0�i, i, i�

C 3.SC 2.S

��9 � 3i, 2 � 14i���5 � i, �4���8 � 4i, 6 � 12i��3i, 9 � 3i�

2�

2;

−1−2

−2

−3

−3 1 2 3

Realaxis

Imaginaryaxis

z /i = 1 − 3 i

z = 3 + i

iz =−1 + 3 i

cos�2�� � i sin�2��r2

32 �

12i

θ

θ

θ=

=

=

11

76

2

6

3

1

2

−2−2

2

2

r = 4Realaxis

Imaginaryaxis

π

π

π

4�cos 11�

6� i sin

11�

6

4�cos 7�

6� i sin

7�

6

4�cos �

2� i sin

2

1 2

2 θθ

θ

θθ

1

2

3

5

4

=

=

= π

==

3

9

7

5

5

5

5

−2

−2 −1

r = 3Realaxis

Imaginaryaxis

π

π

π

π

3�cos 9�

5� i sin

9�

5

3�cos 7�

5� i sin

7�

5 3�cos � � i sin ��

3�cos 3�

5� i sin

3�

5

3�cos �

5� i sin

5

Answer Key A5

9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A5

Page 6: 9781133110873 08 ANS · A2 Answer Key Section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 15 2 8 15 2 8 i 1 2 3 θ= 4 r = 3.75 −2 123 −2 −3 Real axis Imaginary axis

A6 Answer Key

17. (a)

(b)

19. (a)

(b)

21. is Hermitian because 23. is not Hermitian because the entry on the main

diagonal, is not a real number. So,25. is not Hermitian because the matrix is not square.27. 29. 31.

33. 35.

37.

39.

41.

43.

Therefore and is unitary.

45. 47–53. Proofs

55. (a)

(b)

(c)

(d)

Review Exercises (page 430)1. 2 3. 20 5. 7.

9. 11.

13. 15.

17. 19. 21.

23. 25. 27. 29. is singular.

31.

33.

35.

37. 39. 41.

43.

45. 47.

49.

51.

53.

55. 57.59. 61. 4 63. 65.

67.

69.

71. Unitary 73. Not unitary 75. Not Hermitian77.

79–87. Proofs 89. True2 � 5, v2 � �5, 2 � i�1 � �1, v1 � ��2 � i, 5�

A* � �5

2 � i3 � 2i

2 � 2i3 � 2i

�i

�3i2 � i

�1 � 2i�A* � ��1 � 4i

3 � i3 � i2 � i�

13 � i�14�38��5, 1 � 2i���3 � 28i, 14 � 8i�

cos 3�

2� i sin

3�

2� �i

cos 5�

6� i sin

5�

6� �

�3

2�

1

2i

cos �

6� i sin

6�

�3

2�

1

2i

5�cos 4�

3� i sin

4�

3 � �5

2�

5�3

2i

5�cos �

3� i sin

3 �5

2�

5�3

2i

8�2 �cos��5�

6 � i sin��5�

6 �

4�cos � � i sin ��3

2�cos���

6 � i sin���

6 �

12�cos 2�

3� i sin

2�

3

�7i�2�2 � 2�2 i5�3

2�

5

2i

�65 cos��0.519� � i sin��0.519��

r � 2, � ��

6, z � 2�cos

6� i sin

6 4�2�cos

4� i sin

4 A5

6 �56i3

5 �45i8 � 4i

�8�1 � 2i�15 � 18i24 � 33i

3 � 15i15 � 27i�

�1 � 5i��2 � 2i�4

�2�2 � 4i�

� 53 � 2i

2 � i5 � 2i�1, �2, ±�5 i

±�2 i�2,43 �

23i

� i �1

12

12

�2�� 0

2 �i2

2 �i2

1�A �

A � A*

2� i

A � A*

2i

�21

13� � i� 0

�110�

A �A � A

2� i

A � A

2i

A �1�2

��1

�i

�1

i�InA* � A�1

AA* � �100...

010...

001...

. . .. . .. . .� � In

A* � �100...

010...

001...

. . .. . .. . .�

A � �100...

010...

001

...

. . .. . .. . .�

P �1�6 �

�600

0�1 � i

2

02

1 � i�

P �12 �

�2�i

i

0�2�2

�2i

�i �P �

1�2�

1

�i

1

i�v3 � �6 � 4i, 3i, 2�v2 � �2 � 2i, �1, 0�v2 � �1, i�v1 � �1, 0, 0�v1 � �1, �i�

3 � 2 � i2 � i2 � 42 � �11 � 11 � 11 � 1

AA � A*.

a22,AA � A*.A

A�1 �1

2�2 ��3 � i

1 � �3i�3 � i

1 � �3i��r1� � 1, �r2� � 1, r1 � r2 � 0

r2 �1

2�2��3 � i, 1 � �3 i�

r1 �1

2�2��3 � i, 1 � �3 i�,

A�1 � ��45

�35i

35

�45i�

�r1� � 1, �r2� � 1, r1 � r2 � 0

r1 � ��45, 35i�, r2 � �3

5, 45i�

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