9781133110873 08 ans · a2 answer key section 8.3 (page 411) 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21....
TRANSCRIPT
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
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
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
2θ
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
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
5π
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
1θ
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
6θ
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
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
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�
9781133110873_08_ANS.qxd 3/10/12 6:58 AM Page A6