266 Solutions to Problems from Linear Algebra 4th ed., Friedberg, Insel, Spence PDF

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266 Solutions to Problems from “Linear Algebra” 4th ed., Friedberg, Insel, Spence Daniel Callahan 3 2016 c Daniel Callahan. All rights reserved. ISBN-13: 978-1533013033 Email the author: [email protected] Previously published as “The Unauthorized Solutions Manual to “Linear Al- gebra” 4th ed. by...

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266 Solutions to Problems from Linear Algebra 4th ed., Friedberg, Insel, Spence Daniel Callahan

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Friedberg, Insel, and Spence Linear algebra, 4t h ed. SOLUT IONS REFERENCE Rohit kumar Maurya Set Linear Algebra and Set Fuzzy Linear Algebra Florent in Smarandache LINEAR ALGEBRA Paola Vázquez Rizo

266 Solutions to Problems from “Linear Algebra” 4th ed., Friedberg, Insel, Spence

Daniel Callahan

3

c

2016 Daniel Callahan. All rights reserved. ISBN-13: 978-1533013033 Email the author: [email protected] Previously published as “The Unauthorized Solutions Manual to “Linear Algebra” 4th ed. by Friedberg, Insel, Spence”.

Contents Chapter 1. Vector Spaces 1.1. Section 1.2, #9 1.2. Section 1.2, #12 1.3. Section 1.3, #3

15 15 16 16

1.4. 1.5.

Section 1.3, #18 Section 1.3, #19

17 17

1.6.

Section 1.3, #20

18

1.7. 1.8.

Section 1.3, #23 Section 1.3, #24

18 19

1.9.

Section 1.3, #30

20

1.10. Section 1.3, #31 1.11. Section 1.4, #11

21 23

1.12. Section 1.4, #13

24

1.13. Section 1.4, #14 1.14. Section 1.4, #15

24 25

1.15. Section 1.4, #16

26

1.16. Section 1.5, #9 1.17. Section 1.5, #12

26 27

1.18. Section 1.5, #13 1.19. Section 1.5, #14

27 28

1.20. Section 1.5, #16

29

1.21. Section 1.5, #20 1.22. Section 1.6, #11

30 30 5

6

CONTENTS

1.23. Section 1.6, #19 1.24. Section 1.6, #20

31 32

1.25. Section 1.6, #21

33

1.26. Section 1.6, #30 1.27. Section 1.6, #35

34 35

Chapter 2.

Linear Transformations and Matrices

37

2.1. Section 2.1, #11 2.2. Section 2.1, #12

37 37

2.3.

Section 2.1, #13

38

2.4. 2.5.

Section 2.1, #14 Section 2.1, #15

38 39

2.6. 2.7.

Section 2.1, #16 Section 2.1, #17

40 41

2.8. 2.9.

Section 2.1, #18 Section 2.1, #19

41 42

2.10. Section 2.1, #20

43

2.11. Section 2.1, #24 2.12. Section 2.1, #25

43 44

2.13. Section 2.1, #26

45

2.14. Section 2.1, #27 2.15. Section 2.1, #28

46 47

2.16. Section 2.1, #29

47

2.17. Section 2.1, #30 2.18. Section 2.1, #31(a,b)

47 48

2.19. Section 2.1, #32 2.20. Section 2.1, #35

49 50

2.21. Section 2.1, #37

51

2.22. Section 2.1, #38 2.23. Section 2.1, #40(a,b)

51 52

2.24. Section 2.2, #8

52

CONTENTS

7

2.25. Section 2.2, #11 2.26. Section 2.2, #13

53 54

2.27. Section 2.2, #14

54

2.28. Section 2.2, #15 2.29. Section 2.2, #16

55 56

2.30. Section 2.3, #3

56

2.31. Section 2.3, #5 2.32. Section 2.3, #6

58 59

2.33. Section 2.3, #7

59

2.34. Section 2.3, #9 2.35. Section 2.3, #11

60 60

2.36. Section 2.3, #12

61

2.37. Section 2.3, #13 2.38. Section 2.3, #14

62 62

2.39. Section 2.3, #15 2.40. Section 2.3, #16(a)

63 64

2.41. Section 2.4, #4

65

2.42. Section 2.4, #5 2.43. Section 2.4, #6

65 66

2.44. Section 2.4, #7

66

2.45. Section 2.4, #8 2.46. Section 2.4, #9

67 68

2.47. Section 2.4, #10

68

2.48. Section 2.4, #12 2.49. Section 2.4, #13 2.50. Section 2.4, #15

69 70 70

2.51. Section 2.4, #16

71

2.52. Section 2.4, #17 2.53. Section 2.4, #20

72 72

2.54. Section 2.4, #21

73

8

CONTENTS

2.55. Section 2.4, #24 2.56. Section 2.4, #25

75 76

2.57. Section 2.5, #7

77

2.58. Section 2.5, #8 2.59. Section 2.5, #9

79 79

2.60. Section 2.5, #10

80

2.61. Section 2.5, #11 2.62. Section 2.5, #12

81 81

2.63. Section 2.5, #13

82

2.64. Section 2.5, #14

83

Chapter 3.

Elementary Matrix Operations & Systems of Linear Equations

85

3.1. 3.2.

Section 3.2, #3 Section 3.2, #6 (a,b,e only)

85 85

3.3.

Section 3.2, #8

87

3.4. 3.5.

Section 3.2, #14 Section 3.2, #16

87 88

3.6.

Section 3.2, #19

89

3.7. 3.8.

Section 3.2, #21 Section 3.2, #22

90 90

3.9.

Section 3.3, #2g

91

3.10. Section 3.3, #3g 3.11. Section 3.3, #6

92 93

3.12. Section 3.3, #9 3.13. Section 3.3, #10

93 94

3.14. Section 3.4, #3

94

3.15. Section 3.4, #10 3.16. Section 3.4, #14

95 96

3.17. Section 3.4, #15

96

CONTENTS

Chapter 4. Determinants 4.1. Section 4.1, #9

9

97 97

4.2.

Section 4.2, #23

97

4.3. 4.4.

Section 4.2, #24 Section 4.2, #25

98 99

4.5.

Section 4.2, #29

100

4.6. 4.7.

Section 4.2, #30 Section 4.3, #9

100 101

4.8.

Section 4.3, #10

101

4.9. Section 4.3, #11 4.10. Section 4.3, #12

101 102

4.11. Section 4.3, #13(a)

102

4.12. Section 4.3, #15 4.13. Section 4.3, #16

103 103

4.14. Section 4.3, #17 4.15. Section 4.3, #20

103 104

4.16. Section 4.3, #21

105

Chapter 5.

Diagonalization

107

5.1.

Section 5.1, #2(b,d)

107

5.2. 5.3.

Section 5.1, #3(b,d) Section 5.1, #4(b)

107 109

5.4.

Section 5.1, #5

110

5.5. 5.6.

Section 5.1, #6 Section 5.1, #7

111 112

5.7. 5.8.

Section 5.1, #8(a,b) Section 5.1, #9

114 115

5.9.

Section 5.1, #10

115

5.10. Section 5.1, #11 5.11. Section 5.1, #12

116 117

5.12. Section 5.1, #13

118

10

CONTENTS

5.13. Section 5.1, #14 5.14. Section 5.1, #15

119 120

5.15. Section 5.1, #19

121

5.16. Section 5.1, #20 5.17. Section 5.1, #21(a)

121 122

5.18. Section 5.1, #22

123

5.19. Section 5.1, #24 5.20. Section 5.1, #25

125 126

5.21. Section 5.2, #2(b,d)

126

5.22. Section 5.2, #3(a,b,c) 5.23. Section 5.2, #4

128 131

5.24. Section 5.2, #5

131

5.25. Section 5.2, #7 5.26. Section 5.2, #9

132 133

5.27. Section 5.2, #10 5.28. Section 5.2, #12

133 134

5.29. Section 5.2, #22

135

5.30. Section 5.4, #3 5.31. Section 5.4, #4

135 136

5.32. Section 5.4, #5

136

5.33. Section 5.4, #6(b,d) 5.34. Section 5.4, #7

137 137

5.35. Section 5.4, #11

138

5.36. Section 5.4, #12 5.37. Section 5.4, #23 5.38. Section 5.4, #24

138 139 139

5.39. Section 5.4, #27

140

5.40. Section 5.4, #28 5.41. Section 5.4, #29

141 142

5.42. Section 5.4, #30

142

CONTENTS

5.43. Section 5.4, #34 5.44. Section 5.4, #35 Chapter 6.

Inner Product Spaces

11

143 143 145

6.1.

Section 6.1, #5

145

6.2. 6.3.

Section 6.1, #6 Section 6.1, #7

147 148

6.4. 6.5.

Section 6.1, #8(c) Section 6.1, #10

149 149

6.6.

Section 6.1, #12

150

6.7. 6.8.

Section 6.1, #13 Section 6.1, #17

150 151

6.9. Section 6.1 #19 6.10. Section 6.1, #22(a)

152 153

6.11. Section 6.1, #26 6.12. Section 6.1, #28

155 156

6.13. Section 6.1, #29

158

6.14. Section 6.2, #2(a) 6.15. Section 6.2, #3

159 159

6.16. Section 6.2, #6

160

6.17. Section 6.2, #7 6.18. Section 6.2, #13

160 161

6.19. Section 6.2, #15(a)

162

6.20. Section 6.2, #19(c) 6.21. Section 6.2, #20(c)

162 163

6.22. Section 6.2, #21 6.23. Section 6.2, #22

163 164

6.24. Section 6.3, #4

165

6.25. Section 6.3, #6 6.26. Section 6.3, #7

166 166

6.27. Section 6.3, #8

166

12

CONTENTS

6.28. Section 6.3, #11 6.29. Section 6.3, #13

167 168

6.30. Section 6.3, #18

169

6.31. Section 6.3, #22(c) 6.32. Section 6.4, #2(b,d)

169 169

6.33. Section 6.4, #4

170

6.34. Section 6.4, #5 6.35. Section 6.4, #6

170 171

6.36. Section 6.4, #7

173

6.37. Section 6.4, #8 6.38. Section 6.4, #11

174 175

6.39. Section 6.4, #12

176

6.40. Section 6.4, #17(a) 6.41. Section 6.4, #19

177 178

6.42. Section 6.5, #2 6.43. Section 6.5, #3

179 179

6.44. Section 6.5, #6

179

6.45. Section 6.5, #7 6.46. Section 6.5, #17

180 180

6.47. Section 6.5, #18

181

6.48. Section 6.5, #31 6.49. Section 6.6, #4

181 183

6.50. Section 6.6, #6

184

Chapter 7.

Canonical Forms

185

7.1. 7.2.

Section 7.1, #4 Section 7.1, #5

185 186

7.3.

Section 7.1, #8

187

7.4. 7.5.

Section 7.1, #12 Section 7.2, #11

187 188

7.6.

Section 7.2, #12

188

CONTENTS

7.7. 7.8.

Section 7.2, #20 Section 7.3, #6

13

189 190

CHAPTER 1

Vector Spaces 1.1. Section 1.2, #9 Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). P ROOF. Let V be a vector space. Corollary 1: “The vector 0 described in (VS 3) is unique”. Suppose 0′ is also a zero vector. If a, b ∈ V such that a + b = 0, then 0 = a + b = 0′ , and so 0 = 0′ . Corollary 2: “The vector y described in (VS 4) is unique.” Suppose that there exists x, y, z ∈ V such that y + x = 0 and z + x = 0. Then we have that y + x = z + x, and by Theorem 1.1, y = z. Theorem 1.2(c): “a0 = 0 for each a ∈ F.” By (VS 8), (VS 3), and (VS 1), it follows that a0 + a0 = a(0 + 0) = a0 = a0 + 0 By Theorem 1.1, a0 = 0.



15

16

1. VECTOR SPACES

1.2. Section 1.2, #12 A real-valued function f defined on the real line is called an even function if f (−t) = f (t) for each real number t. Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space.

P ROOF. (VS 1), (VS 2), (VS 5), (VS 6), (VS 7), and (VS 8) are obvious. Let z(t) be the zero function. Then z(t) = 0 = z(−t), and so z is an even function and (VS 3) is satisfied. Define g(t) = g(−t) = − f (t). Then g is an even, real function and (VS 4) is satisfied.



1.3. Section 1.3, #3 Prove that (aA+bB)t = aAt +bBt for any A, B ∈ Mn×n (F) and any a, b ∈ F. P ROOF. Let A, B ∈ Mn×n (F) and a, b ∈ F be chosen arbitrarily. Consider the matrix C = aA + bB. Then each entry of C can be written (C)i j where 1 ≤ i ≤ n, 1 ≤ j ≤ n, and (C)i j = aAi j + bBi j . It follows that (Ct )i j = aA ji + bB ji = a(At )i j + b(Bt )i j Hence, (aA + bB)t = aAt + bBt .



1.5. SECTION 1.3, #19

17

1.4. Section 1.3, #18 Prove that a subset W of a vector space V is a subspace of V iff 0 ∈ W and ax + y ∈ W whenever a ∈ F and x, y ∈ W . P ROOF. Let W be a subset of a vector space V . Suppose W is also a subspace of V . Then if a ∈ F and x, y ∈ W where our choices of a, x, y are arbitrary, by Theorem 1.3(a), 0 ∈ W . Also, by Theorem 1.3(c), ax ∈ W ; furthermore, by Theorem 1.3(b), ax + y ∈ W . Now suppose that if a ∈ F and x, y ∈ W , then 0 ∈ W and ax + y ∈ W . We wish to show that W is a subspace. Since V is a vector space, 1 ∈ F, and it follows that x + y ∈ W . Also, notice that since 0 ∈ W , ax + 0 = ax ∈ W . Thus, 0 ∈ W , x + y ∈ W whenever x, y ∈ W , and ax ∈ W whenever x ∈ W and a ∈ F. By Theorem 1.3, W is a vector space of V .



1.5. Section 1.3, #19 S

Let W1 and W2 be subspaces of a vector space V . Prove that W1 W2 is a subspace of V if and only if W1 ⊆ W2 or W2 ⊆ W1 . S

P ROOF. “⇐” If W2 ⊆ W1 , then W1 W2 = W1 . Since W1 is a subspace S of V , W1 W2 is also a subspace of V . A similar result follows if W1 ⊆ W2 , mutatis mutandis. S

“⇒” Suppose that W1 W2 is a subspace of V and neither W1 * W2 nor W1 * T

W2 . Partition W1 , W2 such that G = W1 W2 , H = W2 \W1 , and K = W1 \W2 . Since W1 , W2 are subspaces of V , 0 ∈ G, so G is nonempty. Let x ∈ G, y ∈ K, and z ∈ H. Since W1 is a subspace, x − y ∈ W1 . Similarly, since W2 is a subspace, x − z ∈ W2 .

18

1. VECTOR SPACES

Suppose that y − z ∈ W1 . Then (x − y) + (y − z) = x − z; however, x − y ∈ W1 , y − z ∈ W1 , but x − z ∈ W2 . This is a contradiction, since W1 is a subspace of V. Now suppose that y − z ∈ W2 . Since W2 is a subspace, we also have z − y ∈ W2 . Then (x − z) + (z − y) = x − y; however, x − z ∈ W2 , z − y ∈ W2 , but x − y ∈ W1 . This is also a contradiction, since W2 is a subspace of V . Hence it is not the case that neither W1 * W2 nor W1 * W2 ; instead, we have that either W1 ⊆ W2 or W2 ⊆ W1 , as required.



1.6. Section 1.3, #20 Prove that if W is a subspace of a vector space V and w1 , w2 , ..., wn ∈ W , then a1 w1 + a2 w2 + ... + an wn ∈ W for any scalars a1 , a2 , ..., an . P ROOF. Suppose the above and that a1 , a2 , ..., an ∈ F. By Theorem 1.3(c), ai wi ∈ W for i = 1, ..., n. And by Theorem 1.3(b), a1 w1 + a2 w2 ∈ W . By n − 1 repeated applications of Theorem 1.3(b), we have that a1 w1 + a2 w2 + ... + an wn ∈ W .



1.7. Section 1.3, #23 Let W1 and W2 be subspaces of a vector space V . (a) Prove that W1 +W2 is a subspace of V that contains both W1 and W2 . (b) Prove that the subspace of V that contains both W1 and W2 must also contain W1 +W2 .

1.8. SECTION 1.3, #24

19

P ROOF. (a) Since W1 ,W2 are subspaces of V , 0 is a member of each set, and so 0 ∈ W1 +W2 . Suppose ax ∈ W1 and y ∈ W2 . Then ax = a(x + 0) ∈ W1 +W2 and y = 0 + y ∈ W1 + W2 ; hence, ax + y ∈ W1 + W2 . By 1.3, #18, W1 + W2 is a subspace of V . Also by the above, it is clear that W1 ⊆ W1 +W2 and W2 ⊆ W1 +W2 since a, x, y were arbitrary elements. (b) Let U ⊆ V be a subspace of V such that W1 ⊆ U and W2 ⊆ U. We wish to show that W1 +W2 ⊆ U. Suppose ax + y ∈ W1 + W2 where ax ∈ W1 and y ∈ W2 . It follows from the above that ax ∈ U and y ∈ U. Since U is a subspace, ax + y ∈ U, and so W1 +W2 ⊆ U.



For #24, see the definition of a direct sum on p.22. 1.8. Section 1.3, #24 Show that F n is the direct sum of the subspace W1 = {(a1 , a2 , ..., an ) ∈ F n : an = 0}

and W2 = {(a1 , a2 , ..., an ) ∈ F n : a1 = a2 = ... = an−1 = 0}

P ROOF. W1 is a subspace of F n since: 1) If a1 = a2 = ... = an−1 = 0, then 0 ∈ W1 . 2) If (a1 , a2 , ..., 0), (b1 , b2 , ..., 0) ∈ W1 , then ,(a1 + b1 , a2 + b2 , ..., 0) ∈ W1 . 3) If k ∈ F and (a1 , a2 , ..., 0) ∈ W1 , then (ka1 , ka2 , ..., 0) ∈ W1 .

20

1. VECTOR SPACES

W2 is also a subspace of F n since: 1) If an = 0, then 0 ∈ W2 . 2) If (0, 0, ..., b), (0, 0, ..., c) ∈ W2 , then (0, 0, ..., b + c) ∈ W2 . 3) If k ∈ F and (0, 0, ..., b) ∈ W2 ), then (0, 0, ..., kb) ∈ W2 . T

T

Suppose y ∈ W1 W2 . Then a1 = a2 = ... = an = 0, and so y = 0 (or W1 W2 = {0}). Suppose (c1 , c2 , ..., cn ) ∈ F n . Then (c1 , c2 , ..., cn ) = (c1 , c2 , ..., 0)+(0, 0, ..., cn ) where (c1 , c2 , ..., 0) ∈ W1 and (0, 0, ..., cn ) ∈ W2 . Hence, F n = W1 + W2 . By p.22, F n is the direct sum of W1 and W2 .



1.9. Section 1.3, #30 Let W1 and W2 be subspaces of a vector space V . Prove that V is the direct sum of W1 and W2 if and only if each vector in V can be uniquely written as x1 + x2 where x1 ∈ W1 and x2 ∈ W2 . P ROOF. Let W1 and W2 be subspaces of a vector space V . Suppose that there exists some y ∈ V such that y = x1 + x2 = t1 + t2 where x1 ,t1 ∈ W1 , x2 ,t2 ∈ W2 , x1 6= t1 , and x2 6= t2 ; that is, there exists a vector in V that cannot be written uniquely from elements in W1 and W2 . Since 0 ∈ V , we have that: 0 = y − y = (x1 + x2 ) − (t1 + t2 ) = (x1 − t1 ) + (x2 − t2 ) = k1 + k2 ∈ V where k1 ∈ W1 , k2 ∈ W2 . But since 0 = k1 + k2 , k1 = −k2 , and so k2 ∈ W1 , T

k1 ∈ W2 . It follows that W1 W2 = {0, k1 , k2 }, and V is not a direct sum of subspaces W1 and W2 . Now suppose that all vectors in V can be written uniquely from elements in W1 and W2 ; that is, W1 + W2 = V . Since V is a vector space and W1 , W2 are

1.10. SECTION 1.3, #31

21

subspaces of V , 0 ∈ W1 ,W2 and so 0 + 0 = 0 ∈ V ; it follows that the only sum of elements from W1 and W2 that equal 0 are 0 ∈ W1 ,W2 . It follows that if a ∈ W1 , then −a ∈ / W2 ; i.e., W1 and W2 share no other additive inverses. However, if a subspace contains a vector, then it must also contain the vector’s additive inverse; it follows that W1 and W2 share no other common vectors, T

T

i.e., W1 W2 = {0}. Thus we have that W1 W2 = {0} and W1 + W2 = V , which is the definition of a direct sum. Hence, the proof.



1.10. Section 1.3, #31 Let W be a subspace of a vector space V over a field F. For any v ∈ V the set {v +W } = {v + w : w ∈ W } is called the coset of W containing v. (We denote this coset by v +W rather than {v +W }.) (1) Prove that v +W is a subspace of V iff v ∈ W . (2) Prove that v1 +W = v2 +W iff v1 − v2 ∈ W . (3) Addition and scalar multiplication by scalars of F can be defined in the collection S = {v + W : v ∈ V } of all cosets of W as on p.23. Prove that these operations on are well-defined. (4) Prove that the set S = {v + W : v ∈ W } of all cosets of W is a vector space with the operations defined on p. 23. P ROOF. (1) Suppose that v + W is a subspace of V . Since W is a subspace, 0 ∈ W , and so v + 0 = v ∈ v +W . N...