Exercises in Algebra

, by Kostrikin; Alexandra I.

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ISBN: 9782884490290 | 2884490299
Cover: Hardcover
Copyright: 2/9/1996

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Supplemental Materials

Preface to the Russian edition

(2)

Foreword to the English edition

PART ONE: FOUNDATIONS OF ALGEBRA

(98)

1 Sets and maps

(12)

1. Operations on subsets. Calculation of the number of elements

(1)

2. Calculation of the number of maps and of the number of subsets. Binomial coefficients

(2)

3. Permutations

(5)

4. Recurrence relations. Induction

(2)

5. Summations

(2)

2 Arithmetic spaces and linear equations

(16)

6. Arithmetic spaces

(4)

7. Rank of a matrix

(3)

8. Systems of linear equations

(9)

3 Determinants

(18)

9. Determinants of sizes two and three

(1)

10. Expanding a determinant. Inductive definition

(1)

11. Basic properties of a determinant

(2)

12. Expanding a determinant according to the elements of a row or a column

(2)

13. Calculating a determinant with the help of elementary operations

(3)

14. Calculating special determinants

(2)

15. Determinant of a product of matrices

(1)

16. Additional exercises

(6)

4 Matrices

(12)

17. Operations on matrices

(4)

18. Matrix equations. Inverse matrix

(4)

19. Special matrices

(4)

5 Complex numbers

(16)

20. Complex numbers in algebraic form

(2)

21. Complex numbers in trigonometric form

(2)

22. Roots of complex numbers. Cyclotomic polybomials

(4)

23. Calculation of sums and products with the help of complex numbers

(3)

24. Complex numbers and geometry of a plane

(5)

6 Polynomials

(22)

25. Division with a remainder. Euclidean algorithm

(2)

26. Simple and multiple roots over fields of characteristic zero

(2)

27. Prime decomposition over R and C

(2)

28. Polynomials over the field of rationals and over finite fields

(3)

29. Rational fractions

(1)

30. Interpolation

(2)

31. Symmetric polynomials. Vieta formulas

(5)

32. Resultant and discriminant

(2)

33. Isolation of roots

(3)

PART TWO: LINEAR ALGEBRA AND GEOMETRY

(106)

7 Vector spaces

101

(14)

34. Concept of a vector space. Bases

101

(3)

35. Subspaces

104

(6)

36. Linear functions and mappings

110

(5)

8 Bilinear and quadratic functions

115

(16)

37. General bilinear and sesquilinear functions

115

(9)

38. Symmetric bilinear, Hermitian and quadratic functions

124

(7)

9 Linear operators

131

(20)

39. Definition of a linear operator. The image, the kernel, the matrix of a linear operator

131

(4)

40. Eigenvectors, invariant subspaces, root subspaces

135

(5)

41. Jordan canoncial form and its applications. Minimal polynomial

140

(6)

42. Normed vector spaces and algebras. Non-negative matrices

146

(5)

10 Metric vector spaces

151

(20)

43. Geometry of metric spaces

151

(7)

44. Adjoint and normal operators

158

(5)

45. Selfadjoint operators. Reduction of quadratic functions to principal axes

163

(3)

46. Orthogonal and unitary operators. Polar factorization

166

(5)

11 Tensors

171

(6)

47. Basic concepts

171

(3)

48. Symmetric and skew-symmetric tensors

174

(3)

12 Affine, Euclidean, and projective geometry

177

(28)

49. Affine spaces

177

(6)

50. Convex sets

183

(5)

51. Euclidean spaces

188

(6)

52. Hypersurfaces of second degree

194

(7)

53. Projective spaces

201

(4)

PART THREE: BASIC ALGEBRAIC STRUCTURES

205

(244)

13 Groups

207

(40)

54. Algebraic operations. Semigroups

207

(1)

55. Concept of a group. Isomorphisms of groups

208

(5)

56. Subgroups. Orders of elements of groups. Cosets

213

(6)

57. Action of a group on a set. Relation of conjugacy

219

(6)

58. Homomorphisms and normal subgroups. Factorgroups and centers

225

(5)

59. Sylow subgroups. Groups of small orders

230

(3)

60. Direct products and direct sums. Abelian groups

233

(6)

61. Generators and defining relations

239

(4)

62. Solvable groups

243

(4)

14 Rings

247

(42)

63. Rings and algebras

247

(7)

64. Ideals, homomorphisms, factor-rings

254

(10)

65. Special classes of algebras

264

(5)

66. Fields

269

(5)

67. Extensions of fields. Galois theory

274

(11)

68. Finite fields

285

(4)

15 Elements of representation theory

289

(24)

69. Representations of groups. Basic concepts

289

(5)

70. Representations of finite groups

294

(5)

71. Group algebras and their modules

299

(5)

72. Characters of representations

304

(5)

73. Initial information on representations of continuous groups

309

(4)

Answers and hints

313

(126)

Theoretical material

439

(10)

1. Affine and Euclidean geometry

439

(2)

2. Hypersurfaces of the second order

441

(2)

3. Projective space

443

(1)

4. Tensors

444

(1)

5. Elements of representation theory

445

(4)

List of definitions

449

(8)

List of symbols

457

(4)

Index

461

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Exercises in Algebra

Table of Contents

Supplemental Materials

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