INTRODUCTION TO HIGHER ALGEBRAPDF电子书下载

CHAPTER Ⅰ POLYNOMIALS AND THEIR MOST FUNDAMENTAL PROPERTIES 1

1．Polynomials in One Variable 1

2．Polynomials in More than One Variable 4

3．Geometric Interpretations 8

4．Homogeneous Coordinates 11

5．The Continuity of Polynomials 14

6．The Fundamental Theorem of Algebra 16

CHAPTER Ⅱ A FEW PROPERTIES OF DETERMINANTS 20

7．Some Definitions 20

8．Laplace＇s Development 24

9．The Multiplication Theorem 26

10．Bordered Determinants 28

11．Adjoint Determinants and their Minors 30

CHAPTER Ⅲ THE THEORY OF LINEAR DEPHNDENCE 34

12．Definitions and Preliminary Theorems 34

13．The Condition for Linear Dependence of Sets of Constants 36

14．The Linear Dependence of Polynomials 38

15．Geometric Illustrations 39

CHAPTER Ⅳ LINEAR EQUATIONS 43

16．Non-Homogeneous Linear Equations 43

17．Homogeneous Linear Equations 47

18．Fundamental Systems of Solutions of Homogeneous Linear Equations 49

CHAPTER Ⅴ SOME THEOREMS CONCERNING THE RANK OF A MATRIX 54

19．General Matrices 54

20．Symmetrical Matrices 56

CHAPTER Ⅵ LINEAR TRANSFORMATIONS AND THE COMBINATION OF MATRICES 60

21．Matrices as Complex Quantities 60

22．The Multiplication of Matrices 62

23．Linear Transformation 66

24．Collineation 68

25．Further Development of the Algebra of Matrices 74

26．Sets，Systems，and Groups 80

27．Isomorphism 83

CHAPTER Ⅶ INVARIANTS．FIRST PRINCIPLES AND ILLUSTRATIONS 88

28．Absolute Invariants；Geometric，Algebraic，and Arithmetical 88

29．Equivalence 92

30．The Rank of a System of Points or a System of Linear Forms as an Invariant 94

31．Relative Invariants and Covariants 95

32．Some Theorems Concerning Linear Forms 100

33．Cross-Ratio and Harmonic Division 102

34．Plane-Coordinates and Contragredient Variables 107

35．Line-Coordinates in Space 110

CHAPTER Ⅷ BILINEAR FORMS 114

36．The Algebraic Theory 114

37．A Geometric Application 116

CHAPTER Ⅸ GEOMETRIC INTRODUCTION TO THE STUDY OF QUADRATIC FORMS 118

38．Quadric Surfaces and their Tangent Lines and Planes 118

39．Conjugate Points and Polar Planes 121

40．Classification of Quadric Surfaces by Means of their Rank 123

41．Reduction of the Equation of a Quadric Surface to a Normal Form 124

CHAPTER Ⅹ QUADRATIC FORMS 127

42．The General Quadratic Form and its Polar 127

43．The Matrix and the Discriminant of a Quadratic Form 128

44．Vertices of Quadratic Forms 129

45．Reduction of a Quadratic Form to a Sum of Squares 131

46．A Normal Form，and the Equivalence of Quadratic Forms 134

47．Reducibility 136

48．Integral Rational Invariants of a Quadratic Form 137

49．A Second Method of Reducing a Quadratic Form to a Sum of Squares 139

CHAPTER Ⅺ REAL QUADRATIC FORMS 144

50．The Law of Inertia 144

51．Classification of Real Quadratic Forms 147

52．Definite and Indefinite Forms 150

CHAPTER Ⅻ THE SYSTEM OF A QUADRATIC FORM AND ONE OR MORE LINEAR FORMS 155

53．Relations of Planes and Lines to a Quadric Surface 155

54．The Adjoint Quadratic Form and Other Invariants 159

55．The Rank of the Adjoint Form 161

CHAPTER ⅩⅢ PAIRS OF QUADRATIC FORMS 163

56．Pairs of Conics 163

57．Invariants of a Pair of Quadratic Forms．Their λ-Equation 165

58．Reduction to Normal Form when the λ-Equation has no Multiple Roots 167

59．Reduction to Normal Form when ψ is Definite and Non-Singular 170

CHAPTER ⅩⅣ SOME PROPERTIES OF POLYNOMIALS IN GENERAL 174

60．Factors and Reducibility 174

61．The Irreducibility of the General Determinant and of the Symmetrical Determinant 176

62．Corresponding Homogeneous and Non-Homogeneous Polynomials 178

63．Division of Polynomials 180

64．A Special Transformation of a Polynomial 184

CHAPTER ⅩⅤ FACTORS AND COMMON FACTORS OF POLYNOMIALS IN ONE VARIABLE AND OF BINARY FORMS 187

65．Fundamental Theorems on the Factoring of Polynomials in One Variable and of Binary Forms 187

66．The Greatest Common Divisor of Positive Integers 188

67．The Greatest Common Divisor of Two Polynomials in One Variable 191

68．The Resultant of Two Polynomials in One Variable 195

69．The Greatest Common Divisor in Determinant Form 197

70．Common Roots of Equations．Elimination 198

71．The Cases a0=0 and b0=0 200

72．The Resultant of Two Binary Forms 201

CHAPTER ⅩⅥ FACTORS OF POLYNOMIALS IN TWO OR MORE VARIABLES 203

73．Factors Involving only One Variable of Polynomials in Two Variables 203

74．The Algorithm of the Greatest Common Divisor for Polynomials in Two Variables 206

75．Factors of Polynomials in Two Variables 208

76．Factors of Polynomials in Three or More Variables 212

CHAPTER ⅩⅦ GENERAL THEOREMS ON INTEGRAL RATIONAL INVARIANTS 218

77．The Invariance of the Factors of Invariants 218

78．A More General Method of Approach to the Subject of Relative Invariants 220

79．The Isobaric Character of Invariants and Covariants 222

80．Geometric Properties and the Principle of Homogeneity 226

81．Homogeneous Invariants 230

82．Resultants and Discriminants of Binary Forms 236

CHAPTER ⅩⅧ SYMMETRIC POLYNOMIALS 240

83．Fundamental Conceptions．∑ and S Functions 240

84．Elementary Symmetric Functions 242

85．The Weights and Degrees of Symmetric Polynomials 245

86．The Resultant and the Discriminant of Two Polynomials in One Variable 248

CHAPTER ⅩⅨ POLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES 252

87．Fundamental Conceptions．∑ and S Functions 252

88．Elementary Symmetric Functions of Pairs of Variables 253

89．Binary Symmetric Functions 255

90．Resultants and Discriminants of Binary Forms 257

CHAPTER ⅩⅩ ELEMENTARY DIVISORS AND THE EQUIVALENCE OF λ-MATRICES 262

91．λ-Matrices and their Elementary Transformations 262

92．Invariant Factors and Elementary Divisors 269

93．The Practical Determination of Invariant Factors and Elementary Divisors 272

94．A Second Definition of the Equivalence of λ-Matrices 274

95．Multiplication and Division of λ-Matrices 277

CHAPTER ⅩⅪ THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF BILINEAR FORMS AND OF COLLINEATIONS 279

96．The Equivalence of Pairs of Matrices 279

97．The Equivalence of Pairs of Bilinear Forms 283

98．The Equivalence of Collineations 284

99．Classification of Pairs of Bilinear Forms 287

100．Classification of Collineations 292

CHAPTER ⅩⅫ THE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF QUADRATIC FORMS 296

101．Two Theerems in the Theory of Matrices 296

102．Symmetric Matrices 299

103．The Equivalence of Pairs of Quadratic Forms 302

104．Classification of Pairs of Quadratic Forms 305

105．Pairs of Quadratic Equations，and Pencils of Forms or Equations 307

106．Conclusion 313

INDEX 317