《INTRODUCTION TO HIGHER ALGEBRA》

CHAPTER ⅠPOLYNOMIALS AND THEIR MOST FUNDAMENTAL PROPERTIES1

1．Polynomials in One Variable1

2．Polynomials in More than One Variable4

3．Geometric Interpretations8

4．Homogeneous Coordinates11

5．The Continuity of Polynomials14

6．The Fundamental Theorem of Algebra16

CHAPTER ⅡA FEW PROPERTIES OF DETERMINANTS20

7．Some Definitions20

8．Laplace＇s Development24

9．The Multiplication Theorem26

10．Bordered Determinants28

11．Adjoint Determinants and their Minors30

CHAPTER ⅢTHE THEORY OF LINEAR DEPHNDENCE34

12．Definitions and Preliminary Theorems34

13．The Condition for Linear Dependence of Sets of Constants36

14．The Linear Dependence of Polynomials38

15．Geometric Illustrations39

CHAPTER ⅣLINEAR EQUATIONS43

16．Non-Homogeneous Linear Equations43

17．Homogeneous Linear Equations47

18．Fundamental Systems of Solutions of Homogeneous Linear Equations49

CHAPTER ⅤSOME THEOREMS CONCERNING THE RANK OF A MATRIX54

19．General Matrices54

20．Symmetrical Matrices56

CHAPTER ⅥLINEAR TRANSFORMATIONS AND THE COMBINATION OF MATRICES60

21．Matrices as Complex Quantities60

22．The Multiplication of Matrices62

23．Linear Transformation66

24．Collineation68

25．Further Development of the Algebra of Matrices74

26．Sets，Systems，and Groups80

27．Isomorphism83

CHAPTER ⅦINVARIANTS．FIRST PRINCIPLES AND ILLUSTRATIONS88

28．Absolute Invariants；Geometric，Algebraic，and Arithmetical88

29．Equivalence92

30．The Rank of a System of Points or a System of Linear Forms as an Invariant94

31．Relative Invariants and Covariants95

32．Some Theorems Concerning Linear Forms100

33．Cross-Ratio and Harmonic Division102

34．Plane-Coordinates and Contragredient Variables107

35．Line-Coordinates in Space110

CHAPTER ⅧBILINEAR FORMS114

36．The Algebraic Theory114

37．A Geometric Application116

CHAPTER ⅨGEOMETRIC INTRODUCTION TO THE STUDY OF QUADRATIC FORMS118

38．Quadric Surfaces and their Tangent Lines and Planes118

39．Conjugate Points and Polar Planes121

40．Classification of Quadric Surfaces by Means of their Rank123

41．Reduction of the Equation of a Quadric Surface to a Normal Form124

CHAPTER ⅩQUADRATIC FORMS127

42．The General Quadratic Form and its Polar127

43．The Matrix and the Discriminant of a Quadratic Form128

44．Vertices of Quadratic Forms129

45．Reduction of a Quadratic Form to a Sum of Squares131

46．A Normal Form，and the Equivalence of Quadratic Forms134

47．Reducibility136

48．Integral Rational Invariants of a Quadratic Form137

49．A Second Method of Reducing a Quadratic Form to a Sum of Squares139

CHAPTER ⅪREAL QUADRATIC FORMS144

50．The Law of Inertia144

51．Classification of Real Quadratic Forms147

52．Definite and Indefinite Forms150

CHAPTER ⅫTHE SYSTEM OF A QUADRATIC FORM AND ONE OR MORE LINEAR FORMS155

53．Relations of Planes and Lines to a Quadric Surface155

54．The Adjoint Quadratic Form and Other Invariants159

55．The Rank of the Adjoint Form161

CHAPTER ⅩⅢPAIRS OF QUADRATIC FORMS163

56．Pairs of Conics163

57．Invariants of a Pair of Quadratic Forms．Their λ-Equation165

58．Reduction to Normal Form when the λ-Equation has no Multiple Roots167

59．Reduction to Normal Form when ψ is Definite and Non-Singular170

CHAPTER ⅩⅣSOME PROPERTIES OF POLYNOMIALS IN GENERAL174

60．Factors and Reducibility174

61．The Irreducibility of the General Determinant and of the Symmetrical Determinant176

62．Corresponding Homogeneous and Non-Homogeneous Polynomials178

63．Division of Polynomials180

64．A Special Transformation of a Polynomial184

CHAPTER ⅩⅤFACTORS AND COMMON FACTORS OF POLYNOMIALS IN ONE VARIABLE AND OF BINARY FORMS187

65．Fundamental Theorems on the Factoring of Polynomials in One Variable and of Binary Forms187

66．The Greatest Common Divisor of Positive Integers188

67．The Greatest Common Divisor of Two Polynomials in One Variable191

68．The Resultant of Two Polynomials in One Variable195

69．The Greatest Common Divisor in Determinant Form197

70．Common Roots of Equations．Elimination198

71．The Cases a0=0 and b0=0200

72．The Resultant of Two Binary Forms201

CHAPTER ⅩⅥFACTORS OF POLYNOMIALS IN TWO OR MORE VARIABLES203

73．Factors Involving only One Variable of Polynomials in Two Variables203

74．The Algorithm of the Greatest Common Divisor for Polynomials in Two Variables206

75．Factors of Polynomials in Two Variables208

76．Factors of Polynomials in Three or More Variables212

CHAPTER ⅩⅦGENERAL THEOREMS ON INTEGRAL RATIONAL INVARIANTS218

77．The Invariance of the Factors of Invariants218

78．A More General Method of Approach to the Subject of Relative Invariants220

79．The Isobaric Character of Invariants and Covariants222

80．Geometric Properties and the Principle of Homogeneity226

81．Homogeneous Invariants230

82．Resultants and Discriminants of Binary Forms236

CHAPTER ⅩⅧSYMMETRIC POLYNOMIALS240

83．Fundamental Conceptions．∑ and S Functions240

84．Elementary Symmetric Functions242

85．The Weights and Degrees of Symmetric Polynomials245

86．The Resultant and the Discriminant of Two Polynomials in One Variable248

CHAPTER ⅩⅨPOLYNOMIALS SYMMETRIC IN PAIRS OF VARIABLES252

87．Fundamental Conceptions．∑ and S Functions252

88．Elementary Symmetric Functions of Pairs of Variables253

89．Binary Symmetric Functions255

90．Resultants and Discriminants of Binary Forms257

CHAPTER ⅩⅩELEMENTARY DIVISORS AND THE EQUIVALENCE OF λ-MATRICES262

91．λ-Matrices and their Elementary Transformations262

92．Invariant Factors and Elementary Divisors269

93．The Practical Determination of Invariant Factors and Elementary Divisors272

94．A Second Definition of the Equivalence of λ-Matrices274

95．Multiplication and Division of λ-Matrices277

CHAPTER ⅩⅪTHE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF BILINEAR FORMS AND OF COLLINEATIONS279

96．The Equivalence of Pairs of Matrices279

97．The Equivalence of Pairs of Bilinear Forms283

98．The Equivalence of Collineations284

99．Classification of Pairs of Bilinear Forms287

100．Classification of Collineations292

CHAPTER ⅩⅫTHE EQUIVALENCE AND CLASSIFICATION OF PAIRS OF QUADRATIC FORMS296

101．Two Theerems in the Theory of Matrices296

102．Symmetric Matrices299

103．The Equivalence of Pairs of Quadratic Forms302

104．Classification of Pairs of Quadratic Forms305

105．Pairs of Quadratic Equations，and Pencils of Forms or Equations307

106．Conclusion313

INDEX317