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PREFACE 15

PART Ⅰ:SET THEORY 19

0．Logical reasoning 20

1．The concept of logical perfection 20

2．The real language of mathematics 22

3．Elementary logical operations 24

4．Axioms and theorems 25

5．Logical axioms and tautologies 26

6．Substitution in a relation 30

7．Quantifiers 31

8．Rules for quantifiers 32

9．The Hilbert operation.Criteria of formation 35

Exercises on 0 38

1．The relations of equality and membership 41

1．The relation of equality 41

2．The relation of membership 42

3．Subsets of a set 43

4．The empty set 45

5．Sets of one and two elements 46

6．The set of subsets of a given set 47

Exercises on 1 49

2．The notion of a function 50

1．Ordered pairs 50

2．The Cartesian product of two sets 51

3．Graphs and functions 53

4．Direct and inverse images 56

5．Restrictions and extensions of functions 57

6．Composition of mappings 58

7．Injective mappings 61

8．Surjective and bijective mappings 62

9．Functions of several variables 65

Exercises on 2 68

3．Unions and intersections 70

1．The union and intersection of two sets 70

2．The union of a family of sets 71

3．The intersection of a family of sets 72

Exercises on 3 75

4．Equivalence relations 77

1．Equivalence relations 77

2．Quotient of a set by an equivalence relation 79

3．Functions defined on a quotient set 82

Exercises on 4 86

5．Finite sets and integers 88

1．Equipotent sets 88

2．The cardinal of a set 89

3．Operations on cardinals 92

4．Finite sets and natural numbers 95

5．The set N of the natural numbers 96

6．Mathematical induction 98

7．Combinatorial analysis 99

8．The rational integers 102

9．The rational numbers 107

Exercises on 5 108

PART Ⅱ:GROUPS,RINGS,FIELDS 113

6．Laws of composition 114

1．Laws of composition;associativity and commutativity 114

2．Reflexible elements 117

7．Groups 120

1．Definition of a group.Examples 120

2．Direct product of groups 122

3．Subgroups of a group 124

4．The intersection of subgroups.Generators 127

5．Permutations and transpositions 129

6．Cosets 130

7．The number of permutations of n objects 133

8．Homomorphisms 134

9．The kernel and image of a homomorphism 136

10．Application to cyclic groups 138

11．Groups operating on a set 139

Exercises on 7 142

8．Rings and fields 148

1．Definition of a ring.Examples 148

2．Integral domains and fields 151

3．The ring of integers modulo p 153

4．The binomial theorem 154

5．Expansion of a product of sums 157

6．Ring homomorphisms 158

Exeroises on 8 160

9．Complex numbers 168

1．Square roots 168

2．Preliminaries 168

3．The ring K[？] 169

4．Units in a quadratic extension 172

5．The case of a field 174

6．Geometrical representation of complex numbers 175

7．Multiplication formulae for trigonometric functions 178

Exercises on 9 181

PART Ⅲ:MODULES OVER A RING 187

10．Modules and vector spaces 188

1．Definition of a module over a ring 188

2．Examples of modules 189

3．Submodules;vector subspaces 191

4．Right modules and left modules 192

11．Linear relations in a module 194

1．Linear combinations 194

2．Finitely generated modules 196

3．Linear relations 196

4．Free modules.Bases 198

5．Infinite linear combinations 201

Exercises on 10 and 11 203

12．Linear mappings.Matrices 208

1．Homomorphisms 208

2．Homomorphisms of a finitely generated free module into an arbitrary module 210

3．Homomorphisms and matrices 212

4．Examples of homomorphisms and matrices 215

13．Addition of homomorphisms and matrices 220

1．The additive group Hom(L,M) 220

2．Addition of matrices 221

1．The ring of endomorphisms of a module 223

14．Products of matrices 223

2．The product of two matrices 224

3．Rings of matrices 226

4．Matrix notation for homomorphisms 228

Exercises on 12,13 and 14 230

15．Invertible matrices and change of basis 235

1．The group of automorphisms of a module 235

2．The groups GL(n,K) 235

3．Examples:the groups GL(1,K)and GL(2,K) 236

4．Change of basis.Transition matrices 238

5．Effect of change of bases on the matrix of a homomorphism 241

Exercises on 15 244

16．The transpose of a linear mapping 249

1．The dual of a module 249

2．The dual of a finitely generated free module 250

3．The bidual of a module 252

4．The transpose of a homomorphism 254

5．The transpose of a matrix 255

Exercises on 16 259

17．Sums of submodules 261

1．The sum of two submodules 261

2．Direct product of modules 262

3．Direct sum of submodules 263

4．Direct sums and projections 265

Exercises on 17 268

PART Ⅳ:FINITE-DIMENSIONAL VECTOR SPACES 271

1．Homomorphisms whose kernel and image are finitely generated 272

18．Finiteness theorems 272

2．Finitely generated modules over a Noetherian ring 273

3．Submodules of a free module over a principal ideal domain 274

4．Applications to systems of linear equations 275

5．Other characterizations of Noetherian rings 277

Exercises on 18 279

19．Dimension 282

1．Existence of bases 282

2．Definition of a vector subspace by means of linear equations 284

3．Conditions for consistency of a system of linear equations 285

4．Existence of linear relations 287

5．Dimension 289

6．Characterizations of bases and dimension 291

7．Dimensions of the kernel and image of a homomorphism 292

8．Rank of a homomorphism;rank of a family of vectors;rank of a matrix 294

9．Computation of the rank of a matrix 295

10．Calculation of the dimension of a vector subspace from its equations 298

Exercises on 19 300

20．Linear equations 306

1．Notation and terminology 306

2．The rank of a system of linear equations.Conditions for the existence of solutions 307

3．The associated homogeneous system 308

4．Cramer systems 308

5．Systems of independent equations:reduction to a Cramer system 310

Exercises on 20 313

PART Ⅴ:DETERMINANTS 317

1．Definition of multilinear mappings 318

21．Multilinear functions 318

2．The tensor product of multilinear mappings 321

3．Some algebraic identities 323

4．The case of finitely generated free modules 326

5．The effect of a change of basis on the components of a tenso？ 333

Exercises on 21 336

22．Alternating bilinear mappings 341

1．Alternating bilinear mappings 341

2．The case of finitely generated free modules 342

3．Alternating trilinear mappings 345

4．Expansion with respect to a basis 346

Exercises on 22 350

23．Alternating multilinear mappings 353

1．The signature of a permutation 353

2．Antisymmetrization of a function of several variables 357

3．Alternating multilinear mappings 359

4．Alternating p-linear functions on a module isomorphic to Kp 361

5．Determinants 362

6．Characterization of bases of a finite-dimensional vector space 366

7．Alternating multilinear mappings:the general case 369

8．The criterion for linear independence 371

9．Conditions for consistency of a system of linear equations 373

Exercises on 23 376

24．Determinants 380

1．Fundamental properties of determinants 380

2．Expansion of a determinant along a row or column 382

3．The adjugate matrix 386

4．Cramer's formulae 387

Exercises on 24 390

25．Affine spaces 396

1．The vector space of translations 396

2．Affine spaces associated with a vector space 397

3．Barycentres in an affine space 399

4．Linear varieties in an affine space 402

5．Generation of a linear variety by means of lines 405

6．Finite-dimensional affine spaces.Affine bases 406

7．Calculation of the dimension of a linear variety 408

8．Equations of a linear variety in affine coordinates 410

PART Ⅵ:POLYNOMIALS AND ALGEBRAIC EQUATIONS 413

26．Algebraic relations 414

1．Monomials and polynomials in the elements of a ring 414

2．Algebraic relations 416

3．The case of fields 417

Exercises on 26 420

1．Preliminaries on the case of one variable 423

27．Polynomial rings 423

2．Polynomials in one indeterminate 424

3．Polynomial notation 426

4．Polynomials in several indeterminates 428

5．Partial and total degrees 429

6．Polynomials with coefficients in an integral domain 430

28．Polynomial functions 432

1．The values of a polynomial 432

2．The sum and product of polynomial functions 433

3．The case of an infinite field 435

Exercises on 27 and 28 438

1．The field of fractions of an integral domain:preliminaries 446

29．Rational fractions 446

2．Construction of the field of fractions 447

3．Verification of the field axioms 450

4．Embedding the ring K in its field of fractions 451

5．Rational fractions with coefficients in a field 452

6．Values of a rational fraction 454

Exercises on 29 458

1．Derivations of a ring 463

30．Derivations.Taylor's formula 463

2．Derivations of a polynomial ring 464

3．Partial derivatives 466

4．Derivation of composite functions 467

5．Taylor's formula 468

6．The characteristic of a field 470

7．Multiplicities of the roots of an equation 471

Exercises on 30 475

1．Highest common factor 479

31．Principal ideal domains 479

2．Coprime elements 480

3．Least common multiple 481

4．Existence of prime divisors 482

5．Properties of extremal elements 484

6．Uniqueness of the decomposition into prime factors 485

7．Calculation of h.c.f.and l.c.m.by means of prime factorization 486

8．Decomposition into partial fractions over a principal ideal domain 488

Exercises on 31 491

32．Division of polynomials 497

1．Division of polynomials in one variable 497

2．Idealsin a polynomial ring in one iadeterminate 500

3．The h．c．f．and l．c．m．of several polynomials．Irreducible polynomials 501

4．Application to rational fractions 503

Exercises on 32 506

33．The roots of on algebraic equation 515

1．The maximum number of roots 515

2．Algebraically closed fields 517

3．Number of roots of an equation with coefficients in an algebraically closed field 519

4．Irreducible polynomials with coefficients in an algebraically closed field 521

5．Irreducible polynomials with real coefficients 522

6．Relations between the coefficients and the roots of an equation 524

Exercises on 33 526

PART Ⅶ:REDUCTION OF MATRICES 537

34．Eigenvalues 538

1．Definition of eigenvectors and eigenvalues 538

2．The characteristic polynomial of a matrix 539

3．The form of the characteristic polynomial 540

4．The existence of eigenvalues 541

5．Reduction to triangular form 541

6．The case in which all the eigenvalues are simple 545

7．Characterization of diagonalizable endomorphisms 548

Exercises on 34 551

35．The canonical form of a matrix 565

1．The Cayley-Hamilton theorem 565

2．Decomposition into nilpotent endomorphisms 567

3．The structure of nilpotent endomorphisms 569

4．Jordan's theorem 572

Exercises on 35 575

36．Hermitian forms 583

1．Sesquilinear forms;hermitian forms 583

2．Non-degenerate forms 586

3．The adjoint of ahomomorphism 588

4．Orthogonality with respect to a non-degenerate hermitian form 591

5．Orthogonal bases 596

6．Orthonormal bases 598

7．Automorphisms of a hermitian form 600

8．Automorphisms of a positive definite hermitian form．Reduction to diagonal form 602

9．Isotropic vectors and indefinite forms 606

10．The Cauchy-Schwarz inequality 607

Exercises on 36 610

BIBLIOGRAPHY 622

INDEX OF NOTATION 626

INDEX OF TERMINOLOGY 629