# ABSTRACT ALGEBRA WITH APPLICATIONS

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1 ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong

2 Contents f Volume I Preface v VECTOR SPACES 1. First introduction: Affine geometry 1 Examples from geometry 1/ Vectors as equivalence classes of arrows 2/ Addition and scalar multiplication 3/ Expressing geometrical concepts by vectors 7/ Applications of the vector concept to geometrical problems 9/ Coordinate Systems 15/ Reformulating geometrical problems as Systems of linear equations 17/ Exercises Second introduction: Linear equations 22 Examples from electrical engineering and economics 22/ Fields 24/ Matrices 25/ Matrix Operations 26/ The Gaussian algorithm 30/ Matrix calculus 33/ Matrices as linear mappings 37/ Homogeneous and inhomogeneous Systems of linear equations 40/ Exercises Vector Spaces 49 Vector spaces and subspaces 49/ Examples 50/ Free and generating sets 54/ Bases 55/ Dimension 58/ Dimension formula for subspaces 58/ Direct decompositions 59/ Quotient spaces 62/ Dimension formula for quotient Spaces 63/Exercises Linear and affine mappings 72 Characterization of linear and affine mappings 72/ Examples 74/ Invariant subspaces 76/ Semisimplicity 77/ Isomorphisms 80/ Image, kernel and cokernel 81/ Homomorphism theorem 82/ Dimension formula for linear mappings 83/ Vector spaces of linear mappings 85/ Dual spaces and dual mappings 86/ Exercises Abstract affine geometry 95 Affine spaces and subspaces 95/ Dimension formula for affine subspaces 99/ Affine mappings 100/ Convexity 101/ The natural topology of a finite-dimensional real or complex vector space 103/ Norms 104/ Exercises Representation of linear mappings by matrices 111 Parameterizations, basis isomorphisms and coordinate transformations 111/ Matrix representations of linear mappings 112/ Matrix inversion 114/ Conceptual meaning of the transpose 117/ Conceptual meaning of block decompositions 118/ Change of basis 119/ Equivalence of matrices 121/ Rank of a matrix 121/ Classification of matrices up to equivalence 122/ Similarity of Square matrices 124/ Canonical forms for projections, reflections and nilpotent endomorphisms 124/ Exercises 130 t Headings below section titles refer to topics covered, much like an index, rather than to discrete subsections. ix

3 7. Determinants 138 Determinants as natural constructions in elimination theory 138/ Existence and uniqueness of determinant functions 142/ Significance of determinants for Systems of linear equations 145/ Properties of determinants 146/ Examples 147/ The adjunct of a matrix 150/ Cramer's rule 152/ Exercises Volume functions 159 Motivation: Oriented volumes 159/ Volume functions 160/ Existence and uniqueness 162/ Closed form for the determinant of a matrix 164/ Expressing minors of a matrix product by minors of the individual factors 165/ Determinant of an endomorphism as a distortion factor for volumes 166/ Trace of an endomorphism 168/ Basis deformations 170/ Orientations 170/ Exercises Eigenvectors and eigenvalues 177 Definition of eigenvalues and eigenvectors 177/ Linear independence of eigenvectors 178/ Eigenspaces and generalized eigenspaces 178/ Characteristic polynomial 179/ Gershgorin's theorem 183/ Geometrie and algebraic multiplicity 185/ Diagonalizability 186/ Simultaneous diagonalizability 187/ Triagonalizability 188/ Simultaneous'triagonalizability 189/ Spectral mapping theorem 190/ Perron-Frobenius theorem 194/ Exercises' Classification of endomorphisms up to similarity 205 Algebras 205/ Polynomial expressions of an endomorphism 205/ Matrix polynomials 208/ Hamilton-Cayley theorem 209/ Minimal polynomial 210/ Jordan canonical form 211/ Additive and multiplicative Jordan decomposition 215/ Frobenius' theorem 217/ Determinantal and elementary divisors 219/ Exercises Tensor produets and base-field extensions 230 Complexification 230/ Tensor product of vector Spaces 231/ Properties 232/ Base-field extensions 238/ Fitting decomposition 240/ Semisimplicity 241/ Jordan decomposition over fields which are not algebraically closed 243/ Exercises Metrie geometry 249 Cartesian coordinate Systems 249/ Scalar product 249/ Applications in geometry 252/ Oriented areas and volumes 255/ Vector product 258/ Applications in geometry and mechanics 260/ Exercises Euclidean Spaces 273 Scalar produets 273/ Orthogonality 276/ Orthogonal decompositions 279/ Orthonormalization 281/ Riesz' representation theorem 282/ Gram's determinant 283/ Normed volume functions 284/ Measure theory 285/ Exercises Linear mappings between Euclidean Spaces 303 Adjoint of a linear mapping 303/ Properties 304/ Trace and Operator norm 306/ Normal, self-adjoint, skew-adjoint, orthogonal and unitary endomorphisms 308/ Characterization 308/ Canonical forms 312/ Positive definite endomorphisms 315/ Pseudoinverses 318/ Kaiman filter 322/ Exercises 324 x / Contents

4 15. Bilinear forms 331 Bilinear and sesquilinear forms 331/ Representation by matrices 332/ Congruence of matrices 333/ Symmetrie, antisymmetric and alternating forms 335/ Orthogonality 336/ Isotropy, rank and degeneraey 337/ Orthogonal complements 337/ Classification of alternating forms 341/ Pfaffian 342/ Classification of real Symmetrie forms by rank and signature 344/ Quadratic forms and quadrics 347/ Metrical Classification of real Symmetrie forms; Sylvester's law of inertia 349/ Hyperbolic planes 352/ Witt's theorem 354/ Bruck-Chowla-Ryser theorem 356/ Exercises Groups of automorphisms 366 Linear transformation groups 366/ Examples 366/ Structure of orthogonal groups 370/ Group-theoretical interpretation of the Gaussian algorithm 373/ Gaussian decomposition of GL(n, K) 374/ Bruhat decomposition of GL(ra, K) 376/ Iwasawa, polar and Cartan decomposition of GL(n, R) and GL(n, C) 378/ Parameterizations of automorphism groups 380/ Cayley transform 382/ Exercises Application: Markov chains 393 Markov processes 393/ Transition matrices and graphs 393/ Examples 394/ Long-term probabilities 396/ Communicating classes 399/ Canonical form of a Markov process 400/ Fundamental matrix 401/ Regulär Markov chains 403/ Ergodic Markov chains 405/ Cyclic classes 407/ Mean first passage matrix 411/ Examples in genetics 413/ Exercises Application: Matrix calculus and differential equations Examples from electrical engineering and economics 424/ Existence and uniqueness of Solutions for linear initial value problems 426/ Wronskian determinant 427/ Fundamental System 428/ Exponential funetion for matrices 430/ Matrix differential equations with constant coefficients 432/ Higher-order differential equations 435/ Characteristic polynomial 438/ Fundamental system 438/ Method of judicious guessing 440/ Fulmer's algorithm to exponentiate a matrix 442/ Exercises 444 GROUPS 19. Introduction: Symmetries of geometric figures 451 Symmetries in art and nature 451/ Symmetry transformations 453/ Types of one- and two-dimensional ornaments 454/ Exercises Groups 461 Semigroups, monoids and groups 461/ Elementary properties 462/ Group table 463/ Integers 463/ Additive and multiplicative group of a field 463/ Additive group of a vector space 463/ Permutation groups 463/ Symmetrie groups 464/ Dihedral groups 464/ Automorphism group of a graph 465/ Matrix groups 466/ Groups of residue classes 468/ Direct produets 470/ Exercises 472 Contents / xi

5 21. Subgroups and cosets 476 Subgroups 476/ Centralizers and normalizers 478/ Subgroup generated by a set 478/ Cyclic groups 480/ Order of an dement 480/ Cosets 486/ Index of a subgroup 486/ Lagrange's theorem 487/ Applications in number theory 488/ Exercises Symmetrie and alternating groups 497 Cycle decomposition 498/ Cycle-structure and conjugacy 498/ Generating sets for Sym n 500/ Even and odd permutations 503/ Alternating groups 503/ Exercises Group homomorphisms 511 Isomorphisms 512/ Conjugation; inner and outer automorphisms 514/ Homomorphisms 515/ Kernel and image 518/ Embeddings 521/ Cayley's theorem 521/ Representations 522/ Exercises Normal subgroups and factor groups 529 Characteristic and normal subgroups 529/ Quotient groups 531/ Isomorphism theorems 532/ Abelianization of a group 535/ Simple groups 536/ Simplicity of the alternating groups 536/ Semidirect produets 537/ Exercises Free groups; generators and relations 548 Free groups 549/ Universal property 550/ Generators and relations 553/ Von Dyck's theorem 553/ Free abelian groups 554/ Universal property 555/ Linearly independent and generating sets; bases 555/ Direct sums 556/ Rank of a free abelian group 557/ Torsion subgroup of an abelian group 557/ Classification of finitely generated abelian groups 561/ Character group of an abelian group 567/ Exercises Group actions 575 Group actions 575/ Examples 575/ Orbits and stabilizers 577/ Transitivity 577/ Automorphism groups of graphs 580/ Burnside's lemma 581/ Pölya's theorem 587/ Applications in combinatorics 588/ Exercises Group-theoretical applications of group actions 596 Actions of groups on groups and coset spaces 596/ Class equation 597/ Deriving structural properties of a finite group from its order 597/ Sylow's theorems 600/ Applications 602/ Exercises Nilpotent and solvable groups 608 Commutators 608/ Commutator series and descending central series 608/ Examples 609/ Solvability and nilpotency 613/ Ascending central series 615/ Characterization of finite nilpotent groups 617/ Affine subgroups of Sym n 617/ Characterization of the solvable transitive subgroups of Sym p 620/ Exercises Topological methods in group theory 624 Topological groups and their subgroups 624/ Continuous homomorphisms 628/ Isomorphism theorems 630/ Baire's theorem 633/ Open mapping theorem 633/ Identity component 635/ Examples 635/ Krull topology 639/ Exercises 641 xii / Contents

6 30. Analytical methods in group theory 644 Tangent space of a matrix group 645/ One-parameter subgroups 647/ Lie object of a matrix group 648/ Lie algebras 650/ Examples 650/ Relation between a closed matrix group and its Lie algebra 654/ Adjoint action 655/ Baker-Campbell-Hausdorff formula 660/ Analytic matrix groups and their Lie group topology 662/ Exercises Groups in topology 671 Basic ideas in algebraic topology 671/ Homotopy 673/ Homotopy groups 677/ Fixed point theorems 679/ Commutativity of the higher homotopy groups 682/ Homology 683/ Homology groups 688/ Hi(X) as the abelianization of iti(x) 689/ Knot groups 693/ Wirtinger presentation 693/ Exercises 697 Appendix 701 A. Sets and functions 701/ B. Relations 709/ C. Cardinal and ordinal numbers 718/ D. Point-set topology 728/ E. Categories and functors 737 Bibliography 743 Index 745 Volume II Preface v RINGS AND FIELDS 1. Introduction: The art of doing arithmetic 1 Euler's and Fermat's theorem 2/ Divisibility rules 2/ Other examples for the use of congruence classes to obtain number-theoretical results 3/ Wilson's theorem 4/ Exercises 6 2. Rings and ring homomorphisms 10 Rings, commutative rings and unital rings 10/ Subrings 11/ Examples 11/ Power series rings 12/ Polynomial rings 13/ Matrix rings 15/ Rings of functions 15/ Convolution rings 16/ Direct products and sums 17/ Ring homomorphisms, isomorphisms and embeddings 17/ Exercises Integral domains and fields 29 Zero-divisors 29/ Nilpotent elements 29/ Units 31/ Examples 31/ Divisibility 35/ Integral domains 36/ Fields and skew-fields 38/ Quotient fields 40/ Application: Mikusinski's Operator calculus 41/ Exercises 48 Contents / xiii

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