Linear Algebra and Projective Geometry (eBook)

Front Cover 1
Linear Algebra and Projective Geometry 4
Copyright Page 5
Preface 6
Contents 8
Chapter I. Motivation 10
I.1. The Three-Dimensional Affine Space as Prototype of Linear Manifolds 11
I.2. The Real Projective Plane as Prototype of the Lattice of Subspaces of a Linear Manifold 14
Chapter II. The Basic Properties of a Linear Manifold 16
II.1. Dedekind's Law and the Principle of Complementation 16
II.2. Linear Dependence and Independence Rank
II.3. The Adjoint Space 34
Appendix I. Application to Systems of Linear Homogeneous Equations 42
Appendix II. Paired Spaces 43
II.4. The Adjunct Space 45
Appendix III. Fano's Postulate 46
Chapter III. Projectivities 48
III.1. Representation of Projectivities by Semi-linear Transformations 49
Appendix I. Projective Construction of the Homothetic Group 61
III.2. The Group of Collineations 71
III.3. The Second Fundamental Theorem of Projective Geometry 75
Appendix II. The Theorem of Pappus 78
III.4. The Projective Geometry of a Line in Space Cross Ratios
Appendix III. Projective Ordering of a Space 102
Chapter IV. Dualities 104
IV.1. Existence of Dualities Semi-bilinear Forms
IV.2. Null Systems 115
IV.3. Representation of Polarities 118
IV.4. Isotropic and Non-isotropic Subspaces of a Polarity Index and Nullity
Appendix I. Sylvester's Theorem of Inertia 136
Appendix II. Projective Relations between Lines Induced by Polarities 140
Appendix III. The Theorem of Pascal 147
IV.5. The Group of a Polarity 153
Appendix IV. The Polarities with Transitive Group 162
IV.6. The Non-isotropic Subspaces of a Polarity 167
Chapter V. The Ring of a Linear Manifold 176
V.1. Definition of the Endomorphism Ring 177
V.2. The Three Cornered Galois Theory 181
V.3. The Finitely Generated Ideals 186
V.4. The Isomorphisms of the Endomorphism Ring 191
V.5. The Anti-isomorphisms of the Endomorphism Ring 197
Appendix I. The Two-sided Ideals of the Endomorphism Ring 206
Chapter VI. The Groups of a Linear Manifold 209
VI.1. The Center of the Full Linear Group 210
VI.2. First and Second Centralizer of an Involution 212
VI.3. Transformations of Class 2 216
VI.4. Cosets of Involutions 228
VI.5. The Isomorphisms of the Full Linear Group 237
Appendix I. Groups of Involutions 246
VI.6. Characterization of the Full Linear Group within the Group of Semi-linear Transformations 253
VI.7. The Isomorphisms of the Group of Semi-linear Transformations 256
Chapter VII. Internal Characterization of the System of Subspaces 266
A Short Bibliography of the Principles of Geometry 266
VII.1. Basic Concepts, Postulates and Elementary Properties 267
VII.2. Dependent and Independent Points 272
VII.3. The Theorem of Desargues 275
VII.4. The Imbedding Theorem 277
VII.5. The Group of a Hyperplane 301
VII.6. The Representation Theorem 311
VII.7. The Principles of Affine Geometry 312
Appendix S. A Survey of the Basic Concepts and Principles of the Theory of Sets 317
A Selection of Suitable Introductions into the Theory of Sets 317
Sets and Subsets 317
Mappings 318
Partially Ordered Sets 318
Well Ordering 319
Ordinal Numbers 319
Cardinal Numbers 320
Bibliography 322
Index 324