Preface
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I. |
Motivation |
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I.1 |
The Three-Dimensional Affine Space as Prototype of Linear Manifolds |
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I.2 |
The Real Projective Plane as Prototype of the Lattice of Subspaces of a Linear Manifold |
II. |
The Basic Properties of a Linear Manifold |
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II.1 |
Dedekind's Law and the Principle of Complementation |
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II.2 |
Linear Dependence and Independence; Rank |
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II.3 |
The Adjoint Space |
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Appendix I. |
Application to Systems of Linear Homogeneous Equations |
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Appendix II. |
Paired Spaces |
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II.4 |
The Adjunct Space |
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Appendix III. |
Fano's Postulate |
III. |
Projectivities |
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III.1 |
Representation of Projectivities by Semi-linear Transformations |
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Appendix I. |
Projective Construction of the Homothetic Group |
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III.2 |
The Group of Collineations |
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III.3 |
The Second Fundamental Theorem of Projective Geometry |
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Appendix II. |
The Theorem of Pappus |
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III.4 |
The Projective Geometry of a Line in Space; Cross Ratios |
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Appendix III. |
Projective Ordering of a Space |
IV. |
Dualities |
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IV.1 |
Existence of Dualities; Semi-bilinear Forms |
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IV.2 |
Null Systems |
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IV.3 |
Representation of Polarities |
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IV.4 |
Isotropic and Non-isotropic Subspaces of a Polarity; Index and Nullity |
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Appendix I. |
Sylvester's Theorem of Inertia |
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Appendix II. |
Projective Relations between Lines Induced by Polarities |
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Appendix III. |
The Theorem of Pascal |
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IV.5 |
The Group of a Polarity |
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Appendix IV. |
The Polarities with Transitive Group |
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IV.6 |
The Non-isotropic Subspaces of a Polarity |
V. |
The Ring of a Linear Manifold |
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V.1 |
Definition of the Endomorphism Ring |
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V.2 |
The Three Cornered Galois Theory |
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V.3 |
The Finitely Generated Ideals |
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V.4 |
The Isomorphisms of the Endomorphism Ring |
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V.5 |
The Anti-isomorphisms of the Endomorphism Ring |
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Appendix I. |
The Two-sided Ideals of the Endomorphism Ring |
VI. |
The Groups of a Linear Manifold |
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VI.1 |
The Center of the Full Linear Group |
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VI.2 |
First and Second Centralizer of an Involution |
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VI.3 |
Transformations of Class 2 |
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VI.4 |
Cosets of Involutions |
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VI.5 |
The Isomorphisms of the Full Linear Group |
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Appendix I. |
Groups of Involutions |
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VI.6 |
Characterization of the Full Linear Group within the Group of Semi-linear Transformations |
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VI.7 |
The Isomorphisms of the Group of Semi-linear Transformations |
VII. |
Internal Characterization of the System of Subspaces |
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A Short Bibliography of the Principles of Geometry |
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VII.1 |
Basic Concepts, Postulates and Elementary Properties |
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VII.2 |
Dependent and Independent Points |
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VII.3 |
The Theorem of Desargues |
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VII.4 |
The Imbedding Theorem |
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VII.5 |
The Group of a Hyperplane |
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VII.6 |
The Representation Theorem |
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VII.7 |
The Principles of Affine Geometry |
Appendix S. |
A Survey of the Basic Concepts and Principles of the Theory of Sets |
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A Selection of Suitable Introductions into the Theory of Sets |
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Sets and Subsets |
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Mappings |
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Partially Ordered Sets |
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Well Ordering |
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Ordinal Numbers |
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Cardinal Numbers |
Bibliography |
Index |