Introduction to projective geometry and modern algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Reading, Mass.
Addison-Wesley
1963
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| Schriftenreihe: | Addison-Wesley series in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VIII, 344 S. |
Internformat
MARC
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| 100 | 1 | |a Rosenbaum, Robert A. |d 1926- |e Verfasser |0 (DE-588)143909789 |4 aut | |
| 245 | 1 | 0 | |a Introduction to projective geometry and modern algebra |c by Robert A. Rosenbaum |
| 264 | 1 | |a Reading, Mass. |b Addison-Wesley |c 1963 | |
| 300 | |a VIII, 344 S. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Addison-Wesley series in mathematics | |
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| 650 | 4 | |a Géométrie projective | |
| 650 | 7 | |a Géométrie projective |2 ram | |
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Datensatz im Suchindex
| _version_ | 1804126130043617280 |
|---|---|
| adam_text | CONTENTS
CHAPTER 1. GEOMETRICAL INTRODUCTION (I)
1 1 Classification of geometry 1
1 2 Modern geometry 6
1 3 The notion of projection 10
1 4 Critique of Section 1 3 13
1 5 Ideal elements 15
CHAPTER 2. GEOMETRICAL INTRODUCTION (II)
2 1 Homogeneous coordinates 19
2 2 Equations in homogeneous coordinates 23
2 3 The imaginary 29
2 4 Higher space 35
2 5 Summary 36
CHAPTER 3. AXIOMS OF SOME ALGEBRAIC SYSTEMS
3 1 Axiomatic systems 37
3 2 Notations, conventions, and abbreviations; equivalence
relations 40
3 3 Axioms and elementary properties of a field .... 45
3 4 Further discussion of fields and related systems; ordering 52
3 5 The real and complex number systems; isomorphism;
summary 56
3 6 Groups 62
CHAPTER 4. VECTOR SPACES
4 1 Vectors; vector spaces 68
4 2 Basic results on subspaces of Vn{C) 72
4 3 Linear dependence and independence 74
4 4 Dimension 80
4 5 Change of basis 84
4 6 Length and angle; orthogonality 86
4 7 Normal orthogonal bases 90
4 8 Further results on subspaces 94
4 9 Other definitions of vector spaces 96
v
vi CONTENTS
CHAPTER 5. PROJECTIVE SPACES
5 1 Projective spaces; homogeneous coordinates .... 100
5 2 Change of coordinate frame; subspaces; hyperplanes . 109
5 3 Duality 115
5 4 Symbolic representations 120
5 5 Desargues theorem and related results 124
CHAPTER 6. INTRODUCTION TO PROJECTIVITIES AND
MATRICES
6 1 Projection from one line onto a second in the real,
extended plane; cross ratio in terms of distance . . . 130
6 2 Cross ratios in Si 133
6 3 Projectivities and linear transformations 137
6 4 Projectivities in n dimensions 143
6 5 Matrices: basic definitions, operations, and properties . 145
6 6 Further properties of matrices 159
CHAPTER 7. SYSTEMS OF LINEAR EQUATIONS;
RANK OF MATRICES
7 1 Equivalent systems of linear equations; elementary
matrices 166
7 2 Solution of systems of linear equations 170
7 3 Row rank and column rank of a matrix 177
7 4 Rank and nullity , 180
7 5 Sylvester s law of nullity 185
7 6 Nonsingular matrices 186
7 7 Rational equivalence of matrices 188
7 8 Solution of a system of nonhomogeneous linear
equations 191
7 9 A general definition of linear transformations . . . 193
7 10 The adjoint of a matrix 194
CHAPTER 8. PROJECTIVE GEOMETRY IN ONE AND TWO
DIMENSIONS
8 1 Definitions and notation 196
8 2 Pappus theorem 200
8 3 Cross ratio and harmonic division 203
8 4 Some theorems on the triangle 207
8 5 Change of coordinate system; reduction to canonical
form 210
CONTENTS vii
8 6 Reduction to canonical form applied to an algebraic
problem 219
8 7 Invariant cross ratio of a collineation 221
8 8 Involutory transformations; involutions of Si; cyclic
collineations; metric involutions 223
8 9 Quadratic forms; invariants 230
CHAPTER 9. GROUPS
9 1 Introduction 236
9 2 Subgroups; coset decomposition 239
9 3 Conjugate elements; automorphisms 245
9 4 Orthogonal transformations 250
CHAPTER 10. CONICS
10 1 Definition of conies 254
10 2 Protective generation of conies 259
10 3 Parametric representation of a conic; tangents . . . 264
10 4 Polars 268
10 5 Polars; tangential equations; poles 273
10 6 Geometry on a conic 275
10 7 Correlations and polar reciprocation 280
10 8 Applications of polar reciprocation 282
10 9 Involutions on a conic 286
10 10 Involutions in the plane 288
10 11 Noninvolutory projectivities on a conic; summary . . 289
10 12 Reduction of real symmetric matrices by orthogonal
matrices 292
CHAPTER 11. PAIRS AND PENCILS OF CONICS
11 1 The intersection of two conies 294
11 2 Pencils of conies 297
11 3 Classification of pencils 300
11 4 Some theorems on pairs and pencils of conies . . . 307
11 5 Poles and polars with respect to pencils and conies . . 309
11 6 Invariants of ternary quadratic forms 313
CHAPTER 12. THE JORDAN CANONICAL FORM
12 1 Review of operations on matrices 316
12 2 The case of distinct characteristic roots; fixed and
invariant subspaces 317
viii CONTENTS
12 3 Reducible and irreducible subspaces 318
12 4 Some theorems on self correspondence and reducibility 320
12 5 The Jordan form for an irreducible space .... 322
12 6 The Jordan canonical form of a nonsingular matrix . 325
12 7 Collineations in S2 325
Appendix 1. Summary of some results of three dimensional analytic
geometry 328
Appendix 2. Summary of results on determinants 330
Appendix 3. Supplementary problems and suggestions for projects . 335
BlBLIOGBAPHY 338
Index 340
|
| any_adam_object | 1 |
| author | Rosenbaum, Robert A. 1926- |
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| author_facet | Rosenbaum, Robert A. 1926- |
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| dewey-ones | 516 - Geometry |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-09T18:12:33Z |
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| language | English |
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| physical | VIII, 344 S. |
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| spelling | Rosenbaum, Robert A. 1926- Verfasser (DE-588)143909789 aut Introduction to projective geometry and modern algebra by Robert A. Rosenbaum Reading, Mass. Addison-Wesley 1963 VIII, 344 S. txt rdacontent n rdamedia nc rdacarrier Addison-Wesley series in mathematics Algèbre Algèbre ram Géométrie projective Géométrie projective ram HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007814551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rosenbaum, Robert A. 1926- Introduction to projective geometry and modern algebra Algèbre Algèbre ram Géométrie projective Géométrie projective ram |
| title | Introduction to projective geometry and modern algebra |
| title_auth | Introduction to projective geometry and modern algebra |
| title_exact_search | Introduction to projective geometry and modern algebra |
| title_full | Introduction to projective geometry and modern algebra by Robert A. Rosenbaum |
| title_fullStr | Introduction to projective geometry and modern algebra by Robert A. Rosenbaum |
| title_full_unstemmed | Introduction to projective geometry and modern algebra by Robert A. Rosenbaum |
| title_short | Introduction to projective geometry and modern algebra |
| title_sort | introduction to projective geometry and modern algebra |
| topic | Algèbre Algèbre ram Géométrie projective Géométrie projective ram |
| topic_facet | Algèbre Géométrie projective |
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