Linear Algebra and Projective Geometry

by Reinhold Baer

Overview

Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point. 1952 edition.

Product Details

ISBN-13:	9780486154664
Publisher:	Dover Publications
Publication date:	05/14/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	336
File size:	19 MB
Note:	This product may take a few minutes to download.

Table of Contents

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