Linear Algebra and Projective Geometry

Reinhold Baer

Publisher:

Dover Publications

Publication Date:

2005

Number of Pages:

318

Format:

Paperback

Price:

18.95

ISBN:

0486454658

Category:

Textbook

MAA Review
Table of Contents

[Reviewed by

P. N. Ruane

, on

05/25/2006

]

This Dover edition, published in 2005, is an unabridged version of the work first issued by Academic Press in 1952. At that time, the majority of texts on projective geometry based their approach on synthetic methods, so this book would then have been out of the ordinary due to its heavy reliance on algebraic methods.

It certainly isn't a book for those new to the subject and it is aimed at upper-level undergraduates and graduate students (more likely the latter, I feel). In span of 300 pages, there are only 19 diagrams and, although the ideas from linear algebra are invoked throughout the book, matrices are hardly used at all. The main thrust is a series of theorems on the representation of projective geometries by linear manifolds and of collineations by linear transformations and of dualities by semilinear forms. Such theorems are used to present the geometry within algebraic structures such as the endomorphism of the underlying manifold or the full linear group.

This, however, is one of those books in which the author states that Little actual knowledge is presupposed, but those without good prior knowledge of linear algebra, groups, rings and transfinite set theory and elementary projective geometry will find it extremely heavy going.

Peter Ruane (ruane.p@blueyonder.co.uk) is retired from university teaching, where his interests lay predominantly within the field of mathematics education

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