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Linear Algebra and Projective Geometry

Reinhold Baer
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

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 ( is retired from university teaching, where his interests lay predominantly within the field of mathematics education


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
  Partially Ordered Sets
  Well Ordering
  Ordinal Numbers
  Cardinal Numbers