Linear Algebra and Projective Geometry

Reinhold Baer

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486445656
• eBook ISBN: 9780486154664
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

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 focus on the representation of projective geometries by linear manifolds, of projectivities by semilinear transformations, of collineations by linear transformations, and of dualities by semilinear forms. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point, within algebraic structures such as the endomorphism ring of the underlying manifold or the full linear group.
Restricted to topics of an algebraic nature, the text shows how far purely algebraic methods may extend. It assumes only a familiarity with the basic concepts and terms of algebra. The methods of transfinite set theory frequently recur, and for readers unfamiliar with this theory, the concepts and principles appear in a special appendix.

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Chapter I. Motivation
• Chapter II. The Basic Properties of a Linear Manifold
• Chapter III. Projectivities
• Chapter IV. Dualities
• Chapter V. The Ring of a Linear Manifold
• Chapter VI. The Groups of a Linear Manifold
• Chapter VII. Internal Characterization of the System of Subspaces
• Appendix S. A Survey of the Basic Concepts and Principles of the Theory of Sets
• Bibliography
• Index