Linear Algebra and Geometry

✍ Scribed by Igor R. Shafarevich, Alexey O. Remizov (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2013
• Tongue: English
• Leaves: 536
• Edition: 1
• Category: Library

✦ Synopsis

This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.

✦ Table of Contents

Front Matter....Pages I-XXI
Linear Equations....Pages 1-23
Matrices and Determinants....Pages 25-77
Vector Spaces....Pages 79-131
Linear Transformations of a Vector Space to Itself....Pages 133-160
Jordan Normal Form....Pages 161-189
Quadratic and Bilinear Forms....Pages 191-212
Euclidean Spaces....Pages 213-288
Affine Spaces....Pages 289-317
Projective Spaces....Pages 319-347
The Exterior Product and Exterior Algebras....Pages 349-383
Quadrics....Pages 385-432
Hyperbolic Geometry....Pages 433-465
Groups, Rings, and Modules....Pages 467-495
Elements of Representation Theory....Pages 497-514
Back Matter....Pages 515-526

✦ Subjects

Linear and Multilinear Algebras, Matrix Theory; Algebra; Geometry; Associative Rings and Algebras