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

Chapter 1: Linear Equations

Chapter 2: Matrices and Determinants

Chapter 3: Vector Spaces

Chapter 4: Linear Transformations of a Vector Space to Itself

Chapter 5: Jordan Normal Form

Chapter 6: Quadratic and Bilinear Forms

Chapter 7: Euclidean Spaces

Chapter 8: Affine Spaces

Chapter 9: Projective Spaces

Chapter 10: The Exterior Product and Exterior Algebras

Chapter 11: Quadrics

Chapter 12: Hyperbolic Geometry

Chapter 13: Groups, Rings, and Modules

Chapter 14: Elements of Representation Theory

Review

From the reviews:

“Shafarevich (Russian Academy of Sciences) and Remizov (École Polytechnique, CNRS, France) provide insightful comments that apply not only to linear algebra but also to mathematics in general. … The book is quite readable … . Summing Up: Recommended. Upper-division undergraduates, graduate students, researchers/faculty, and professionals.” (J. R. Burke, Choice, Vol. 50 (8), April, 2013)

“This beautiful textbook not only reflects I. R. Shafarevich’s unrivalled mastery of mathematical teaching and expository writing, but also the didactic principles of the Russian mathematical school in teaching basic courses such as linear algebra and analytic geometry. … made accessible to a wide audience of international readers, and to further generations of students, too. … this book may be regarded as a historical document in the relevant textbook literature … .” (Werner Kleinert, Zentralblatt MATH, Vol. 1256, 2013)