Linear Algebra and Geometry

Main subject categories: • Linear algebra • Geometry • Analytic geometryMathematics Subject Classification (2010): • 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra • 51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometryThis 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. 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.

Linear Equations -- Matrices and Determinants -- Vector Spaces -- Linear Transformations of a Vector Space to Itself -- Jordan Normal Form -- Quadratic and Bilinear Forms -- Euclidean Spaces -- Affine Spaces -- Projective Spaces -- The Exterior Product and Exterior Algebras -- Quadrics -- Hyperbolic Geometry -- Groups, Rings, and Modules -- Elements of Representation Theory.