Linear Algebra and Geometry

Description

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 Algebra and Geometry Preface Acknowledgements Contents Preliminaries Sets and Mappings Some Topological Notions Chapter 1: Linear Equations 1.1 Linear Equations and Functions 1.2 Gaussian Elimination 1.3 Examples* Chapter 2: Matrices and Determinants 2.1 Determinants of Orders 2 and 3 2.2 Determinants of Arbitrary Order 2.3 Properties that Characterize Determinants 2.4 Expansion of a Determinant Along Its Columns 2.5 Cramer's Rule 2.6 Permutations, Symmetric and Antisymmetric Functions 2.7 Explicit Formula for the Determinant 2.8 The Rank of a Matrix 2.9 Operations on Matrices 2.10 Inverse Matrices Chapter 3: Vector Spaces 3.1 The Definition of a Vector Space 3.2 Dimension and Basis 3.3 Linear Transformations of Vector Spaces 3.4 Change of Coordinates 3.5 Isomorphisms of Vector Spaces 3.6 The Rank of a Linear Transformation 3.7 Dual Spaces 3.8 Forms and Polynomials in Vectors Chapter 4: Linear Transformations of a Vector Space to Itself 4.1 Eigenvectors and Invariant Subspaces 4.2 Complex and Real Vector Spaces 4.3 Complexification 4.4 Orientation of a Real Vector Space Chapter 5: Jordan Normal Form 5.1 Principal Vectors and Cyclic Subspaces 5.2 Jordan Normal Form (Decomposition) 5.3 Jordan Normal Form (Uniqueness) 5.4 Real Vector Spaces 5.5 Applications* Chapter 6: Quadratic and Bilinear Forms 6.1 Basic Definitions 6.2 Reduction to Canonical Form 6.3 Complex, Real, and Hermitian Forms Chapter 7: Euclidean Spaces 7.1 The Definition of a Euclidean Space 7.2 Orthogonal Transformations 7.3 Orientation of a Euclidean Space* 7.4 Examples* 7.5 Symmetric Transformations 7.6 Applications to Mechanics and Geometry* 7.7 Pseudo-Euclidean Spaces 7.8 Lorentz Transformations Chapter 8: Affine Spaces 8.1 The Definition of an Affine Space 8.2 Affine Spaces 8.3 Affine Transformations 8.4 Affine Euclidean Spaces and Motions Chapter 9: Projective Spaces 9.1 Definition of a Projective Space 9.2 Projective Transformations 9.3 The Cross Ratio 9.4 Topological Properties of Projective Spaces* Chapter 10: The Exterior Product and Exterior Algebras 10.1 Plücker Coordinates of a Subspace 10.2 The Plücker Relations and the Grassmannian 10.3 The Exterior Product 10.4 Exterior Algebras* 10.5 Appendix* Chapter 11: Quadrics 11.1 Quadrics in Projective Space 11.2 Quadrics in Complex Projective Space 11.3 Isotropic Subspaces 11.4 Quadrics in a Real Projective Space 11.5 Quadrics in a Real Affine Space 11.6 Quadrics in an Affine Euclidean Space 11.7 Quadrics in the Real Plane* Chapter 12: Hyperbolic Geometry 12.1 Hyperbolic Space* 12.2 The Axioms of Plane Geometry* 12.3 Some Formulas of Hyperbolic Geometry* Chapter 13: Groups, Rings, and Modules 13.1 Groups and Homomorphisms 13.2 Decomposition of Finite Abelian Groups 13.3 The Uniqueness of the Decomposition 13.4 Finitely Generated Torsion Modules over a Euclidean Ring* Chapter 14: Elements of Representation Theory 14.1 Basic Concepts of Representation Theory 14.2 Representations of Finite Groups 14.3 Irreducible Representations 14.4 Representations of Abelian Groups Historical Note References Index