DOCTOR SWAP COMPLETE SERIES: SELF GENDER SWAPPING & BIMBO EXPERIMENT BUNDLED COLLECTION

In this volume in his exceptional series of translations of Russian mathematical texts, Richard Silverman has taken Shilov's course in linear algebra and has made it even more accessible and more useful for English language readers. Georgi E. Shilov, Professor of Mathematics at the Moscow State University, covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional algebras and their representations, with an appendix on categories of finite-dimensional spaces. The author begins with elementary material and goes easily into the advanced areas, covering all the standard topics of an advanced undergraduate or beginning graduate course. The material is presented in a consistently clear style. Problems are included, with a full section of hints and answers in the back. Keeping in mind the unity of algebra, geometry and analysis in his approach, and writing practically for the student who needs to learn techniques, Professor Shilov has produced one of the best expositions on the subject. Because it contains an abundance of problems and examples, the book will be useful for self-study as well as for the classroom. Chapter 1 DETERMINANTS 1 1.1. Number Fields 1 1.2. Problems of the Theory of Systems of Linear Equations 3 1.3. Determinants of Order a 5 1.4. Properties of Determinants 8 1.5. Co-factors and Minors RB 1.6. Practical Evaluation of Determinants t6 1.7. Cramer's Rule (8 1.8. Minors of Arbitrary Order. Laplace's Theorem 20 1.9. Linear Dependence between Columns 23 Problems 28 contents vii Chapter 2 LINEAR SPACES 31 2.1. Definitions 31 2.2. Linear Dependence 36 2.3. Bases, Components, Dimension 38 2.4. Subspaces 42 2.5. Linear Manifolds 49 2.6. Hyperplane 51 2.7. Morphisms of Linear Spaces 33 Problems 56 Chapter 3 SYSTEMS OF LINEAR EQUATIONS 58 3.1. More on the Rank of a Matrix 8 3.2. Nontrivial Compatibility of 0 Homogeneous Linear System 60. 3.3. The Compatibility Condition for a General Linear System 61 3.4. The General Solution of a Linear System 63 3.5. Geometric Properties of the Solution Space 65 3.6. Methods for Calculating the Rank of a Matrix 67 Problems 71 Chapter 4 LINEAR FUNCTIONS OF A VECTOR ARGUMENT 75 4.1. Linear Forms 75 4.2. Linear Operators 77 4.3. Sums and Products of Linear Operators 82 4.4. Corresponding Operations on Matrices 84 4.5. Further Properties of Matrix Multiplication 88 4.6. The Range and Null Space of a Linear Operator 93 4.7. Linear Operators Mapping a Space K_n into Itself 98 4.2. Invariant Subspaces 106 4.9. Eigenvectors and Eigenvalues 108 Problems 113 contents ix Chapter 5 COORDINATE TRANSFORMATIONS 18 5.1. Transformation to a New Basis 118 5.2. Consecutive Transformations 120 5.3. Transformation of the Components of a Vector 121 5.4. Transformation of the Coefficients of a Linear Form 123 5.5. Transformation of the Matrix of a Linear Operator 124 5.6. Tensors 126 Problems 131 Chapter 6 THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR 133 6.1. Canonical Form of the Matrix of a Nilpotent Operator 133 6.2. Algebras. The Algebra of Polynomials 136 6.3. Canonical Form of the Matrix of an Arbitrary Operator 142 6.4. Elementary Divisors 147 6.5. Further Implications 183 6.6. The Real Jordan Canonical Form 185 6.7. Spectra, Jets and Polynomials 1460 6.8. Operator Functions and Their Matrices 169 Problems 176 Chapter 7 BILINEAR AND QUADRATIC FORMS 179 7.1. Bilinear Forms 179 7.2. Quadratic Forms 183 7.3. Reduction of a Quadratic Form to Canonical Form 185 7.4. The Canonical Basis of a Bilinear Form 190 7.8. Construction of a Canonical Basis by Jacobi’s Method 192 7.6. Adjoint Linear Operators 196 7.7. Isomorphism of Spaces Equipped with a Bilinear Form 199 7.8. Multilinear Forms 202 7.9. Bilinear and Quadratic Forms in a Real Space 204 Problems 210 x contents Chapter 8 EUCLIDEAN SPACES 24 8.1. Introduction 24 8.2. Definition of a Euclidean Space 2ts 8.3. Basic Metric Concepts 216 8.4. Orthogonal Bases 222 8.5. Perpendiculars 223 8.6. The Orthogonalization Theorem 226 8.7. The Gram Determinant 230 8.8. Incompatible Systems and the Method of Least Squares 234 8.9. Adjoint Operators and Isometry 237 Problems 241 Chapter 9 UNITARY SPACES 247 9.1. Hermitian Forms 247 9.2. The Scalar Product in a Complex Space 254 9.3. Normal Operators 259 9.4. Applications to Operator Theory in Euclidean Space 263 Problems 271 Chapter 10 QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES 273 10.1. Basic Theorem on Quadratic Forms in a Euclidean Space 273 10.2. Extremal Properties of a Quadratic Form 276 10.3. Simultaneous Reduction of Two Quadratic Forms 283 10.4. Reduction of the General Equation of a Quadric Surface 287 10.5. Geometric Properties of a Quadric Surface 289 10.6. Analysis of a Quadric Surface from Its General Equation 300 10.7. Hermitian Quadratic Forms 308 Problems 310