# Linear Algebra (Book)

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Linear Algebra | |

Information | |
---|---|

Author | Georgi Shilov |

Language | English |

Publisher | Dover Publications |

Publication Date | 1 June 1977 |

Pages | 400 |

ISBN-10 | 048663518X |

ISBN-13 | 978-0486635187 |

The textbook * Linear Algebra* by Georgi Shilov thoroughly covers all major aspects of linear algebra, in addition it covers more geometrically motivated linear algebra in the latter half.

## Table of Contents[edit | edit source]

Chapter/Section # | Title | Page # |
---|---|---|

Chapter 1: DETERMINANTS | ||

1.1 | Number Fields | 1 |

1.2 | Problems of the Theory of Systems of Linear Equations | 3 |

1.3 | Determinants of Order \(n\) | 5 |

1.4 | Properties of Determinants | 8 |

1.5 | Cofactors and Minors | 12 |

1.6 | Practical Evaluation of Determinants | 16 |

1.7 | Cramer's Rule | 18 |

1.8 | Minors of Arbitrary Order. Laplace's Theorem | 20 |

1.9 | Multiplicative inverses | 23 |

Problems | 28 | |

Chapter 2: LINEAR SPACES | ||

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 | Hyperplanes | 51 |

2.7 | Morphisms of Linear Spaces | 53 |

Problems | 56 | |

Chapter 3: SYSTEMS OF LINEAR EQUATIONS | ||

3.1 | More on the Rank of a Matrix | 58 |

3.2 | Nontrivial Compatibility of a 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 |

3.4 | Geometric Properties of the Solution Space | 65 |

3.4 | Methods for Calculating the Rank of a Matrix | 67 |

Problems | 71 | |

Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT | ||

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 |

4.8 | Invariant Subspaces | 106 |

4.9 | Eigenvectors and Eigenvalues | 108 |

Problems | 113 | |

Chapter 5: COORDINATE TRANSFORMATIONS | ||

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 | ||

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 | 153 |

6.6 | The Real Jordan Canonical Form | 155 |

6.7 | Spectra, Jets and Polynomials | 160 |

6.8 | Operator Functions and Their Matrices | 169 |

Problems | 176 | |

Chapter 7: BILINEAR AND QUADRATIC FORMS | ||

7.1 | Bilinear Forms | 179 |

7.2 | Quadratic Forms | 183 |

7.3 | Reduction of a Quadratic Form to Canonical Form | 183 |

7.4 | The Canonical Basis of a Bilinear Form | 183 |

7.5 | Construction of a Canonical Basis by Jacobi's Method | 183 |

7.6 | Adjoint Linear Operators | 183 |

7.7 | Isomorphism of Spaces Equipped with a Bilinear Form | 183 |

*7.8 | Multilinear Forms | 183 |

7.9 | Bilinear and Quadratic Forms in a Real Space | 183 |

Problems | 210 | |

Chapter 8: EUCLIDEAN SPACES | ||

8.1 | Introduction | 214 |

8.2 | Definition of a Euclidean Space | 215 |

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 | ||

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 | ||

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 | |

Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS | ||

11.1 | More on Algebras | 312 |

11.2 | Representations of Abstract Algebras | 313 |

11.3 | Irreducible Representations and Schur's Lemma | 314 |

11.4 | Basic Types of Finite-Dimensional Algebras | 315 |

11.5 | The Left Regular Representation of a Simple Algebra | 318 |

11.6 | Structure of Simple Algebras | 320 |

11.7 | Structure of Semisimple Algebras | 323 |

11.8 | Representations of Simple and Semisimple Algebras | 327 |

11.9 | Some Further Results | 331 |

Problems | 332 | |

*Appendix | ||

CATEGORIES OF FINITE-DIMENSIONAL SPACES | ||

A.1 | Introduction | 335 |

A.2 | The Case of Complete Algebras | 338 |

A.3 | The Case of One-Dimensional Algebras | 340 |

A.4 | The Case of Simple Algebras | 345 |

A.5 | The Case of Complete Algebras of Diagonal Matrices | 353 |

A.6 | Categories and Direct Sums | 357 |

HINTS AND ANSWERS | 361 | |

BIBLIOGRAPHY | 379 | |

INDEX | 381 |