Difference between revisions of "Linear Algebra (Book)"

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The textbook '''''Linear Algebra''''' by [https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov] provides a thorough introduction to linear algebra.
The textbook '''''Linear Algebra''''' by [https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov] thoroughly covers all major aspects of linear algebra, in addition it covers more geometrically motivated linear algebra in the latter half.
 
 
This text can be viewed as an introduction to reading mathematics texts; The initial proofs are on arithmetic and properties of linear equations, and are more approachable than the manipulations of functions typical of analysis texts. It is difficult to overstate the universal importance of this subject, but it can be seen through reading. Linear algebra is the basis of all quantities of interest in physics, geometry, number theory, and the same techniques appear in engineering disciplines through physics, numerical computing, or machine learning.


== Table of Contents ==
== Table of Contents ==
Line 21: Line 24:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|-  
|-  
! colspan="3" | Chapter 1: Determinants
! colspan="3" | Chapter 1: DETERMINANTS
|-
|-
| 1.1 || Number Fields || 1
| 1.1 || Number Fields || 1
Line 43: Line 46:
| || Problems || 28
| || Problems || 28
|-  
|-  
! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | 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
|-
! colspan="3" | 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
|-
! colspan="3" | 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
|-
|-
| 1 || Equations in two unknowns || 53
| 4.7 || Linear Operators Mapping a Space \(K_n\) into Itself || 98
|-
|-
| 2 || Equations in three unknowns || 57
| 4.8 || Invariant Subspaces || 106
|-
|-
! colspan="3" | Chapter 3: Real Numbers
| 4.9 || Eigenvectors and Eigenvalues || 108
|-
|-
| 1 || Addition and multiplication || 61
| || Problems|| 113
|-
|-
| 2 || Real numbers: positivity || 64
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
|-
|-
| 3 || Powers and roots || 70
| 5.1 || Transformation to a New Basis || 118
|-
|-
| 4 || Inequalities || 75
| 5.2 || Consecutive Transformations || 120
|-
|-
! colspan="3" | Chapter 4: Quadratic Equations
| 5.3 || Transformation of the Components of a Vector  || 121
|-
|-
! colspan="3" | Interlude: On Logic and Mathematical Expressions
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
|-
|-
| 1 || On reading books || 93
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
|-
|-
| 2 || Logic || 94
| *5.6 || Tensors || 126
|-
|-
| 3 || Sets and elements || 99
| || Problems || 131
|-
|-
| 4 || Notation || 100
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
|-
|-
! colspan="3" | PART II: INTUITIVE GEOMETRY
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
|-
|-
! colspan="3" | Chapter 5: Distance and Angles
| 6.2 || Algebras. The Algebra of Polynomials || 136
|-
|-
| 1 || Distance || 107
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
|-
|-
| 2 || Angles || 110
| 6.4 || Elementary Divisors || 147
|-
|-
| 3 || The Pythagoras theorem || 120
| 6.5 || Further Implications || 153
|-
|-
! colspan="3" | Chapter 6: Isometries
| 6.6 || The Real Jordan Canonical Form || 155
|-
|-
| 1 || Some standard mappings of the plane || 133
| 6.7 || Spectra, Jets and Polynomials || 160
|-
|-
| 2 || Isometries || 143
| 6.8 || Operator Functions and Their Matrices || 169
|-
|-
| 3 || Composition of isometries || 150
| || Problems || 176
|-
|-
| 4 || Inverse of isometries || 155
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
|-
|-
| 5 || Characterization of isometries || 163
| 7.1 || Bilinear Forms || 179
|-
|-
| 6 || Congruences || 166
| 7.2 || Quadratic Forms || 183
|-
|-
! colspan="3" | Chapter 7: Area and Applications
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
|-
|-
| 1 || Area of a disc of radius ''r'' || 173
| 7.4 || The Canonical Basis of a Bilinear Form || 183
|-
|-
| 2 || Circumference of a circle of radius ''r'' || 180
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
|-
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
| 7.6 || Adjoint Linear Operators || 183
|-
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
|-
|-
| 1 || Coordinate systems || 191
| *7.8 || Multilinear Forms || 183
|-
|-
| 2 || Distance between points || 197
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
|-
|-
| 3 || Equation of a circle || 203
| || Problems || 210
|-
|-
| 4 || Rational points on a circle || 206
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
|-
|-
! colspan="3" | Chapter 9: Operations on Points
| 8.1 || Introduction || 214
|-
|-
| 1 || Dilations and reflections || 213
| 8.2 || Definition of a Euclidean Space || 215
|-
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
| 8.3 || Basic Metric Concepts || 216
|-
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
| 8.4 || Orthogonal Bases || 222
|-
|-
| 1 || Segments || 229
| 8.5 || Perpendiculars || 223
|-
|-
| 2 || Rays || 231
| 8.6 || The Orthogonalization Theorem || 226
|-
|-
| 3 || Lines || 236
| 8.7 || The Gram Determinant || 230
|-
|-
| 4 || Ordinary equation for a line || 246
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
|-
|-
! colspan="3" | Chapter 11: Trigonometry
| 8.9 || Adjoint Operators and Isometry || 237
|-
|-
| 1 || Radian measure || 249
| || Problems || 241
|-
|-
| 2 || Sine and cosine || 252
! colspan="3" | Chapter 9: UNITARY SPACES
|-
|-
| 3 || The graphs || 264
| 9.1 || Hermitian Forms || 247
|-
|-
| 4 || The tangent || 266
| 9.2 || The Scalar Product in a Complex Space || 254
|-
|-
| 5 || Addition formulas || 272
| 9.3 || Normal Operators || 259
|-
|-
| 6 || Rotations || 277
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
|-
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
| || Problems || 271
|-
|-
| 1 || The straight line again || 281
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
|-
|-
| 2 || The parabola || 291
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
|-
|-
| 3 || The ellipse || 297
| 10.2 || Extremal Properties of a Quadratic Form || 276
|-
|-
| 4 || The hyperbola || 300
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
|-
|-
| 5 || Rotation of hyperbolas || 305
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
|-
|-
! colspan="3" | PART IV: MISCELLANEOUS
| 10.5 || Geometric Properties of a Quadric Surface || 289
|-
|-
! colspan="3" | Chapter 13: Functions
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
|-
|-
| 1 || Definition of a function || 313
| 10.7 || Hermitian Quadratic Forms || 308
|-
|-
| 2 || Polynomial functions || 318
| || Problems || 310
|-
|-
| 3 || Graphs of functions || 330
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
|-
|-
| 4 || Exponential function || 333
| 11.1 || More on Algebras || 312
|-
|-
| 5 || Logarithms || 338
| 11.2 || Representations of Abstract Algebras || 313
|-
|-
! colspan="3" | Chapter 14: Mappings
| 11.3 || Irreducible Representations and Schur's Lemma || 314
|-
|-
| 1 || Definition || 345
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
|-
|-
| 2 || Formalism of mappings || 351
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
|-
|-
| 3 || Permutations || 359
| 11.6 || Structure of Simple Algebras || 320
|-
|-
! colspan="3" | Chapter 15: Complex Numbers
| 11.7 || Structure of Semisimple Algebras || 323
|-
|-
| 1 || The complex plane || 375
| 11.8 || Representations of Simple and Semisimple Algebras || 327
|-
|-
| 2 || Polar form || 380
| 11.9 || Some Further Results || 331
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
| || Problems || 332
|-
|-
| 1 || Induction || 383
| *Appendix || ||  
|-
|-
| 2 || Summations || 388
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
|-
|-
| 3 || Geometric series || 396
| A.1 || Introduction || 335
|-
|-
! colspan="3" | Chapter 17: Determinants
| A.2 || The Case of Complete Algebras || 338
|-
|-
| 1 || Matrices || 401
| A.3 || The Case of One-Dimensional Algebras || 340
|-
|-
| 2 || Determinants of order 2 || 406
| A.4 || The Case of Simple Algebras || 345
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| A.5 || The Case of Complete Algebras of Diagonal Matrices || 353
|-
|-
| 4 || Determinants of order 3 || 414
| A.6 || Categories and Direct Sums || 357
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
! colspan="2" | HINTS AND ANSWERS || 361
|-
|-
| 6 || Cramer's Rule || 424
! colspan="2" | BIBLIOGRAPHY || 379
|-
|-
! colspan="2" | Index || 429
! colspan="2" | INDEX || 381
|-
|-
|}
|}


[[Category:Mathematics]]
[[Category:Mathematics]]

Latest revision as of 15:15, 15 February 2023

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Linear Algebra
Shilov Linear Algebra Cover.jpg
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.


This text can be viewed as an introduction to reading mathematics texts; The initial proofs are on arithmetic and properties of linear equations, and are more approachable than the manipulations of functions typical of analysis texts. It is difficult to overstate the universal importance of this subject, but it can be seen through reading. Linear algebra is the basis of all quantities of interest in physics, geometry, number theory, and the same techniques appear in engineering disciplines through physics, numerical computing, or machine learning.

Table of Contents

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