Linear Algebra (Book): Difference between revisions

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! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
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! colspan="3" | Chapter 1: Determinants
! colspan="3" | Chapter 1: DETERMINANTS
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| 1.1 || Number Fields || 1
| 1.1 || Number Fields || 1
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| || Problems || 28
| || Problems || 28
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! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | Chapter 2: LINEAR SPACES
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| 1 || Equations in two unknowns || 53
| 2.1 || Definitions || 31
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| 2 || Equations in three unknowns || 57
| 2.2 || Linear Dependence || 36
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! colspan="3" | Chapter 3: Real Numbers
| 2.3 || Bases, Components, Dimension || 38
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| 1 || Addition and multiplication || 61
| 2.4 || Subspaces || 42
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| 2 || Real numbers: positivity || 64
| 2.5 || Linear Manifolds || 49
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| 3 || Powers and roots || 70
| 2.6 || Hyperplanes || 51
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| 4 || Inequalities || 75
| 2.7 || Morphisms of Linear Spaces || 53
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! colspan="3" | Chapter 4: Quadratic Equations
| || Problems || 56
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! colspan="3" | Interlude: On Logic and Mathematical Expressions
! colspan="3" | Chapter 3: SYSTEMS OF LINEAR EQUATIONS
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| 1 || On reading books || 93
| 3.1 || More on the Rank of a Matrix || 58
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| 2 || Logic || 94
| 3.2 || Nontrivial Compatibility of a Homogeneous Linear System || 60
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| 3 || Sets and elements || 99
| 3.3 || The Compatibility Condition for a General Linear System || 61
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| 4 || Notation || 100
| 3.4 || The General Solution of a Linear System || 63
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! colspan="3" | PART II: INTUITIVE GEOMETRY
| 3.4 || Geometric Properties of the Solution Space || 65
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! colspan="3" | Chapter 5: Distance and Angles
| 3.4 || Methods for Calculating the Rank of a Matrix || 67
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| 1 || Distance || 107
| || Problems || 71
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| 2 || Angles || 110
! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT
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| 3 || The Pythagoras theorem || 120
| 4.1 || Linear Forms || 75
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! colspan="3" | Chapter 6: Isometries
| 4.2 || Linear Operators || 77
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| 4.3 || Sums and Products of Linear Operators || 82
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| 4.4 || Corresponding Operations on Matrices || 84
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| 4.5 || Further Properties of Matrix Multiplication || 88
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| 4.6 || The Range and Null Space of a Linear Operator || 93
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| 4.7 || Linear Operators Mapping a Space \(K_n\) into Itself || 98
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| 4.8 || Invariant Subspaces || 106
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| 4.9 || Eigenvectors and Eigenvalues || 108
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| || Problems|| 113
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! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
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| 5.1 || Transformation to a New Basis || 118
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| 5.2 || Consecutive Transformations || 120
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| 5.3 || Transformation of the Components of a Vector  || 121
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| 5.4 || Transformation of the Coefficients of a Linear Form || 123
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| 5.5 || Transformation of the Matrix of a Linear Operator || 124
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| *5.6 || Tensors || 126
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| || Problems || 131
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! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
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| 1 || Some standard mappings of the plane || 133
| 1 || Some standard mappings of the plane || 133

Revision as of 16:26, 21 September 2021

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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 provides a thorough introduction to linear algebra.

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
1 Some standard mappings of the plane 133
2 Isometries 143
3 Composition of isometries 150
4 Inverse of isometries 155
5 Characterization of isometries 163
6 Congruences 166
Chapter 7: Area and Applications
1 Area of a disc of radius r 173
2 Circumference of a circle of radius r 180
PART III: COORDINATE GEOMETRY
Chapter 8: Coordinates and Geometry
1 Coordinate systems 191
2 Distance between points 197
3 Equation of a circle 203
4 Rational points on a circle 206
Chapter 9: Operations on Points
1 Dilations and reflections 213
2 Addition, subtraction, and the parallelogram law 218
Chapter 10: Segments, Rays, and Lines
1 Segments 229
2 Rays 231
3 Lines 236
4 Ordinary equation for a line 246
Chapter 11: Trigonometry
1 Radian measure 249
2 Sine and cosine 252
3 The graphs 264
4 The tangent 266
5 Addition formulas 272
6 Rotations 277
Chapter 12: Some Analytic Geometry
1 The straight line again 281
2 The parabola 291
3 The ellipse 297
4 The hyperbola 300
5 Rotation of hyperbolas 305
PART IV: MISCELLANEOUS
Chapter 13: Functions
1 Definition of a function 313
2 Polynomial functions 318
3 Graphs of functions 330
4 Exponential function 333
5 Logarithms 338
Chapter 14: Mappings
1 Definition 345
2 Formalism of mappings 351
3 Permutations 359
Chapter 15: Complex Numbers
1 The complex plane 375
2 Polar form 380
Chapter 16: Induction and Summations
1 Induction 383
2 Summations 388
3 Geometric series 396
Chapter 17: Determinants
1 Matrices 401
2 Determinants of order 2 406
3 Properties of 2 x 2 determinants 409
4 Determinants of order 3 414
5 Properties of 3 x 3 determinants 418
6 Cramer's Rule 424
Index 429
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