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{{InfoboxBook
{{InfoboxBook
|title=Linear Algebra
|title=Linear Algebra
|image=[[File:Shilov Linear Algebra Cover.jpg]]
|image=[[Shilov Linear Algebra Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|language=English
|language=English
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|isbn13=978-0486635187
|isbn13=978-0486635187
}}
}}
{{NavContainerFlex
The textbook '''''Linear Algebra''''' by [https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov] provides a thorough introduction to linear algebra.
|content=
{{NavButton|link=[[Read#Basic_Mathematics|Read]]}}
}}
Β 
The textbook [https://cosmathclub.files.wordpress.com/2014/10/georgi-shilov-linear-algebra4.pdf '''''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.
Β 
Β 
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.
Β 
Another similarly famous Russian text on linear algebra would be [https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/manin.pdf Kostrikin and Manin's Linear Algebra and Geometry] but better as a second casual read since it is more advanced. It contains more information about the interaction between linear algebra and quantum mechanics and relativity, and classcial projective geometry such as the Hopf fibration and PlΓΌcker coordinates. Projective geometries are treated concretely here, but are eventually characterized much more abstractly and generally in algebraic topology.


== Table of Contents ==
== Table of Contents ==
Line 30: Line 21:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|- Β 
|- Β 
! colspan="3" | Chapter 1: DETERMINANTS
! colspan="3" | Chapter 1: Determinants
|-
|-
| 1.1 || Number Fields || 1
| 1.1 || Number Fields || 1
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| 1.2 || Problems of the Theory of Systems of Linear Equations || 3
| 1.2 || Problems of the Theory of Systems of Linear Equations || 3
|-
|-
| 1.3 || Determinants of Order <math>n</math> || 5
| 1.3 || Determinants of Order \(n\) || 5
|-
|-
| 1.4 || Properties of Determinants || 8
| 1.4 || Properties of Determinants || 8
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| || Problems || 28
| || Problems || 28
|- Β 
|- Β 
! colspan="3" | Chapter 2: LINEAR SPACES
! colspan="3" | Chapter 2: Linear Equations
|-
| 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
|-
|Β  || Problems || 71
|-
|-
! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT
| 1 || Equations in two unknowns || 53
|-
|-
| 4.1 || Linear Forms || 75
| 2 || Equations in three unknowns || 57
|-
|-
| 4.2 || Linear Operators || 77
! colspan="3" | Chapter 3: Real Numbers
|-
|-
| 4.3 || Sums and Products of Linear Operators || 82
| 1 || Addition and multiplication || 61
|-
|-
| 4.4 || Corresponding Operations on Matrices || 84
| 2 || Real numbers: positivity || 64
|-
|-
| 4.5 || Further Properties of Matrix Multiplication || 88
| 3 || Powers and roots || 70
|-
|-
| 4.6 || The Range and Null Space of a Linear Operator || 93
| 4 || Inequalities || 75
|-
|-
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
! colspan="3" | Chapter 4: Quadratic Equations
|-
|-
| 4.8 || Invariant Subspaces || 106
! colspan="3" | Interlude: On Logic and Mathematical Expressions
|-
|-
| 4.9 || Eigenvectors and Eigenvalues || 108
| 1 || On reading books || 93
|-
|-
| || Problems|| 113
| 2 || Logic || 94
|-
|-
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
| 3 || Sets and elements || 99
|-
|-
| 5.1 || Transformation to a New Basis || 118
| 4 || Notation || 100
|-
|-
| 5.2 || Consecutive Transformations || 120
! colspan="3" | PART II: INTUITIVE GEOMETRY
|-
|-
| 5.3 || Transformation of the Components of a VectorΒ  || 121
! colspan="3" | Chapter 5: Distance and Angles
|-
|-
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
| 1 || Distance || 107
|-
|-
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
| 2 || Angles || 110
|-
|-
| *5.6 || Tensors || 126
| 3 || The Pythagoras theorem || 120
|-
|-
| || Problems || 131
! colspan="3" | Chapter 6: Isometries
|-
|-
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
| 1 || Some standard mappings of the plane || 133
|-
|-
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
| 2 || Isometries || 143
|-
|-
| 6.2 || Algebras. The Algebra of Polynomials || 136
| 3 || Composition of isometries || 150
|-
|-
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
| 4 || Inverse of isometries || 155
|-
|-
| 6.4 || Elementary Divisors || 147
| 5 || Characterization of isometries || 163
|-
|-
| 6.5 || Further Implications || 153
| 6 || Congruences || 166
|-
|-
| 6.6 || The Real Jordan Canonical Form || 155
! colspan="3" | Chapter 7: Area and Applications
|-
|-
| 6.7 || Spectra, Jets and Polynomials || 160
| 1 || Area of a disc of radius ''r'' || 173
|-
|-
| 6.8 || Operator Functions and Their Matrices || 169
| 2 || Circumference of a circle of radius ''r'' || 180
|-
|-
| || Problems || 176
! colspan="3" | PART III: COORDINATE GEOMETRY
|-
|-
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
! colspan="3" | Chapter 8: Coordinates and Geometry
|-
|-
| 7.1 || Bilinear Forms || 179
| 1 || Coordinate systems || 191
|-
|-
| 7.2 || Quadratic Forms || 183
| 2 || Distance between points || 197
|-
|-
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
| 3 || Equation of a circle || 203
|-
|-
| 7.4 || The Canonical Basis of a Bilinear Form || 183
| 4 || Rational points on a circle || 206
|-
|-
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
! colspan="3" | Chapter 9: Operations on Points
|-
|-
| 7.6 || Adjoint Linear Operators || 183
| 1 || Dilations and reflections || 213
|-
|-
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
| 2 || Addition, subtraction, and the parallelogram law || 218
|-
|-
| *7.8 || Multilinear Forms || 183
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
|-
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
| 1 || Segments || 229
|-
|-
| || Problems || 210
| 2 || Rays || 231
|-
|-
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
| 3 || Lines || 236
|-
|-
| 8.1 || Introduction || 214
| 4 || Ordinary equation for a line || 246
|-
|-
| 8.2 || Definition of a Euclidean Space || 215
! colspan="3" | Chapter 11: Trigonometry
|-
|-
| 8.3 || Basic Metric Concepts || 216
| 1 || Radian measure || 249
|-
|-
| 8.4 || Orthogonal Bases || 222
| 2 || Sine and cosine || 252
|-
|-
| 8.5 || Perpendiculars || 223
| 3 || The graphs || 264
|-
|-
| 8.6 || The Orthogonalization Theorem || 226
| 4 || The tangent || 266
|-
|-
| 8.7 || The Gram Determinant || 230
| 5 || Addition formulas || 272
|-
|-
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
| 6 || Rotations || 277
|-
|-
| 8.9 || Adjoint Operators and Isometry || 237
! colspan="3" | Chapter 12: Some Analytic Geometry
|-
|-
| || Problems || 241
| 1 || The straight line again || 281
|-
|-
! colspan="3" | Chapter 9: UNITARY SPACES
| 2 || The parabola || 291
|-
|-
| 9.1 || Hermitian Forms || 247
| 3 || The ellipse || 297
|-
|-
| 9.2 || The Scalar Product in a Complex Space || 254
| 4 || The hyperbola || 300
|-
|-
| 9.3 || Normal Operators || 259
| 5 || Rotation of hyperbolas || 305
|-
|-
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
! colspan="3" | PART IV: MISCELLANEOUS
|-
|-
| || Problems || 271
! colspan="3" | Chapter 13: Functions
|-
|-
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
| 1 || Definition of a function || 313
|-
|-
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
| 2 || Polynomial functions || 318
|-
|-
| 10.2 || Extremal Properties of a Quadratic Form || 276
| 3 || Graphs of functions || 330
|-
|-
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
| 4 || Exponential function || 333
|-
|-
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
| 5 || Logarithms || 338
|-
|-
| 10.5 || Geometric Properties of a Quadric Surface || 289
! colspan="3" | Chapter 14: Mappings
|-
|-
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
| 1 || Definition || 345
|-
|-
| 10.7 || Hermitian Quadratic Forms || 308
| 2 || Formalism of mappings || 351
|-
|-
| || Problems || 310
| 3 || Permutations || 359
|-
|-
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
! colspan="3" | Chapter 15: Complex Numbers
|-
|-
| 11.1 || More on Algebras || 312
| 1 || The complex plane || 375
|-
|-
| 11.2 || Representations of Abstract Algebras || 313
| 2 || Polar form || 380
|-
|-
| 11.3 || Irreducible Representations and Schur's Lemma || 314
! colspan="3" | Chapter 16: Induction and Summations
|-
|-
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
| 1 || Induction || 383
|-
|-
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
| 2 || Summations || 388
|-
|-
| 11.6 || Structure of Simple Algebras || 320
| 3 || Geometric series || 396
|-
|-
| 11.7 || Structure of Semisimple Algebras || 323
! colspan="3" | Chapter 17: Determinants
|-
|-
| 11.8 || Representations of Simple and Semisimple Algebras || 327
| 1 || Matrices || 401
|-
|-
| 11.9 || Some Further Results || 331
| 2 || Determinants of order 2 || 406
|-
|-
| || Problems || 332
| 3 || Properties of 2 x 2 determinants || 409
|-
|-
| *Appendix || || Β 
| 4 || Determinants of order 3 || 414
|-
|-
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
| 5 || Properties of 3 x 3 determinants || 418
|-
|-
| A.1 || Introduction || 335
| 6 || Cramer's Rule || 424
|-
|-
| A.2 || The Case of Complete Algebras || 338
! colspan="2" | Index || 429
|-
| 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
|-
! colspan="2" | HINTS AND ANSWERS || 361
|-
! colspan="2" | BIBLIOGRAPHY || 379
|-
! colspan="2" | INDEX || 381
|-
|-
|}
|}


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