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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Linear Algebra
|title=Linear Algebra
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|isbn13=978-0486635187
|isbn13=978-0486635187
}}
}}
The textbook '''''Linear Algebra''''' by [https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov] provides a thorough introduction to linear algebra.
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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.
 
 
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.
 
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 Plücker 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 ==
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! 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 27: Line 36:
| 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 \(n\) || 5
| 1.3 || Determinants of Order <math>n</math> || 5
|-
|-
| 1.4 || Properties of Determinants || 8
| 1.4 || Properties of Determinants || 8
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| || 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
|-
|-
| 1 || Equations in two unknowns || 53
! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT
|-
|-
| 2 || Equations in three unknowns || 57
| 4.1 || Linear Forms || 75
|-
|-
! colspan="3" | Chapter 3: Real Numbers
| 4.2 || Linear Operators || 77
|-
|-
| 1 || Addition and multiplication || 61
| 4.3 || Sums and Products of Linear Operators || 82
|-
|-
| 2 || Real numbers: positivity || 64
| 4.4 || Corresponding Operations on Matrices || 84
|-
|-
| 3 || Powers and roots || 70
| 4.5 || Further Properties of Matrix Multiplication || 88
|-
|-
| 4 || Inequalities || 75
| 4.6 || The Range and Null Space of a Linear Operator || 93
|-
|-
! colspan="3" | Chapter 4: Quadratic Equations
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
|-
|-
! colspan="3" | Interlude: On Logic and Mathematical Expressions
| 4.8 || Invariant Subspaces || 106
|-
|-
| 1 || On reading books || 93
| 4.9 || Eigenvectors and Eigenvalues || 108
|-
|-
| 2 || Logic || 94
| || Problems|| 113
|-
|-
| 3 || Sets and elements || 99
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
|-
|-
| 4 || Notation || 100
| 5.1 || Transformation to a New Basis || 118
|-
|-
! colspan="3" | PART II: INTUITIVE GEOMETRY
| 5.2 || Consecutive Transformations || 120
|-
|-
! colspan="3" | Chapter 5: Distance and Angles
| 5.3 || Transformation of the Components of a Vector  || 121
|-
|-
| 1 || Distance || 107
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
|-
|-
| 2 || Angles || 110
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
|-
|-
| 3 || The Pythagoras theorem || 120
| *5.6 || Tensors || 126
|-
|-
! colspan="3" | Chapter 6: Isometries
| || Problems || 131
|-
|-
| 1 || Some standard mappings of the plane || 133
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
|-
|-
| 2 || Isometries || 143
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
|-
|-
| 3 || Composition of isometries || 150
| 6.2 || Algebras. The Algebra of Polynomials || 136
|-
|-
| 4 || Inverse of isometries || 155
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
|-
|-
| 5 || Characterization of isometries || 163
| 6.4 || Elementary Divisors || 147
|-
|-
| 6 || Congruences || 166
| 6.5 || Further Implications || 153
|-
|-
! colspan="3" | Chapter 7: Area and Applications
| 6.6 || The Real Jordan Canonical Form || 155
|-
|-
| 1 || Area of a disc of radius ''r'' || 173
| 6.7 || Spectra, Jets and Polynomials || 160
|-
|-
| 2 || Circumference of a circle of radius ''r'' || 180
| 6.8 || Operator Functions and Their Matrices || 169
|-
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
| || Problems || 176
|-
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
|-
|-
| 1 || Coordinate systems || 191
| 7.1 || Bilinear Forms || 179
|-
|-
| 2 || Distance between points || 197
| 7.2 || Quadratic Forms || 183
|-
|-
| 3 || Equation of a circle || 203
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
|-
|-
| 4 || Rational points on a circle || 206
| 7.4 || The Canonical Basis of a Bilinear Form || 183
|-
|-
! colspan="3" | Chapter 9: Operations on Points
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
|-
|-
| 1 || Dilations and reflections || 213
| 7.6 || Adjoint Linear Operators || 183
|-
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
|-
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
| *7.8 || Multilinear Forms || 183
|-
|-
| 1 || Segments || 229
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
|-
|-
| 2 || Rays || 231
| || Problems || 210
|-
|-
| 3 || Lines || 236
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
|-
|-
| 4 || Ordinary equation for a line || 246
| 8.1 || Introduction || 214
|-
|-
! colspan="3" | Chapter 11: Trigonometry
| 8.2 || Definition of a Euclidean Space || 215
|-
|-
| 1 || Radian measure || 249
| 8.3 || Basic Metric Concepts || 216
|-
|-
| 2 || Sine and cosine || 252
| 8.4 || Orthogonal Bases || 222
|-
|-
| 3 || The graphs || 264
| 8.5 || Perpendiculars || 223
|-
|-
| 4 || The tangent || 266
| 8.6 || The Orthogonalization Theorem || 226
|-
|-
| 5 || Addition formulas || 272
| 8.7 || The Gram Determinant || 230
|-
|-
| 6 || Rotations || 277
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
|-
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
| 8.9 || Adjoint Operators and Isometry || 237
|-
|-
| 1 || The straight line again || 281
| || Problems || 241
|-
|-
| 2 || The parabola || 291
! colspan="3" | Chapter 9: UNITARY SPACES
|-
|-
| 3 || The ellipse || 297
| 9.1 || Hermitian Forms || 247
|-
|-
| 4 || The hyperbola || 300
| 9.2 || The Scalar Product in a Complex Space || 254
|-
|-
| 5 || Rotation of hyperbolas || 305
| 9.3 || Normal Operators || 259
|-
|-
! colspan="3" | PART IV: MISCELLANEOUS
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
|-
|-
! colspan="3" | Chapter 13: Functions
| || Problems || 271
|-
|-
| 1 || Definition of a function || 313
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
|-
|-
| 2 || Polynomial functions || 318
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
|-
|-
| 3 || Graphs of functions || 330
| 10.2 || Extremal Properties of a Quadratic Form || 276
|-
|-
| 4 || Exponential function || 333
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
|-
|-
| 5 || Logarithms || 338
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
|-
|-
! colspan="3" | Chapter 14: Mappings
| 10.5 || Geometric Properties of a Quadric Surface || 289
|-
|-
| 1 || Definition || 345
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
|-
|-
| 2 || Formalism of mappings || 351
| 10.7 || Hermitian Quadratic Forms || 308
|-
|-
| 3 || Permutations || 359
| || Problems || 310
|-
|-
! colspan="3" | Chapter 15: Complex Numbers
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
|-
|-
| 1 || The complex plane || 375
| 11.1 || More on Algebras || 312
|-
|-
| 2 || Polar form || 380
| 11.2 || Representations of Abstract Algebras || 313
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
| 11.3 || Irreducible Representations and Schur's Lemma || 314
|-
|-
| 1 || Induction || 383
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
|-
|-
| 2 || Summations || 388
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
|-
|-
| 3 || Geometric series || 396
| 11.6 || Structure of Simple Algebras || 320
|-
|-
! colspan="3" | Chapter 17: Determinants
| 11.7 || Structure of Semisimple Algebras || 323
|-
|-
| 1 || Matrices || 401
| 11.8 || Representations of Simple and Semisimple Algebras || 327
|-
|-
| 2 || Determinants of order 2 || 406
| 11.9 || Some Further Results || 331
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| || Problems || 332
|-
|-
| 4 || Determinants of order 3 || 414
| *Appendix || ||  
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
|-
|-
| 6 || Cramer's Rule || 424
| A.1 || Introduction || 335
|-
|-
! colspan="2" | Index || 429
| 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
|-
! colspan="2" | HINTS AND ANSWERS || 361
|-
! colspan="2" | BIBLIOGRAPHY || 379
|-
! colspan="2" | INDEX || 381
|-
|-
|}
|}


[[Category:Mathematics]]
[[Category:Mathematics]]
{{Stub}}
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