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{{InfoboxBook
{{InfoboxBook
|title=Linear Algebra
|title=Linear Algebra
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|isbn13=978-0486635187
|isbn13=978-0486635187
}}
}}
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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.
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{{NavButton|link=[[Read#Basic_Mathematics|Read]]}}
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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.
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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 ==
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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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| 4.6 || The Range and Null Space of a Linear Operator || 93
| 4.6 || The Range and Null Space of a Linear Operator || 93
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|-
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
| 4.7 || Linear Operators Mapping a Space \(K_n\) into Itself || 98
|-
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| 4.8 || Invariant Subspaces || 106
| 4.8 || Invariant Subspaces || 106
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! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
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| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
| 1 || Some standard mappings of the plane || 133
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| 6.2 || Algebras. The Algebra of Polynomials || 136
| 2 || Isometries || 143
|-
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| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
| 3 || Composition of isometries || 150
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| 6.4 || Elementary Divisors || 147
| 4 || Inverse of isometries || 155
|-
|-
| 6.5 || Further Implications || 153
| 5 || Characterization of isometries || 163
|-
|-
| 6.6 || The Real Jordan Canonical Form || 155
| 6 || Congruences || 166
|-
|-
| 6.7 || Spectra, Jets and Polynomials || 160
! colspan="3" | Chapter 7: Area and Applications
|-
|-
| 6.8 || Operator Functions and Their Matrices || 169
| 1 || Area of a disc of radius ''r'' || 173
|-
|-
| || Problems || 176
| 2 || Circumference of a circle of radius ''r'' || 180
|-
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! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
! colspan="3" | PART III: COORDINATE GEOMETRY
|-
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| 7.1 || Bilinear Forms || 179
! colspan="3" | Chapter 8: Coordinates and Geometry
|-
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| 7.2 || Quadratic Forms || 183
| 1 || Coordinate systems || 191
|-
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| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
| 2 || Distance between points || 197
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| 7.4 || The Canonical Basis of a Bilinear Form || 183
| 3 || Equation of a circle || 203
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| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
| 4 || Rational points on a circle || 206
|-
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| 7.6 || Adjoint Linear Operators || 183
! colspan="3" | Chapter 9: Operations on Points
|-
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| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
| 1 || Dilations and reflections || 213
|-
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| *7.8 || Multilinear Forms || 183
| 2 || Addition, subtraction, and the parallelogram law || 218
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| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
|-
| || Problems || 210
| 1 || Segments || 229
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! colspan="3" | Chapter 8: EUCLIDEAN SPACES
| 2 || Rays || 231
|-
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| 8.1 || Introduction || 214
| 3 || Lines || 236
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| 8.2 || Definition of a Euclidean Space || 215
| 4 || Ordinary equation for a line || 246
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| 8.3 || Basic Metric Concepts || 216
! colspan="3" | Chapter 11: Trigonometry
|-
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| 8.4 || Orthogonal Bases || 222
| 1 || Radian measure || 249
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| 8.5 || Perpendiculars || 223
| 2 || Sine and cosine || 252
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| 8.6 || The Orthogonalization Theorem || 226
| 3 || The graphs || 264
|-
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| 8.7 || The Gram Determinant || 230
| 4 || The tangent || 266
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| 8.8 || Incompatible Systems and the Method of Least Squares || 234
| 5 || Addition formulas || 272
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| 8.9 || Adjoint Operators and Isometry || 237
| 6 || Rotations || 277
|-
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| || Problems || 241
! colspan="3" | Chapter 12: Some Analytic Geometry
|-
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! colspan="3" | Chapter 9: UNITARY SPACES
| 1 || The straight line again || 281
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| 9.1 || Hermitian Forms || 247
| 2 || The parabola || 291
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| 9.2 || The Scalar Product in a Complex Space || 254
| 3 || The ellipse || 297
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| 9.3 || Normal Operators || 259
| 4 || The hyperbola || 300
|-
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| 9.4 || Applications to Operator Theory in Euclidean Space || 263
| 5 || Rotation of hyperbolas || 305
|-
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| || Problems || 271
! colspan="3" | PART IV: MISCELLANEOUS
|-
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! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
! colspan="3" | Chapter 13: Functions
|-
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| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
| 1 || Definition of a function || 313
|-
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| 10.2 || Extremal Properties of a Quadratic Form || 276
| 2 || Polynomial functions || 318
|-
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| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
| 3 || Graphs of functions || 330
|-
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| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
| 4 || Exponential function || 333
|-
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| 10.5 || Geometric Properties of a Quadric Surface || 289
| 5 || Logarithms || 338
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| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
! colspan="3" | Chapter 14: Mappings
|-
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| 10.7 || Hermitian Quadratic Forms || 308
| 1 || Definition || 345
|-
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| || Problems || 310
| 2 || Formalism of mappings || 351
|-
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! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
| 3 || Permutations || 359
|-
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| 11.1 || More on Algebras || 312
! colspan="3" | Chapter 15: Complex Numbers
|-
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| 11.2 || Representations of Abstract Algebras || 313
| 1 || The complex plane || 375
|-
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| 11.3 || Irreducible Representations and Schur's Lemma || 314
| 2 || Polar form || 380
|-
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| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
! colspan="3" | Chapter 16: Induction and Summations
|-
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| 11.5 || The Left Regular Representation of a Simple Algebra || 318
| 1 || Induction || 383
|-
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| 11.6 || Structure of Simple Algebras || 320
| 2 || Summations || 388
|-
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| 11.7 || Structure of Semisimple Algebras || 323
| 3 || Geometric series || 396
|-
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| 11.8 || Representations of Simple and Semisimple Algebras || 327
! colspan="3" | Chapter 17: Determinants
|-
|-
| 11.9 || Some Further Results || 331
| 1 || Matrices || 401
|-
|-
| || Problems || 332
| 2 || Determinants of order 2 || 406
|-
|-
| *Appendix || || Β 
| 3 || Properties of 2 x 2 determinants || 409
|-
|-
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
| 4 || Determinants of order 3 || 414
|-
|-
| A.1 || Introduction || 335
| 5 || Properties of 3 x 3 determinants || 418
|-
|-
| A.2 || The Case of Complete Algebras || 338
| 6 || Cramer's Rule || 424
|-
|-
| A.3 || The Case of One-Dimensional Algebras || 340
! colspan="2" | Index || 429
|-
| 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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