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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Linear Algebra
|title=Basic Mathematics
|image=[[File:Shilov Linear Algebra Cover.jpg]]
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|language=English
|language=English
|series=
|series=
|genre=
|genre=
|publisher=Dover Publications
|publisher=Springer
|publicationdate=1 June 1977
|publicationdate=1 July 1988
|pages=400
|pages=496
|isbn10=048663518X
|isbn10=0387967877
|isbn13=978-0486635187
|isbn13=978-0387967875
}}
{{NavContainerFlex
|content=
{{NavButton|link=[[Read#Basic_Mathematics|Read]]}}
}}
}}
The textbook '''''Basic Mathematics''''' by [https://en.wikipedia.org/wiki/Serge_Lang Serge Lang] provides an overview of mathematical topics usually encountered through the end of high school/secondary school, specifically arithmetic, algebra, trigonometry, logic, and geometry. It serves as a solid review no matter how far along one may be in their studies, be it just beginning or returning to strengthen one's foundations.


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.
Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.
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 23:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|- Β 
|- Β 
! colspan="3" | Chapter 1: DETERMINANTS
! colspan="3" | PART I: ALGEBRA
|-
|-
| 1.1 || Number Fields || 1
! colspan="3" | Chapter 1: Numbers
|-
|-
| 1.2 || Problems of the Theory of Systems of Linear Equations || 3
| 1 || The integers || 5
|-
|-
| 1.3 || Determinants of Order <math>n</math> || 5
| 2 || Rules for addition || 8
|-
|-
| 1.4 || Properties of Determinants || 8
| 3 || Rules for multiplication || 14
|-
|-
| 1.5 || Cofactors and Minors || 12
| 4 || Even and odd integers; divisibility || 22
|-
|-
| 1.6 || Practical Evaluation of Determinants || 16
| 5 || Rational numbers || 26
|- Β 
|-
| 1.7 || Cramer's Rule || 18
| 6 || Multiplicative inverses || 42
|-
| 1.8 || Minors of Arbitrary Order. Laplace's Theorem || 20
|- Β 
|- Β 
| 1.9 || Multiplicative inverses || 23
! colspan="3" | Chapter 2: Linear Equations
|-
| || Problems || 28
|-
! 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
| 1 || Equations in two unknowns || 53
|-
|-
| 3.1 || More on the Rank of a Matrix || 58
| 2 || Equations in three unknowns || 57
|-
|-
| 3.2 || Nontrivial Compatibility of a Homogeneous Linear System || 60
! colspan="3" | Chapter 3: Real Numbers
|-
|-
| 3.3 || The Compatibility Condition for a General Linear System || 61
| 1 || Addition and multiplication || 61
|-
|-
| 3.4 || The General Solution of a Linear System || 63
| 2 || Real numbers: positivity || 64
|-
|-
| 3.4 || Geometric Properties of the Solution Space || 65
| 3 || Powers and roots || 70
|-
|-
| 3.4 || Methods for Calculating the Rank of a Matrix || 67
| 4 || Inequalities || 75
|-
|-
| || Problems || 71
! colspan="3" | Chapter 4: Quadratic Equations
|-
|-
! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT
! colspan="3" | Interlude: On Logic and Mathematical Expressions
|-
|-
| 4.1 || Linear Forms || 75
| 1 || On reading books || 93
|-
|-
| 4.2 || Linear Operators || 77
| 2 || Logic || 94
|-
|-
| 4.3 || Sums and Products of Linear Operators || 82
| 3 || Sets and elements || 99
|-
|-
| 4.4 || Corresponding Operations on Matrices || 84
| 4 || Notation || 100
|-
|-
| 4.5 || Further Properties of Matrix Multiplication || 88
! colspan="3" | PART II: INTUITIVE GEOMETRY
|-
|-
| 4.6 || The Range and Null Space of a Linear Operator || 93
! colspan="3" | Chapter 5: Distance and Angles
|-
|-
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
| 1 || Distance || 107
|-
|-
| 4.8 || Invariant Subspaces || 106
| 2 || Angles || 110
|-
|-
| 4.9 || Eigenvectors and Eigenvalues || 108
| 3 || The Pythagoras theorem || 120
|-
|-
| || Problems|| 113
! colspan="3" | Chapter 6: Isometries
|-
|-
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
| 1 || Some standard mappings of the plane || 133
|-
|-
| 5.1 || Transformation to a New Basis || 118
| 2 || Isometries || 143
|-
|-
| 5.2 || Consecutive Transformations || 120
| 3 || Composition of isometries || 150
|-
|-
| 5.3 || Transformation of the Components of a VectorΒ  || 121
| 4 || Inverse of isometries || 155
|-
|-
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
| 5 || Characterization of isometries || 163
|-
|-
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
| 6 || Congruences || 166
|-
|-
| *5.6 || Tensors || 126
! colspan="3" | Chapter 7: Area and Applications
|-
|-
| || Problems || 131
| 1 || Area of a disc of radius ''r'' || 173
|-
|-
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
| 2 || Circumference of a circle of radius ''r'' || 180
|-
|-
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
! colspan="3" | PART III: COORDINATE GEOMETRY
|-
|-
| 6.2 || Algebras. The Algebra of Polynomials || 136
! colspan="3" | Chapter 8: Coordinates and Geometry
|-
|-
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
| 1 || Coordinate systems || 191
|-
|-
| 6.4 || Elementary Divisors || 147
| 2 || Distance between points || 197
|-
|-
| 6.5 || Further Implications || 153
| 3 || Equation of a circle || 203
|-
|-
| 6.6 || The Real Jordan Canonical Form || 155
| 4 || Rational points on a circle || 206
|-
|-
| 6.7 || Spectra, Jets and Polynomials || 160
! colspan="3" | Chapter 9: Operations on Points
|-
|-
| 6.8 || Operator Functions and Their Matrices || 169
| 1 || Dilations and reflections || 213
|-
|-
| || Problems || 176
| 2 || Addition, subtraction, and the parallelogram law || 218
|-
|-
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
|-
| 7.1 || Bilinear Forms || 179
| 1 || Segments || 229
|-
|-
| 7.2 || Quadratic Forms || 183
| 2 || Rays || 231
|-
|-
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
| 3 || Lines || 236
|-
|-
| 7.4 || The Canonical Basis of a Bilinear Form || 183
| 4 || Ordinary equation for a line || 246
|-
|-
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
! colspan="3" | Chapter 11: Trigonometry
|-
|-
| 7.6 || Adjoint Linear Operators || 183
| 1 || Radian measure || 249
|-
|-
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
| 2 || Sine and cosine || 252
|-
|-
| *7.8 || Multilinear Forms || 183
| 3 || The graphs || 264
|-
|-
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
| 4 || The tangent || 266
|-
|-
| || Problems || 210
| 5 || Addition formulas || 272
|-
|-
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
| 6 || Rotations || 277
|-
|-
| 8.1 || Introduction || 214
! colspan="3" | Chapter 12: Some Analytic Geometry
|-
|-
| 8.2 || Definition of a Euclidean Space || 215
| 1 || The straight line again || 281
|-
|-
| 8.3 || Basic Metric Concepts || 216
| 2 || The parabola || 291
|-
|-
| 8.4 || Orthogonal Bases || 222
| 3 || The ellipse || 297
|-
|-
| 8.5 || Perpendiculars || 223
| 4 || The hyperbola || 300
|-
|-
| 8.6 || The Orthogonalization Theorem || 226
| 5 || Rotation of hyperbolas || 305
|-
|-
| 8.7 || The Gram Determinant || 230
! colspan="3" | PART IV: MISCELLANEOUS
|-
|-
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
! colspan="3" | Chapter 13: Functions
|-
|-
| 8.9 || Adjoint Operators and Isometry || 237
| 1 || Definition of a function || 313
|-
|-
| || Problems || 241
| 2 || Polynomial functions || 318
|-
|-
! colspan="3" | Chapter 9: UNITARY SPACES
| 3 || Graphs of functions || 330
|-
|-
| 9.1 || Hermitian Forms || 247
| 4 || Exponential function || 333
|-
|-
| 9.2 || The Scalar Product in a Complex Space || 254
| 5 || Logarithms || 338
|-
|-
| 9.3 || Normal Operators || 259
! colspan="3" | Chapter 14: Mappings
|-
|-
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
| 1 || Definition || 345
|-
|-
| || Problems || 271
| 2 || Formalism of mappings || 351
|-
|-
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
| 3 || Permutations || 359
|-
|-
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
! colspan="3" | Chapter 15: Complex Numbers
|-
|-
| 10.2 || Extremal Properties of a Quadratic Form || 276
| 1 || The complex plane || 375
|-
|-
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
| 2 || Polar form || 380
|-
|-
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
! colspan="3" | Chapter 16: Induction and Summations
|-
|-
| 10.5 || Geometric Properties of a Quadric Surface || 289
| 1 || Induction || 383
|-
|-
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
| 2 || Summations || 388
|-
|-
| 10.7 || Hermitian Quadratic Forms || 308
| 3 || Geometric series || 396
|-
|-
| || Problems || 310
! colspan="3" | Chapter 17: Determinants
|-
|-
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
| 1 || Matrices || 401
|-
|-
| 11.1 || More on Algebras || 312
| 2 || Determinants of order 2 || 406
|-
|-
| 11.2 || Representations of Abstract Algebras || 313
| 3 || Properties of 2 x 2 determinants || 409
|-
|-
| 11.3 || Irreducible Representations and Schur's Lemma || 314
| 4 || Determinants of order 3 || 414
|-
|-
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
| 5 || Properties of 3 x 3 determinants || 418
|-
|-
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
| 6 || Cramer's Rule || 424
|-
|-
| 11.6 || Structure of Simple Algebras || 320
! colspan="2" | Index || 429
|-
| 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 || ||
|-
! colspan="3" | 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
|-
! colspan="2" | HINTS AND ANSWERS || 361
|-
! colspan="2" | BIBLIOGRAPHY || 379
|-
! colspan="2" | INDEX || 381
|-
|-
|}
|}


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