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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Basic Mathematics
|title=Linear Algebra
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|image=[[File:Shilov Linear Algebra Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|language=English
|language=English
|series=
|series=
|genre=
|genre=
|publisher=Springer
|publisher=Dover Publications
|publicationdate=1 July 1988
|publicationdate=1 June 1977
|pages=496
|pages=400
|isbn10=0387967877
|isbn10=048663518X
|isbn13=978-0387967875
|isbn13=978-0486635187
}}
{{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.


Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.
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 ==
Line 23: Line 30:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|-  
|-  
! colspan="3" | PART I: ALGEBRA
! colspan="3" | 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 <math>n</math> || 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
|-
! colspan="3" | Chapter 2: LINEAR SPACES
|-
| 2.1 || Definitions || 31
|-
| 2.2 || Linear Dependence || 36
|-
|-
! colspan="3" | Chapter 1: Numbers
| 2.3 || Bases, Components, Dimension || 38
|-
|-
| 1 || The integers || 5
| 2.4 || Subspaces || 42
|-
|-
| 2 || Rules for addition || 8
| 2.5 || Linear Manifolds || 49
|-
|-
| 3 || Rules for multiplication || 14
| 2.6 || Hyperplanes || 51
|-
|-
| 4 || Even and odd integers; divisibility || 22
| 2.7 || Morphisms of Linear Spaces || 53
|-
|-
| 5 || Rational numbers || 26
| || Problems || 56
|-
|-
| 6 || Multiplicative inverses || 42
! colspan="3" | Chapter 3: SYSTEMS OF LINEAR EQUATIONS
|-  
|-
! colspan="3" | Chapter 2: 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
|-
! colspan="3" | 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
|-
|-
| 1 || Equations in two unknowns || 53
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
|-
|-
| 2 || Equations in three unknowns || 57
| 4.8 || Invariant Subspaces || 106
|-
|-
! colspan="3" | Chapter 3: Real Numbers
| 4.9 || Eigenvectors and Eigenvalues || 108
|-
|-
| 1 || Addition and multiplication || 61
| || Problems|| 113
|-
|-
| 2 || Real numbers: positivity || 64
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
|-
|-
| 3 || Powers and roots || 70
| 5.1 || Transformation to a New Basis || 118
|-
|-
| 4 || Inequalities || 75
| 5.2 || Consecutive Transformations || 120
|-
|-
! colspan="3" | Chapter 4: Quadratic Equations
| 5.3 || Transformation of the Components of a Vector  || 121
|-
|-
! colspan="3" | Interlude: On Logic and Mathematical Expressions
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
|-
|-
| 1 || On reading books || 93
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
|-
|-
| 2 || Logic || 94
| *5.6 || Tensors || 126
|-
|-
| 3 || Sets and elements || 99
| || Problems || 131
|-
|-
| 4 || Notation || 100
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
|-
|-
! colspan="3" | PART II: INTUITIVE GEOMETRY
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
|-
|-
! colspan="3" | Chapter 5: Distance and Angles
| 6.2 || Algebras. The Algebra of Polynomials || 136
|-
|-
| 1 || Distance || 107
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
|-
|-
| 2 || Angles || 110
| 6.4 || Elementary Divisors || 147
|-
|-
| 3 || The Pythagoras theorem || 120
| 6.5 || Further Implications || 153
|-
|-
! colspan="3" | Chapter 6: Isometries
| 6.6 || The Real Jordan Canonical Form || 155
|-
|-
| 1 || Some standard mappings of the plane || 133
| 6.7 || Spectra, Jets and Polynomials || 160
|-
|-
| 2 || Isometries || 143
| 6.8 || Operator Functions and Their Matrices || 169
|-
|-
| 3 || Composition of isometries || 150
| || Problems || 176
|-
|-
| 4 || Inverse of isometries || 155
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
|-
|-
| 5 || Characterization of isometries || 163
| 7.1 || Bilinear Forms || 179
|-
|-
| 6 || Congruences || 166
| 7.2 || Quadratic Forms || 183
|-
|-
! colspan="3" | Chapter 7: Area and Applications
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
|-
|-
| 1 || Area of a disc of radius ''r'' || 173
| 7.4 || The Canonical Basis of a Bilinear Form || 183
|-
|-
| 2 || Circumference of a circle of radius ''r'' || 180
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
|-
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
| 7.6 || Adjoint Linear Operators || 183
|-
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
|-
|-
| 1 || Coordinate systems || 191
| *7.8 || Multilinear Forms || 183
|-
|-
| 2 || Distance between points || 197
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
|-
|-
| 3 || Equation of a circle || 203
| || Problems || 210
|-
|-
| 4 || Rational points on a circle || 206
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
|-
|-
! colspan="3" | Chapter 9: Operations on Points
| 8.1 || Introduction || 214
|-
|-
| 1 || Dilations and reflections || 213
| 8.2 || Definition of a Euclidean Space || 215
|-
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
| 8.3 || Basic Metric Concepts || 216
|-
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
| 8.4 || Orthogonal Bases || 222
|-
|-
| 1 || Segments || 229
| 8.5 || Perpendiculars || 223
|-
|-
| 2 || Rays || 231
| 8.6 || The Orthogonalization Theorem || 226
|-
|-
| 3 || Lines || 236
| 8.7 || The Gram Determinant || 230
|-
|-
| 4 || Ordinary equation for a line || 246
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
|-
|-
! colspan="3" | Chapter 11: Trigonometry
| 8.9 || Adjoint Operators and Isometry || 237
|-
|-
| 1 || Radian measure || 249
| || Problems || 241
|-
|-
| 2 || Sine and cosine || 252
! colspan="3" | Chapter 9: UNITARY SPACES
|-
|-
| 3 || The graphs || 264
| 9.1 || Hermitian Forms || 247
|-
|-
| 4 || The tangent || 266
| 9.2 || The Scalar Product in a Complex Space || 254
|-
|-
| 5 || Addition formulas || 272
| 9.3 || Normal Operators || 259
|-
|-
| 6 || Rotations || 277
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
|-
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
| || Problems || 271
|-
|-
| 1 || The straight line again || 281
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
|-
|-
| 2 || The parabola || 291
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
|-
|-
| 3 || The ellipse || 297
| 10.2 || Extremal Properties of a Quadratic Form || 276
|-
|-
| 4 || The hyperbola || 300
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
|-
|-
| 5 || Rotation of hyperbolas || 305
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
|-
|-
! colspan="3" | PART IV: MISCELLANEOUS
| 10.5 || Geometric Properties of a Quadric Surface || 289
|-
|-
! colspan="3" | Chapter 13: Functions
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
|-
|-
| 1 || Definition of a function || 313
| 10.7 || Hermitian Quadratic Forms || 308
|-
|-
| 2 || Polynomial functions || 318
| || Problems || 310
|-
|-
| 3 || Graphs of functions || 330
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
|-
|-
| 4 || Exponential function || 333
| 11.1 || More on Algebras || 312
|-
|-
| 5 || Logarithms || 338
| 11.2 || Representations of Abstract Algebras || 313
|-
|-
! colspan="3" | Chapter 14: Mappings
| 11.3 || Irreducible Representations and Schur's Lemma || 314
|-
|-
| 1 || Definition || 345
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
|-
|-
| 2 || Formalism of mappings || 351
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
|-
|-
| 3 || Permutations || 359
| 11.6 || Structure of Simple Algebras || 320
|-
|-
! colspan="3" | Chapter 15: Complex Numbers
| 11.7 || Structure of Semisimple Algebras || 323
|-
|-
| 1 || The complex plane || 375
| 11.8 || Representations of Simple and Semisimple Algebras || 327
|-
|-
| 2 || Polar form || 380
| 11.9 || Some Further Results || 331
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
| || Problems || 332
|-
|-
| 1 || Induction || 383
| *Appendix || ||  
|-
|-
| 2 || Summations || 388
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
|-
|-
| 3 || Geometric series || 396
| A.1 || Introduction || 335
|-
|-
! colspan="3" | Chapter 17: Determinants
| A.2 || The Case of Complete Algebras || 338
|-
|-
| 1 || Matrices || 401
| A.3 || The Case of One-Dimensional Algebras || 340
|-
|-
| 2 || Determinants of order 2 || 406
| A.4 || The Case of Simple Algebras || 345
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| A.5 || The Case of Complete Algebras of Diagonal Matrices || 353
|-
|-
| 4 || Determinants of order 3 || 414
| A.6 || Categories and Direct Sums || 357
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
! colspan="2" | HINTS AND ANSWERS || 361
|-
|-
| 6 || Cramer's Rule || 424
! colspan="2" | BIBLIOGRAPHY || 379
|-
|-
! colspan="2" | Index || 429
! colspan="2" | INDEX || 381
|-
|-
|}
|}


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