Editing Linear Algebra (Book)

{{InfoboxBook
|title=Linear Algebra
|image=[[File:Shilov Linear Algebra Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Georgiy_Shilov Georgi Shilov]
|language=English
|series=
|genre=
|publisher=Dover Publications
|publicationdate=1 June 1977
|pages=400
|isbn10=048663518X
|isbn13=978-0486635187
}}
{{NavContainerFlex
|content=
{{NavButton|link=[[Read#Basic_Mathematics|Read]]}}
}}

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 ==

{| class="wikitable"
|-
! Chapter/Section # !! Title !! Page #
|- 
! 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
|-
| 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
|-
! 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
|-
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
|-
| 4.8 || Invariant Subspaces || 106
|-
| 4.9 || Eigenvectors and Eigenvalues || 108
|-
| || Problems|| 113
|-
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
|-
| 5.1 || Transformation to a New Basis || 118
|-
| 5.2 || Consecutive Transformations || 120
|-
| 5.3 || Transformation of the Components of a Vector  || 121
|-
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
|-
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
|-
| *5.6 || Tensors || 126
|-
| || Problems || 131
|-
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
|-
| 1 || Some standard mappings of the plane || 133
|-
| 2 || Isometries || 143
|-
| 3 || Composition of isometries || 150
|-
| 4 || Inverse of isometries || 155
|-
| 5 || Characterization of isometries || 163
|-
| 6 || Congruences || 166
|-
! colspan="3" | Chapter 7: Area and Applications
|-
| 1 || Area of a disc of radius ''r'' || 173
|-
| 2 || Circumference of a circle of radius ''r'' || 180
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
|-
| 1 || Coordinate systems || 191
|-
| 2 || Distance between points || 197
|-
| 3 || Equation of a circle || 203
|-
| 4 || Rational points on a circle || 206
|-
! colspan="3" | Chapter 9: Operations on Points
|-
| 1 || Dilations and reflections || 213
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
| 1 || Segments || 229
|-
| 2 || Rays || 231
|-
| 3 || Lines || 236
|-
| 4 || Ordinary equation for a line || 246
|-
! colspan="3" | Chapter 11: Trigonometry
|-
| 1 || Radian measure || 249
|-
| 2 || Sine and cosine || 252
|-
| 3 || The graphs || 264
|-
| 4 || The tangent || 266
|-
| 5 || Addition formulas || 272
|-
| 6 || Rotations || 277
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
|-
| 1 || The straight line again || 281
|-
| 2 || The parabola || 291
|-
| 3 || The ellipse || 297
|-
| 4 || The hyperbola || 300
|-
| 5 || Rotation of hyperbolas || 305
|-
! colspan="3" | PART IV: MISCELLANEOUS
|-
! colspan="3" | Chapter 13: Functions
|-
| 1 || Definition of a function || 313
|-
| 2 || Polynomial functions || 318
|-
| 3 || Graphs of functions || 330
|-
| 4 || Exponential function || 333
|-
| 5 || Logarithms || 338
|-
! colspan="3" | Chapter 14: Mappings
|-
| 1 || Definition || 345
|-
| 2 || Formalism of mappings || 351
|-
| 3 || Permutations || 359
|-
! colspan="3" | Chapter 15: Complex Numbers
|-
| 1 || The complex plane || 375
|-
| 2 || Polar form || 380
|-
! colspan="3" | Chapter 16: Induction and Summations
|-
| 1 || Induction || 383
|-
| 2 || Summations || 388
|-
| 3 || Geometric series || 396
|-
! colspan="3" | Chapter 17: Determinants
|-
| 1 || Matrices || 401
|-
| 2 || Determinants of order 2 || 406
|-
| 3 || Properties of 2 x 2 determinants || 409
|-
| 4 || Determinants of order 3 || 414
|-
| 5 || Properties of 3 x 3 determinants || 418
|-
| 6 || Cramer's Rule || 424
|-
! colspan="2" | Index || 429
|-
|}

[[Category:Mathematics]]
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