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 \(n\) || 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 \(K_n\) 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
|-
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
|-
| 6.2 || Algebras. The Algebra of Polynomials || 136
|-
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
|-
| 6.4 || Elementary Divisors || 147
|-
| 6.5 || Further Implications || 153
|-
| 6.6 || The Real Jordan Canonical Form || 155
|-
| 6.7 || Spectra, Jets and Polynomials || 160
|-
| 6.8 || Operator Functions and Their Matrices || 169
|-
| || Problems || 176
|-
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
|-
| 7.1 || Bilinear Forms || 179
|-
| 7.2 || Quadratic Forms || 183
|-
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
|-
| 7.4 || The Canonical Basis of a Bilinear Form || 183
|-
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
|-
| 7.6 || Adjoint Linear Operators || 183
|-
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
|-
| *7.8 || Multilinear Forms || 183
|-
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
|-
| || Problems || 210
|-
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
|-
| 8.1 || Introduction || 214
|-
| 8.2 || Definition of a Euclidean Space || 215
|-
| 8.3 || Basic Metric Concepts || 216
|-
| 8.4 || Orthogonal Bases || 222
|-
| 8.5 || Perpendiculars || 223
|-
| 8.6 || The Orthogonalization Theorem || 226
|-
| 8.7 || The Gram Determinant || 230
|-
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
|-
| 8.9 || Adjoint Operators and Isometry || 237
|-
| || Problems || 241
|-
! colspan="3" | Chapter 9: UNITARY SPACES
|-
| 9.1 || Hermitian Forms || 247
|-
| 9.2 || The Scalar Product in a Complex Space || 254
|-
| 9.3 || Normal Operators || 259
|-
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
|-
| || Problems || 271
|-
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
|-
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
|-
| 10.2 || Extremal Properties of a Quadratic Form || 276
|-
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
|-
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
|-
| 10.5 || Geometric Properties of a Quadric Surface || 289
|-
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
|-
| 10.7 || Hermitian Quadratic Forms || 308
|-
| || Problems || 310
|-
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
|-
| 11.1 || More on Algebras || 312
|-
| 11.2 || Representations of Abstract Algebras || 313
|-
| 11.3 || Irreducible Representations and Schur's Lemma || 314
|-
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
|-
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
|-
| 11.6 || Structure of Simple Algebras || 320
|-
| 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]]
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