Linear Algebra (Book)

The textbook Linear Algebra by Georgi Shilov thoroughly covers all major aspects of linear algebra, in addition it covers more geometrically motivated linear algebra in the latter half.

Linear Algebra
Information
Author	Georgi Shilov
Language	English
Publisher	Dover Publications
Publication Date	1 June 1977
Pages	400
ISBN-10	048663518X
ISBN-13	978-0486635187

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 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

Chapter/Section #	Title	Page #
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]\displaystyle{ 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
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
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
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]\displaystyle{ K_n }[/math] into Itself	98
4.8	Invariant Subspaces	106
4.9	Eigenvectors and Eigenvalues	108
	Problems	113
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
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
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
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
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
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
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
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
HINTS AND ANSWERS		361
BIBLIOGRAPHY		379
INDEX		381

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