Linear Algebra (Book)

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The textbook Linear Algebra by Georgi Shilov provides a thorough introduction to linear algebra.

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

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 \(n\)	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
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 \(K_n\) 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
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
Chapter 7: Area and Applications
1	Area of a disc of radius r	173
2	Circumference of a circle of radius r	180
PART III: COORDINATE GEOMETRY
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
Chapter 9: Operations on Points
1	Dilations and reflections	213
2	Addition, subtraction, and the parallelogram law	218
Chapter 10: Segments, Rays, and Lines
1	Segments	229
2	Rays	231
3	Lines	236
4	Ordinary equation for a line	246
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
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
PART IV: MISCELLANEOUS
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
Chapter 14: Mappings
1	Definition	345
2	Formalism of mappings	351
3	Permutations	359
Chapter 15: Complex Numbers
1	The complex plane	375
2	Polar form	380
Chapter 16: Induction and Summations
1	Induction	383
2	Summations	388
3	Geometric series	396
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
Index		429