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| Basic Mathematics
|
|
| Information
|
| Author
|
Georgi Shilov
|
| Language
|
English
|
| Publisher
|
Springer
|
| Publication Date
|
1 July 1988
|
| Pages
|
496
|
| ISBN-10
|
0387967877
|
| ISBN-13
|
978-0387967875
|
The textbook Linear Algebra by Georgi Shilov provides a thorough introduction to linear algebra.
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 Equations
|
| 1 |
Equations in two unknowns |
53
|
| 2 |
Equations in three unknowns |
57
|
| Chapter 3: Real Numbers
|
| 1 |
Addition and multiplication |
61
|
| 2 |
Real numbers: positivity |
64
|
| 3 |
Powers and roots |
70
|
| 4 |
Inequalities |
75
|
| Chapter 4: Quadratic Equations
|
| Interlude: On Logic and Mathematical Expressions
|
| 1 |
On reading books |
93
|
| 2 |
Logic |
94
|
| 3 |
Sets and elements |
99
|
| 4 |
Notation |
100
|
| PART II: INTUITIVE GEOMETRY
|
| Chapter 5: Distance and Angles
|
| 1 |
Distance |
107
|
| 2 |
Angles |
110
|
| 3 |
The Pythagoras theorem |
120
|
| Chapter 6: Isometries
|
| 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
|
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