Linear Algebra With Elements Of Analytic Geometry | A. S. Solodovnikov, G. A. Toropova

Preface 8

Part One. ANALYTIC GEOMETRY 10

Chapter 1. VECTORS IN THE PLANE AND IN SPACE. CARTESI­ AN COORDINATE SYSTEM 10

1.1. Vectors 10
1.2. Vector Basis in the Plane and in Space 20
1.3. Cartesian Coordinate System on a Straight Line, in the Plane, and in Space 28
Exercises to Chapter 1 35

Chapter 2. RECTANGULAR CARTESIAN COORDINATES. SIM­PLE PROBLEMS IN ANALYTIC GEOMETRY 37

2.1. Projection of a Vector on an Axis 37
2.2. Rectangular Cartesian Coordinate System 40
2.3. Scalar Product of Vectors 47
2.4. Polar Coordinates 54
Exercises to Chapter 2 55

Chapter 3. DETERMINANTS 57

3.1. Second-Order Determinants. Cramer’s Rule 57
3.2. Third-Order Determinants 60
3.3. n-th-Order Determinants 62
3.4. Transposition of a Determinant 67
3.5. Expansion of a Determinant by Rows and Columns 69
3.6. Properties of nth-Order Determinants 71
3.7. Minors. Evaluation of nth-Order Determinants 77
3.8. Cramer’s Rule for an n x n System 82
3.9. A Homogeneous n x n System 86
3.10. A Condition for a Determinant to Be Zero 91
Exercises to Chapter 3 95

Chapter 4. THE EQUATION OF A LINE IN THE PLANE. A STRAIGHT LINE IN THE PLANE 100

4.1. The Equation of a Line 100
4.2. Parametric Equations of a Line 105
4.3. A Straight Line in the Plane and Its Equation 107
4.4. Relative Position of Two Straight Lines in the Plane 122
4.5. Parametric Equations of a Straight Line 124
4.6. Distance Between a Point and a Straight Line 125
4.7. Half-Planes Defined by a Straight Line 127
Exercises to Chapter 4 128

Chapter 5. CONIC SECTIONS 131

5.1. The Ellipse 131
5.2. The Hyperbola 140
5.3. The Parabola 148
Exercises to Chapter 5 153

Chapter 6. THE PLANE IN SPACE 156

6.1. The Equation of a Surface in Space. The Equation of a Plane 156
6.2. Special Forms of the Equation of a Plane 163
6.3. Distance Between a Point and a Plane. Angle Between Two Planes 168
6.4. Half-Spaces 169 Exercises to Chapter 6 171

Chapter 7. A STRAIGHT LINE IN SPACE 174

7.1. Equations of a Line in Space. Equations ofa Straight Line 174
7.2. General Equations of a Straight Line 178
7.3. Relative Position of Two Straight Lines 183
7.4. Relative Position of a Straight Line anda Plane 186
Exercises to Chapter 7 189

Chapter 8. QUADRIC SURFACES 192

8.1. The Ellipsoid 192
8.2. The Hyperboloid 195
8.3. The Paraboloid 198


Part Two. LINEAR ALGEBRA 202

Chapter 9. SYSTEMS OF LINEAR EQUATIONS 203

9.1. Elementary Transformations of a System of Linear Equations 203
9.2. Gaussian Elimination 205
Exercises to Chapter 9 216

Chapter 10. VECTOR SPACES 218

10.1. Arithmetic Vectors and Operations with Them 218
10.2. Linear Dependence of Vectors 222
10.3. Properties of Linear Dependence 227
10.4. Bases in Space R^n 230
10.5. Abstract Vector Spaces 233
Exercises to Chapter 10  239

Chapter 11. MATRICES 241

11.1. Rank of a Matrix 242
11.2. Practical Method for Finding the Rank of a Matrix 245
11.3. Theorem on the Rank of a Matrix 247
11.4. Rank of a Matrix and Systems of Linear Equations 249
11.5. Operations with Matrices 250
11.6. Properties of Matrix Multiplication 253
11.7. Inverse of a Matrix 255
11.8. Systems of Linear Equations in Matrix Form 259
Exercises to Chapter 11 263

Chapter 12. EUCLIDEAN VECTOR SPACES 266

12.1. Scalar Product. Euclidean Vector Spaces 266
12.2. Simple Metric Concepts in Euclidean Vector Spaces 269
12.3. Orthogonal System of Vectors. Orthogonal Basis 271
12.4. Orthonormal Basis 274
Exercises to Chapter 12 275

Chapter 13. AFFINE SPACES. CONVEX SETS AND POLY­HEDRONS 277

13.1. The Affine Space A^n 277
13.2. Simple Geometric Figures in A^n 279
13.3. Convex Sets of Points in A^n. Convex Polyhedrons 282
Exercises to Chapter 13 286

Index 288