ФЕДЕРАЛЬНОЕ АГЕНТСТВО ПО ОБРАЗОВАНИЮ
Государственное образовательное учреждение высшего профессионального образования
«ТОМСКИЙ ПОЛИТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ»
V.V. Konev
LINEAR ALGEBRA, VECTOR ALGEBRA AND
ANALYTICAL GEOMETRY
TextBook
Рекомендовано в качестве учебного пособия Редакционно-издательским советом Томского политехнического университета
Издательство Томского политехнического университета
2009
UDС 517
V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp.
This textbook consists of 3 parts devoted to the mathematical methods of Linear Algebra and Analytical Geometry based on the vector analysis technique. The basic concepts are explained by examples and illustrated by figures.
The textbook is helpful for students who want to understand and be able to use matrix operations, solve systems of linear equations, analyze relative positions of figures, transform coordinate systems, and so on.
The textbook is designed to English speaking students.
Reviewed by: V.A. Kilin, Professor of the Higher Mathematics Department, TPU, D.Sc.
©Konev V.V. 2001-2009
©Tomsk Polytechnic University, 2001-2009
PREFACE
This textbook is intended for students who have already studied basic mathematics and need to study the methods of higher mathematics. It covers three content areas: Linear Algebra, Vector Algebra and Analytical Geometry. Each part contains basic mathematical conceptions and explains new mathematical terms. Many useful examples and exercises are presented in the textbook. explained and illustrated by examples and exercises.
The Linear Algebra topics include matrix operations, determinants and systems
of linear equations.
In the section “Vector Algebra”, a main attention is paid to the geometrical applications of vector operations. The vector approach is considered to be basic for discussion of classic problems of Analytical Geometry.
The author welcomes reader’s suggestions for improvement of future editions of this textbook.
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CONTENTS
Preface ……………………..………………………………………… 3 Contents ……………………………………………………………….. 4
LINEAR ALGEBRA
Chapter 1. MATRICES
1.1.Basic Definitions ……………………………………………. . 7
1.2.Matrix Operations ……………………………………………. 8
1.3.Types of Matrices …………………………………………. … 12
1.4.Kronecker Delta Symbol……………………………………… 15
1.5.Properties of Matrix Operations……………………………… 16
Chapter 2. DETERMINANTS
2.1.Permutations and Transpositions……………………………… 20
2.2.Determinant General Definition ……………………………... 23
2.3.Properties of Determinants …………………………………... 25
2.4.Determinant Calculation……………………………………… 31
Chapter 3. INVERSE MATRICES
3.1.Three Lemmas ……………………………………………….. 36
3.2.Theorem of Inverse Matrix …………………………………... 38
3.2.1.Examples ……….…………………………………………. 39
3.3.Calculation of Inverse Matrices by Elementary
Transformations ……………………………………………… 42
Chapter 4. SYSTEMS OF LINEAR EQUATIONS
4.1.Matrix Rank ………………………………………………….. 43
4.2.Basic Concepts ………………………………………………. 45
4.3.Gaussian Elimination ………………………………………… 46
4.3.1.Examples ………………………………………………….. 47
4.4. Homogeneous Systems of Linear Equations………………… 50
4.4.1.Examples …………………………………………………. 51
4.5.Cramer’s Rule ……………………………………………….. 54
4.6. Cramer’s General Rule ……………………………………… 57
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VECTOR ALGEBRA
Chapter 5. VECTORS
5.1. |
Basic Definitions …………………………………………... |
60 |
5.2. |
Geometrical Interpretation ………………………………… |
61 |
5.2.1. Vectors in Three-Dimensional Space …………………… |
61 |
|
5.2.2. Linear Vector Operations ……………………………….. |
62 |
|
5.2.3. Projection of a Vector in a Given Direction ……………. |
63 |
|
5.2.4. Properties of Linear Vector Operations ………………… |
64 |
|
5.3. Resolution of Vectors into Components ………………….. |
65 |
|
5.3.1. Rectangular Orthogonal Basis ………………………….. |
65 |
|
5.3.2.Linear Dependence of Vectors ………………………….. 66
5.3.3.Vector Bases …………………………………………….. 68
5.4.Scalar Product of Vectors …………………………….......... 69
5.4.1.Properties of the Scalar Product …………………………. 70
5.4.2.Some Examples ………………………………………….. 70
5.4.3. Direction Cosines ………………………………………… |
71 |
5.5. Vector Product ……………………………………………… |
72 |
5.5.1.Properties of the Vector Product …………………………. 73
5.5.2.Some Examples …………………………………………… 74
5.6. The Scalar Triple Product …………………………………… |
75 |
5.6.1. Properties of the Scalar Triple Product …………………… |
76 |
5.6.2.Some Examples …………………………………………… 77
5.7.Transformation of Coordinates Under Rotation of the
Coordinate System ………………………………………….. 79 5.7.1. Rotation of the x,y–Plane Around the z-Axis …………….. 81
ANALYTICAL GEOMETRY
Chapter 6. STRAIGHT LINES
6.1. Equations of lines …………………………………………… 82
6.2.Lines in a Plane ……………………………………………... 84
6.3.Angle Between Two Lines ………………………………….. 86
6.3.Distance From a Point to a Line …………………………….. 89
6.4. Relative Position of Lines …………………………………… 90
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Chapter 7. PLANES |
|
7.1. General Equation of a Plane ………………………………… |
91 |
7.2. Equation of a Plane Passing Through Three Points ………… |
93 |
7.3. Other Forms of Equations of a Plane ………………………. |
94 |
7.4. Angle Between Two Planes ………………………………… |
95 |
7.5. Distance Between a Point and a Plane ……………………… |
96 |
7.6.Relative Position of Planes …………………………………. 97
7.7.Relative Position of a Plane and a Line ……………………… 98
7.8.Angle Between a Plane and a Line ………………………….. 98
Chapter 8. Quadratic Curves
8.1.Circles ………………………………………………………. 99
8.2.Ellipses ………………………………………………………. 101
8.2.1.Properties of Ellipses ……………………………………… 102
8.3.Hyperbolas …………………………………………………… 105
8.3.1.Properties of Hyperbolas ………………………………….. 106
8.4.Parabolas …………………………………………………….. 109
8.5.Summary …………………………………………………….. 111
References…………………………………………………………….. 112
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