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G. V. Rudnyeva

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ELEMENTS OF LINEAR ALGEBRA AND ANALYTIC GEOMETRY

The Educational Textbook for Students of Technical Universities

 

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ϲ 2008

 

22.1

83

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ISBN 978-966-593-660-2

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ISBN 978-966-593-660-2 "", 2008

 

CONTENT

 

INTRODUCTION

CHAPTER 1. THE ELEMENTS OF LINEAR ALGEBRA

1.1. ConceptofMatrix

1.2. Operations on Matrices

1.3. The Determinants of the Second and Third Orders

1.4. The Determinant of the nth Order. Preliminary Idea

1.5. Permutations and Substitutions

1.6. Definition of the Determinant of the nth Order

1.7. Properties of the Determinant of the nth Order

1.8. Algebraic Cofactors and Minors

1.9. Rule by Cramer

1.10. InverseMatrix

1.11. Solving of the Matrix Equations by Means of Inverse Matrix

1.12. Concept of Linear Dependence

1.13. Rank of the Matrix

1.14. Elementary Row Operations and Column Operations...

1.15. The Theorem by Kronecker-Kapelly

1.16. Homogeneous Systems. Construction of the Fundamental System of Solutions

1.17. Method by Gauss (Method of Sequential Elimination of the Unknown Variables). Method by Jordan-Gauss

CHAPTER 2. ANALYTIC GEOMETRY

2.1. Vector Algebra

2.1.1. Vectors. Basic Definitions and Concepts

2.1.2. Linear Operations on Vectors

2.1.3. Concept of Linear Space

2.1.4. Concept of Basis. Decomposition of the Vector

2.1.5. Linear Operations on Vectors Given by Their Coordinates in Some Basis

2.1.6. Transition to a New Basis

2.1.7. Projection of the Vector on Axis

2.1.8. Cartesian Coordinate System

2.1.9. Radius-vector of the Point

2.1.10. Division of the Segment in the Given Ratio

2.1.11. Scalar Product

2.1.12. Inner Product Space and Normed Space

2.1.13. Vector Product...

2.1.14. Mixed product

2.2. Surfaces and Lines in Space R3

2.2.1. General Equation of Plane in Space

2.2.2. Equation of Plane with Given Intercepts

2.2.3. Angle Between Two Planes. Parallel and Perpendicular Planes

2.2.4. Distance from Point to Plane

2.2.5. Normal Equation of Plane

2.2.6. Three Particular Cases for Plane Equations

2.2.7. General Equation of Straight Line in Space.

2.2.8. Canonical Equations of Straight Line

2.2.9. Equation of Straight Line Passing through Two Points

2.2.10. Parametric Equations of Straight Line

2.2.11. Distance from Point to Straight Line

2.2.12. Positional Relationship of Straight Lines in Space

2.2.13. Distance between Two Straight Lines

2.2.14. Equation of Common Perpendicular to Skew Straight Lines

2.2.15. Positional Relationship of Plane and Straight Line.

2.2.16. Point of Intersection of Straight Line and Plane

2.3. Straight Lines in Plane

2.3.1. Line in Plane

2.3.2. Straight Line in Plane. General Equation of the Straight Line

2.3.3. Equation of the Straight Line with Given Intercepts

2.3.4. Canonical Equation of the Straight Line

2.3.5. Canonical Equation of the Straight Line Passing Through Two Points

2.3.6. Parametric Equations of the Straight Line.

2.3.7. Distance between Point and Straight Line.

2.3.8. Normal Equation of the Straight Line

2.3.9. Equation of the Straight Line with Given Slope

2.3.10. Angle between Two Straight Lines
2.4. Curves of the Second Order in Plane

2.4.1. Circle

2.4.2. Conic sections.....

2.4.3. Canonical Equation of Parabola

2.4.4. Canonical Equation of Ellipse

2.4.5. Canonical Equation of Hyperbola

2.4.6. Transformation of Cartesian Coordinates in Plane

2.4.7. Reducing the General Equation of the Second Order Curve to a Canonical Form

2.4.8. Invariance of the Expressions AC - B2 and A + C at Turn of Coordinate Axes. Classification of the Second Order Curves

2.4.9. Linear Operator. Matrix of Linear Operator

2.4.10. Eigenvalues and Eigenvectors of Linear Operator...........

2.4.11. Quadratic Form. Reducing the Quadratic Form to a Canonical Form

2.4.12. Surfaces of the Second Order


REFERENCES

CONTENT