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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Lecture 1. Elements of Determinants’ Theory . . . . . . . . . . . . . . . . 5
1.1. Determinants and their properties . . . . . . . . . . . . . . . 5
1.2. Use of determinants to solve systems of linear algebraic equations . 9
Lecture 2. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1. Definition of a matrix . . . . . . . . . . . . . . . . . . . . . 10
2.2. Operations with matrices . . . . . . . . . . . . . . . . . . . 11
2.3. Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . 12
2.4. Using matrices for solving systems of linear algebraic equations . . 13
2.5. Rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . 14
Lecture 3. General Research Into the System of Linear Algebraic Equations . . 16
3.1. General information about systems of linear equations . . . . . . 16
3.2. Theorem on the existence of a system of linear equations solution . 16
3.3. Linear homogeneous systems . . . . . . . . . . . . . . . . . 18
3.4. Gauss’ method of successive exception of unknowns . . . . . . . 18
Lecture 4. Elements of Vector Algebra. . . . . . . . . . . . . . . . . . . 20
4.1. Rectangular system of coordinates in space . . . . . . . . . . . 20
4.2. Definition of a vector . . . . . . . . . . . . . . . . . . . . . 21
4.3. Linear operations with vectors . . . . . . . . . . . . . . . . 21
4.3.1. Addition of vectors . . . . . . . . . . . . . . . . . . . . . 22
4.3.2. Multiplying of a vector by a digit . . . . . . . . . . . . . . . 23
4.4. Projection of a vector onto the axis . . . . . . . . . . . . . . . 23
4.5. Projections of a vector onto the coordinate axis . . . . . . . . . 25
4.6. Direction cosines of a vector . . . . . . . . . . . . . . . . . 26
4.7. Linear dependence of vectors. Basis . . . . . . . . . . . . . . 27
4.8. A rectangular basis. Expansion of a vector by basis. . . . . . . . 29
4.9. Separation of a closed interval in a given ratio . . . . . . . . . . 30
Lecture 5. Multiplication of Vectors . . . . . . . . . . . . . . . . . . . . 31
5.1. Dot product of vectors . . . . . . . . . . . . . . . . . . . . 31
5.2. Cross product of vectors . . . . . . . . . . . . . . . . . . . 34
5.3. Mixed product of vectors . . . . . . . . . . . . . . . . . . . 37
Lecture 6. Analytic geometry on a plane . . . . . . . . . . . . . . . . . . 40
6.1. Polar system of coordinates . . . . . . . . . . . . . . . . . . 40
6.2. Transformation of rectangular coordinates . . . . . . . . . . . 41
6.2.1. Parallel shift of axes . . . . . . . . . . . . . . . . . . . . 41
6.2.2. Turning of axes . . . . . . . . . . . . . . . . . . . . . . 42
6.3. Equation of a line on a plane . . . . . . . . . . . . . . . . . . 42
6.4. Straight line on a plane. Its equation . . . . . . . . . . . . . . 44
6.4.1. General equation of a line . . . . . . . . . . . . . . . . . . 44
6.4.2. Canonical equation of a line. . . . . . . . . . . . . . . . . . 45
6.4.3. Equation of a line, that goes through a given point parallel to the given
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4.4. Equation of a line with the angle coefficient . . . . . . . . . . . 46
6.4.5. Equation of a line that crosses two given points . . . . . . . . . 46
6.4.6. Equation of a line in segments. . . . . . . . . . . . . . . . . 47
6.4.7. Normal equation of a line. . . . . . . . . . . . . . . . . . . 48

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