“Analytical Geometry” - course 2800 rub. from MSU, training 15 weeks. (4 months), Date: November 30, 2023.

Lecture 1. Definition of a vector. Addition of vectors, multiplication of a vector by a number. Vectors on a straight line. Linear dependence of vectors.
Lecture 2. Collinearity and coplanarity of vectors. Geometric meaning of linear dependence. Bases and coordinates. Geometric description of vector coordinates.
Lecture 3. Dot product of vectors. Metric basis coefficients. Dot product in coordinates.

Lecture 4. Affine and rectangular coordinates. Polar coordinates on the plane and in space.
Lecture 5. Matrices and operations on them. Transition from one basis to another. Transition from one affine coordinate system to another.
Lecture 6. Definition of an orthogonal matrix. Transformation of rectangular coordinates.
Lecture 7. Orientation of line, plane and space. Oriented area and oriented volume. Vector and mixed product of vectors.
Lecture 8. Vector equations of a line and a plane. The relative position of two lines in space. Calculation of distances.
Lecture 9. Equation of a straight line on a plane. The relative position of lines on a plane. Half-planes. A straight line on a plane with a rectangular coordinate system.
Lecture 10. Equation of a plane. The relative position of two planes. Half-spaces. Straight in space. Straight line and plane in space with a rectangular coordinate system.
Lecture 11. Algebraic lines on the plane. Quadratic functions and their matrices. Orthogonal invariants of quadratic functions. Transformation of the equation of a second-order line when rotating the coordinate axes.
Lecture 12. Reducing the second order line equation to canonical form. Determination of the equation of a second order line using orthogonal invariants.
Lecture 13. Directorial property of ellipse, hyperbola and parabola. Focal property of ellipse and hyperbola. Second order curves in polar coordinates.
Lecture 14. The intersection of a second-order line with a straight line. Uniqueness theorems for second order lines. Centers of second order lines.
Lecture 15. Asymptotes and conjugate diameters of second-order lines. Conjugate directions.
Lecture 16. Tangents to lines of the second order. Optical properties of ellipse, hyperbola and parabola.
Lecture 17. Principal directions and principal diameters of second-order lines. Axes of symmetry.
Lecture 18. Definition and properties of affine transformations. Analytical notation of affine transformations. Affine classification of second order lines.
Lecture 19. Definition and properties of isometric transformations. Classification of plane movements.
Lecture 20. Second order surfaces and matrices of quadratic functions. The main theorem on second-order surfaces (without proof).
Lecture 21. Ellipsoid and hyperboloids, their plane sections. Rectilinear generators of a one-sheet hyperboloid. Conic sections.
Lecture 22. Paraboloids, their flat sections. Rectilinear generators of a hyperbolic paraboloid. Cylindrical surfaces. Affine classification of second order surfaces.
Lecture 23. Models of the projective plane: augmented plane, copula, their isomorphism. Homogeneous coordinates on the projective plane.
Lecture 24. Arithmetic model of the projective plane. The principle of duality. Desargues's theorem.