Analytical geometry - free course from Open Education, training 13 weeks, about 5 hours per week, Date November 29, 2023.

Awards and achievements Russian Government Prize in the field of education for 2010, Honored Teacher of MIPT.

The course consists of 12 training weeks and one examination week

Week 1. Matrices

01.00 Introduction

01.01 Matrix definition

01.02 Operations with matrices

01.02.01 Problem. Calculation of linear combination of matrices

02/01/02 Problem. Finding the Transposed Matrix

01.03 Product of matrices. Part 1

01.04 Product of matrices. Part 2

04/01/01 Problem. Calculating the product of matrices

04/01/02 Problem. Checking the existence of a product and calculating it 

04/01/03 Problem. Calculating a matrix to the nth power. Example 1

04/01/04 Problem. Calculating a matrix to the nth power. Example 2

04/01/05 Problem. Calculating a Matrix Polynomial

04/01/06 Problem. Checking the validity of matrix equality

04/01/07 Problem. Computing a Matrix to a Numerical Power

01.05 Matrix determinant

01.05.01 Problem. Calculation of the determinant of a matrix

01.06 Cramer's Rule

06/01/01 Problem. Solving a system of linear equations using Cramer's method

Week 2. Vectors

02.01 Determination of a directed segment, vector

02.02 Repetition from the school geometry course 

02.02.01 Problem. Proof of the inequality for a quadrilateral in space

02.02.02 Problem. Proof of equality for an n-gon

02.03 Linear combination of vectors

02.04 Linear dependence and independence of vectors 

02.05 Criterion for linear dependence of a system of vectors

02.06 Basis

02.06.01 Problem. Finding vector coordinates

02.06.02 Problem. Finding the coordinates of a parallelepiped using vectors

02.07 Replacement of basis

07/02/01 Problem. Finding the coordinates of a prism point in a new coordinate system

07/02/02 Problem. Finding the coordinates of a parallelogram point in a new coordinate system

02.08 Cartesian coordinate system (DCS)

02.08.01 Problem. Checking that vectors form a basis

02.09 Replacement of ODSC

02.09.01 Problem. Finding the coordinates of the origin and basis vectors in the new and old coordinate systems

02.09.02 Problem. Finding the coordinates of a vector in the new basis through the coordinates in the old one

Week 3. Product of vectors

03.01 Dot product of vectors

03.02 Projection of a vector onto a non-zero vector

03.03 Properties of the scalar product of vectors. Part 1

03.04 Properties of the scalar product of vectors. Part 2

04/03/01 Problem. Finding the lengths of the sides and angles of a parallelogram using basis vectors

04/03/02 Problem. Finding the orthogonal projection of a vector onto a line

03.05 Orientation of bases. Oriented volumes and areas

03.06 Mixed product of vectors. Part 1

03.07 Mixed product of vectors. Part 2

03.08 Vector product of vectors. Part 1

03.09 Vector product of vectors. Part 2

03.09.01 Problem. Proof of coplanarity of vectors

03.09.02 Problem. Finding the area of ​​a triangle using vector coordinates

09/03/03 Problem. Proof of equality for non-collinear vectors

09/03/04 Problem. Finding the volume of a tetrahedron and its height

03.10 Double cross product

03.10.1 Problem. Proof of identity

03.11 Mutual basis

Week 4 Part 1. Plane in space

04.01 Definition of a plane in space

04.02 Various forms of writing the equation of a plane

04.03 General plane equation

04.03.01 Problem. Plane equation

Week 4 Part 2. Straight on a plane. Straight line and plane in space

04.04 Straight line on a plane

04.04.01 Problem. Finding the radius vector of a point

04.04.02 Problem. Conditions for intersection, parallelism and perpendicularity of lines in a plane

04.05 General equation of a straight line on a plane. Straight line in space

04.05.01 Problem. Finding the radius vector of the point of intersection of lines

04.05.02 Problem. Equation of a line intersecting two skew lines

04.05.03 Problem. Equation of a line passing through a point and parallel to another line

04.05.04 Problem. Condition for the intersection of a line and a plane

04.06 Mutual arrangement of lines and planes

06/04/01 Problem. Equation of a plane passing through a point and parallel to two lines

06/04/02 Problem. Equation of a plane passing through one line and parallel to another line

04.07 Straight line and plane in PDSC

04.07.01 Problem. Equation of lines passing through one point and equidistant from two other points

04.07.02 Problem. Equation of the bisector of the angle between lines

04.08 Some metric problems in PDSC. Part 1

04.08.01 Problem. Equation of lines parallel to another line and separated from a point at some distance

04.08.02 Problem. General equation of a plane passing through some point and a line. Distance from this plane to a given point

04.09 Some metric problems in PDSC. Part 2

04.09.01 Problem. Distance between lines

Week 5. Algebraic lines of second order on the plane

05.01 Definition of algebraic lines and surfaces

05.02 Second order lines on a plane. Ellipse equation

05.03 Equation of an imaginary ellipse, a pair of imaginary intersecting lines, a hyperbola, a pair of intersecting lines

05.04 Equation of a parabola, pairs of parallel lines, pairs of imaginary parallel lines, pairs of coincident lines

05.05 Center of the line. Elliptic and hyperbolic lines

05.05.01 Problem. A type of second-order curve defined by some equation. The canonical equation of a curve and the canonical coordinate system. Example 1

05.05.02 Problem. A type of second-order curve defined by some equation. The canonical equation of a curve and the canonical coordinate system. Example 2

05.05.03 Problem. A type of second-order curve defined by some equation. The canonical equation of a curve and the canonical coordinate system. Example 3

Week 6 Studying the properties of ellipse, hyperbola and parabola

06.01 Ellipse

01/06/01 Problem. Canonical ellipse equation

06.02 Properties of the ellipse

06.03 Equation of a tangent to an ellipse

03/06/01 Problem. Equation of tangents to an ellipse

03/06/02 Problem. Angle between the tangent and the Ox axis

06.04 Hyperbole

04/06/01 Problem. Hyperbola eccentricity

06.05 Geometric properties of a hyperbola

05/06/01 Problem. Proof of the constancy of the product of the distance from any point of a hyperbola to its asymptotes

06.06 Parabola

06.06.01 Problem. Parabola equation

06.06.02 Problem. Equations of tangents to a parabola

06.07 Ellipse, hyperbola and parabola in the polar coordinate system

Week 7 Second order surface

07.01 Surface of rotation

07.02 Ellipsoid

07.03 Second order cone

07.04 Single-sheet hyperboloid

07.05 Rectilinear generators of a one-sheet hyperboloid

07.06 Two-sheet hyperboloid, elliptic and hyperbolic paraboloid

06/07/01 Problem. Determining surface type

06/07/02 Problem. Common points of a line and second-order surfaces

06/07/03 Problem. Parametric equations of rectilinear generators of a given surface

06/07/04 Problem. Type of surface formed by rotating a straight line

Week 8 Mappings and Transformations

08.01 Definition of mapping and transformation

08.02 One-to-one mapping. Product of mappings

08.03 Properties of the product of plane transformations. Coordinate recording of mappings

08.04 Orthogonal plane transformations

08.05 Linear and affine transformations

08.06 Image of a vector during linear transformation. Part 1

08.07 Image of a vector during linear transformation. Part 2

08.08 Geometric properties of affine transformations

08.08.01 Problem. Symmetry about a straight line

08.08.02 Problem. An affine transformation of a plane that takes given lines into itself and a given point into some other point

08.09 Changing areas during affine transformation

08.10 Images of second-order lines under affine transformation

08.10.01 Problem. Second order curve type

08.10.02 Problem. Proof of equality of sums of areas of triangles

08.11 Decomposition of an affine transformation

08.11.01 Problem. Representation of a given affine transformation as products of three transformations

Week 9 Determinants of nth order matrices

09.01 Determinants

01/09/01 Problem. Determinant of order n. Example 1

01/09/02 Problem. Determinant of order n. Example 2

09.02 Properties of determinant. Part 1

09.03 Properties of determinant. Part 2

09.04 Properties of determinant. Part 3

04/09/01 Problem. Vandermonde determinant

04/09/02 Problem. Determinant of order 2n

09.05 Formula for complete development of determinant

05/09/01 Problem. Complete decomposition formula for a fifth-order matrix

09.06 SLAU in a special case

09.07 Cramer's rule in the general case

Week 10 Matrix rank

10.01 Minors of arbitrary order

10.02 Matrix rank

02/10/01 Problem. Rank and basis system of matrix columns

02/10/02 Problem. Estimating the rank of a matrix of order n

02/10/03 Problem. Proof of rank inequality for any matrices of the same size

02/10/04 Problem. Non-zero minor of order r of a matrix of rank r

02/10/05 Problem. Matrix rank estimation

10.03 Reducing the matrix to a simplified form

10.04 Gaussian method

10.05 Basis minor theorem

05/10/01 Problem. Representation of a matrix through the product of matrices

10.06 Matrix rank theorem

06/10/01 Problem. Upper bound for the rank of the product of two matrices

06/10/02 Problem. Proof of equality of the rank of a matrix to the highest order of its minors

Week 11 inverse matrix

11.01 Definition of inverse matrix

11.02 Expressing elements of an inverse matrix through elements of the original matrix

 02/11/01 Problem. Calculation of the inverse matrix. Example 1

02/11/02 Problem. Finding the inverse matrix. Example 2

11.03 Properties of an inverse matrix

03/11/01 Problem. Checking the validity of the identity for matrices

11.04 Another proof of the existence of an inverse matrix for a non-singular square matrix

11.05 Characteristic polynomial of a matrix

05/11/01 Problem. inverse matrix

11.06 Hamilton-Cayley theorem

11.07 Elementary transformations like matrix multiplication

07/11/01 Problem. Calculation of the inverse matrix through elementary transformations. Example 1

07/11/02 Problem. Finding the inverse matrix. Example 2

Week 12 General theory of linear systems

12.01 Kronecker-Capelli theorem

12.02 Fredholm's theorem

12.03 General solution of inhomogeneous SLAE

12.04 Fundamental matrix of a homogeneous SLAE. Part 1

12.05 Fundamental matrix of a homogeneous SLAE. Part 2

05.12.01 Problem. Fundamental matrix of SLAE

05.12.02 Problem. Checking the fundamental matrix of SLAE

05.12.03 Problem. SLAE solution

05.12.04 Problem. General view of an arbitrary fundamental matrix of SLAEs

05/12/05 Problem. Equivalence condition for SLAEs

12.06 General solution of inhomogeneous SLAE

06/12/01 Problem. SLAE solution

06.12.02 Problem. Compatibility of heterogeneous SLAEs

Week 13 final examination