Algebra and geometry - free course from Open Education, Training, Date: November 30, 2023.
Miscellaneous / / December 04, 2023
Currently, Moscow University is one of the leading centers of national education, science and culture. Raising the level of highly qualified personnel, searching for scientific truth, focusing on humanistic ideals of goodness, justice, freedom - this is what we see today as following the best university traditions Moscow State University is the largest classical university in the Russian Federation, a particularly valuable object of cultural heritage of the peoples of Russia. It trains students in 39 faculties in 128 areas and specialties, graduate students and doctoral students in 28 faculties in 18 branches of science and 168 scientific specialties, which cover almost the entire spectrum of modern university education. Currently, more than 40 thousand students, graduate students, doctoral students, as well as specialists in the advanced training system are studying at Moscow State University. In addition, about 10 thousand schoolchildren study at Moscow State University. Scientific work and teaching are carried out in museums, at educational and scientific practice bases, on expeditions, on research vessels, and in advanced training centers.
A new element of the Russian education system - open online courses - can be transferred to any university. We make this a real practice, expanding the boundaries of education for every student. A full range of courses from leading universities. We are systematically working to create courses for the basic part of all areas of training, ensuring that any university can conveniently and profitably integrate the course into its educational programs
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Lecture 1.Chapter I. Fundamentals of matrix theory§ 1. Concept of a matrix Compact form of writing a matrix. Matrices of a special type.§ 2. Operations on matricesLinear operations. Matrix multiplication. Matrix transposition.
Lecture 2.§ 3. Elementary transformations of a matrix and matrix of elementary transformations Reduction to a step form. Matrices of elementary transformations.§ 4. Determinant of a matrixPermutations. Construction of the nth order determinant. The simplest properties. Lecture 3.§ 4. Matrix determinant (continued) Minors and algebraic complements. Laplace's theorem, general scheme of proof. Lecture 4.§ 4. Determinant of the matrix (continued) Proof of Laplace's theorem. Decomposition of the determinant in a row (column). Block matrices. Determinant of the product of matrices. Lecture 5.§ 5. Inverse matrix Definition and simplest properties. Adjoint matrix. Reversibility criterion. Explicit form of the inverse matrix. Chapter II. Set-theoretic concepts§ 6. The concept of set. About the concept of set. Operations on sets. Cartesian product of sets.§ 7. Binary relation. Equivalence relation§ 8. DisplaysDefinition. Bijective (one-to-one) mapping. Reverse mapping. Reversibility criterion. Lecture 6.Chapter III. Geometric vectors§ 9. Directed segments§ 10. Free vector. Linear operations on vectors Definition and terminology. Linear operations on vectors. Sets of vectors on a straight line, on a plane and in space. Lecture 7.Chapter IV. Introduction to the theory of linear spaces§ 11. Real linear space. Definition. Examples: geometric spaces, arithmetic space, matrix space, polynomial spaces.§ 12. Linear dependence§ 13. Geometric meaning of linear dependence
Lecture 8.§ 14. Matrix rank Matrix rank and linear dependence. Matrix rank and elementary transformations. Rank calculation. Equivalent matrices.§ 15. Basis and dimension of linear space Definitions. Vector coordinates. Transition to another basis. Lecture 9.Chapter V. Vector algebra§ 16. Vector coordinates on the axis§ 17. Affine (general Cartesian) coordinate system. Point coordinates§ 18. Projections of a vectorProjections of a vector on a plane. Projections of a vector in space. Projection vectors and coordinates. Lecture 10.§ 19. Dot product Definition and basic properties. Orthonormal basis. Vector coordinates and scalar product in an orthonormal basis.§ 20. Vector and mixed product of vectors Orientation in real space. Basic facts. Vector and mixed products in rectangular coordinates.§ 21. Transformation of a rectangular Cartesian coordinate system. Orthogonal matrix. Transition matrix from one orthonormal basis to another orthonormal basis. Transformation of a rectangular Cartesian coordinate system on a plane. Lecture 11.Chapter VI. Systems of linear algebraic equations§ 22. Main problems of the theory of solving systems of linear algebraic equations Terminology. Compact system recording. Equivalence of systems.§ 23. Systems with a square non-singular matrix§ 24. General systems. General solution of the system System compatibility. Collaborative system research design. General solution of the system. Homogeneous systems.§ 25. Gauss method of studying and solving systems of equationsSystems with a trapezoidal matrix. Elementary transformations of a system of equations. Reducing a general system to a system with an upper trapezoidal matrix. Lecture 12.Chapter VII. Geometric properties of solutions to a system of linear algebraic equations§ 26. Linear subspace of solutions of a homogeneous systemLinear subspace of a linear space. The set of solutions to a homogeneous system of linear algebraic equations as a linear subspace of an arithmetic space. Fundamental system of solutions. General solution of the system.§ 27. Linear manifold of solutions to an inhomogeneous systemLinear manifold in linear space. The set of solutions to a non-homogeneous system of linear algebraic equations as a linear variety in an arithmetic space. General solution of the system
This course is the first of the five-step “Medical English” cycle and is intended for medical professionals who want to expand their knowledge in the field of professional in English. This course is also suitable for translators who want to improve their competencies in medical English.
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