Czech

5

Not applicable.

The succesful course leaver should be able to correctly and precisely define the fundamental concepts of matrix and tensor calcullus; moreover, he or she should be able to demonstrate at least 2-3 concrete examples for each discussed concept. The course leaver should also be able to formulate the most important theorems of the matrix and tensor calcullus, including the precise formulations of their assumptions of validity. The student, passing the course should know and understand to the main principles of the proofs of the discused theorems and should be able to explain them. The successful course leaver should be able to apply the discussed notions and theorems for the solution of concrete tasks and exercises from a given class, covering the most common applications.

General mathematical knowledge. The ability of orientation in the basic problems of higher mathematics and the ability to apply the basic methods. Solving problems in the areas cited in the annotation above by using basic rules. Solving these problems by using modern mathematical software. For more detail, see the content of the subject BMA1 Matematika 1. The previous attendance to the subject BMAS Matematický seminář is recommended.

Not applicable.

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Requirements for completion of a course are specified by a regulation issued by the lecturer responsible for the course and updated annually.

1. Matrices and systems of linear equations. Gaussian elimination and its modifications. Matrix operations. Rank of a matrix.Frobenius Theorem. Special matrices.
2. Determinants. Methods of calcullation. The adjoint matrix and its relationshp to the inverse matrix. Cramerius Rule.
3. Vector space, its base and dimension. Transformation of the basis and the coordinates. Transition matrix.
4. Operations with vector spaces. Subspaces. Sum and intersection of vector spaces.
5. Linear mapping (operator) and its matrix in various bases. The kernel and the image of a linear mapping.
6. Scalar (dot) product. Gram matrix, ortogonalization.
7. Ortogonal projection, ortogonal complement of a vector subspace.
8. Matrix of the ortogonal projection. Appoximation by an orthogonal projection.
9. Eigenvalues and eigenvectors. Diagonal form of a self-adjoint matrix. Spectral reprezentation.
10. Quadratic forms and their definitness. Quadratic surfaces.
11. Tensors on a real vector space. The dual space of linear forms. Various bases.
12. Tensor product. Covariant and contravariant tensors.
13. Antisymmetric tensors and the antisymmetric outer product.

Not applicable.

The aim of the subject is to learn the students to use the foundations of the theory of matrices, vector spaces, linear operators and multilinear forms (tensors) and yield them an introduction to some modern applications of matrix and tensor calcullus in the subject of their own study.

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated annually.

Not applicable.

Not applicable.

Havel V., Holenda J.: Lineární algebra, SNTL, Praha 1984.
Hrůza B., Mrhačová H.: Cvičení z algebry a geometrie. Ediční stř. VUT 1993, skriptum
Schmidtmayer J.: Maticový počet a jeho použití, SNTL, Praha, 1967.
Boček L.: Tenzorový počet, SNTL Praha 1976.
Angot A.: Užitá matematika pro elektroinženýry, SNTL, Praha 1960.
Kolman, B., Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986.
Kolman, B., Introductory Linear Algebra, Macmillan Publ. Comp., New York 1991.
Demlová, M., Nagy, J., Algebra, STNL, Praha 1982.
Krupka D., Musilová J., Lineární a multilineární algebra, Skriptum Př. f. MU, SPN, Praha, 1989.

Not applicable.

8 hours, compulsory

Definition of matrix, fundamental notion. Transposition of matrices.
Determinant of quadratic complex matrix.
Operations with matrices. Special types of matrices. Inverse matrix. Matrices solutions of linear algebraic equations.
Linear, bilinear and quadratic forms. Definite of quadratics forms.
Spectral attributes of matrices.
Linear space, dimension.
Linearn transformations of coordinates of vector.
Covariant and contravariant coordinates of vector.
Definition of tensor. Covariants, contravariants and mixed tensor.
Operations with tensors. Symmetry and antisymmetry of tensors of second order.