Czech

5

Not applicable.

During the examination it is verified if the student is able to:

- define the notions of a matrix, system of linear equations and its solution, vector space, its dimension and base, sum and
intersection of vector spaces, inner product, orthogonal projection, approximation by the orthogonal projection, orthogonal
complement, eigenvector and eigenvalue, quadratic form, dual vector space, covariant and contravariant base, covariant, contravariant
and mixed tensor, tensor and oughter product, antilinear form.

- explain the content of the above mentioned notions and their mutual relationships;
- apply the most important theorems of the matrix and tensor calcululus in solving concrete tasks;
- calculate the solution of a general system of linear equations;
- determine the dimension and base of a vector space;
- calculate the transition matrix between two bases and new coordinates of a vector;
- calculate the intersection and sum of vector spaces;
- calculate the orthogonal projection and its matrix;
- calculate the eigenvectors and eigenvalues of a matrix;
- calculate the diagonal form of a Hermitian (self-adjjoint) matrix and the corresponding orthogonal transformation matrices;
- determine the definiteness of a quadratic form;
- calculate the coordinates of vectors, linear forms and tensors in the standard, covariant and contravariant bases;
- list the most important applications of the matrix and tensor calculus outside mathematics.

Regarding the foundations of mathematics the student should be able to:

- describe the most important number sets (the natural numbers, integers, racionals, real and complex numbers);
- explain the fundamental properties of the above mentioned sets especially with respect to their cardinality and the existence of a
solution of algebraic equations;
- apply the fundamental rules for working with complex numbers;
- apply the fundamental rules for working with algebraic expressions;
- calculate the solution of simple systems of linear equations by the substitution method;
- calculate the roots of a quadratic equation;
- calculate the roots of selected algebraic equations of the $n$-th degree by the Horner's scheme;
- calculate the derivatives and integrals of simple functions of one real variable, in particular, of the elementary functions;
- calculate the solution of a simple ordinary differential equation by the method of separation of variables.

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 for every.

Definition of a matrix, basic notions. Transpose of a matrix.
Determinant of coplex square matrix.
Matrix operations, special forms of matrices. The inverse matrix.
Applications of matrices for solving systems of linear equations.
Linear, bilinear, and quadratic forms. Definiteness of quadratic forms.
Spectral properties of matrices.
Linear spaces. Base and dimension.
Linear transformation of vector coordinates.
Covariant and contravariant coordinates of a vector.
Definition of a tensor.
Covariant, contravariand and mixed tensors.
Operations with tensors.
Symetry and antisymmetry of 2-tensors .

Not applicable.

Master the bases of the matrices and tensors calculus and its applications.

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 for every academic year.

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.