Linear Algebra
Code  Completion  Credits  Range  Language 

BIKLIN  Z,ZK  7  26KP+4KC  Czech 
 Lecturer:
 Karel Klouda (guarantor)
 Tutor:
 Karel Klouda (guarantor)
 Supervisor:
 Department of Applied Mathematics
 Synopsis:

Students understand the theoretical foundation of algebra and mathematical principles of linear models of systems around us, where the dependencies among components are only linear. They know the basic methods for operating with matrices and linear spaces. They are able to perform matrix operations and solve systems of linear equations. They can apply these mathematical principles to solving problems in 2D or 3D analytic geometry. They understand the errordetecting and errorcorrecting codes.
 Requirements:

Secondary school mathematics.
 Syllabus of lectures:

1. Introduction: definition, theorem, proof. Types of proofs.
2. Set operations: Intersection, union, relative complement, Cartesian product. Maps, composite maps, inverse map, bijection, permutation.
3. Polynomials, roots of polynomials, irreducible polynomials. Polynomials in R, C, Q. Greatest common divisor and Euclid's algorithm. Binary operation, its properties. Group, ring, field. Homomorphisms (isomorphisms). Properties of a field. Finite fields.
4. Sets of linear equations. Gaussian elimination method.
5. Linear spaces, linear combination and linear independence.
6. Bases, dimensions, vector coordinates in a base. Coordinate transformations.
7. Matrices, matrix operations.
8. Determinants.
9. Inverse matrix, its calculation.
10. Linear map, linear map matrix. Rotation, projection onto a straight line (plane), symmetry with respect to a straight line (plane) in $R^2$, $R^3$.
11. Eigenvalues and eigenvectors of a matrix or a linear map.
12. Invariant subspaces. Jordan form.
13. Bilinear and quadratic forms. Scalar product, orthogonality. Orthogonal complement. Euclidean and unitary space. Linear map of Euclidean and unitary spaces. Affine space. Affine transformation. Translation.
14. Selfcorrecting codes.
 Syllabus of tutorials:

1. Operations with polynomials. Roots of polynomials. Euclid's algorithm. Greatest common divisor. Sets of linear equations. Gaussian elimination method. Linear dependence and independence. Bases, dimensions, vector coordinates in a base. Coordinate transformations. Matrices, matrix operations. Determinants and their calculation.
2. Inverse matrix and its calculation. Linear map, linear map matrix. Eigenvalues and eigenvectors of a matrix. Jordan form. Bilinear and quadratic forms. Scalar product, orthogonality. Affine transformation. Translation. Selfcorrecting codes.
 Study Objective:

The goal of the module is to build basic mathematical way of thinking and provide students
 Study materials:

1. Pták, P. ''Introduction to Linear Algebra''. ČVUT, Praha, 2005.
 Note:
 Timetable for winter semester 2019/2020:
 Timetable is not available yet
 Timetable for summer semester 2019/2020:
 Timetable is not available yet
 The course is a part of the following study plans:

 Bc. Programme Informatics, Part Time Form of Study, in Czech, Version 2015  2020 (compulsory course in the program)
 Bc Branch Security and Information Technology, PartTime Form, in Czech, Version 2015 to 2020 (compulsory course in the program)
 Bc.Branch WSI, Specialization Software Engineering, PartTime Form, Versionverze 2015  2020 (compulsory course in the program)