Linear Algebra

Relations:

During a review of study plans, the course BIK-LA1.21 can be substituted for the course BIK-LIN.

Course guarantor:

Karel Klouda

Lecturer:

Karel Klouda

Tutor:

Karel Klouda

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 error-detecting and error-correcting 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. Self-correcting 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. Self-correcting 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:

Further information:

https://courses.fit.cvut.cz/BI-LIN/

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bachelor program Informatics, unspecified branch, in Czech, part-time, 2015 – 2021 (compulsory course in the program)
• Bachelor branch Security and Information Technology, in Czech, part-time, 2015 - 2019 (compulsory course in the program)
• Bachelor branch Web and Software Engineering, spec. Software Engin., in Czech, part-time, 2015–2020 (compulsory course in the program)
• Bachelor branch Security and Information Technology, part-time, in Czech, 2020 (compulsory course in the program)