Linear Algebra

Lecturer:

Jaroslav Milota

Tutor:

Tomáš Kalvoda, Jaroslav Milota

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 polynomials 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 error-detecting and error-correcting codes.

Requirements:

High-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 Euclidean algorithm. Binary operation, its properties. Group, ring, field. Homomorphisms (isomorphisms). Properties of a field. Finite fields.

4. Systems 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. Inverse matrices.

9. Linear map, linear map matrix. Rotation, projection onto a straight line (plane), symmetry with respect to a straight line (plane) in $R^2$, $R^3$.

10. Eigenvalues and eigenvectors of a matrix or a linear map.

11. Invariant subspaces. Jordan form.

12. 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.

13. Self-correcting codes.

Syllabus of tutorials:

1. Operations with polynomials.

2. Roots of polynomials. Euclidean algorithm. Greatest common divisor.

3. Sets of linear equations. Gaussian elimination method.

4. Linear dependence and independence.

5. Bases, dimensions, vector coordinates in a base. Coordinate transformations.

6. Matrices, matrix operations.

7. Determinants and their calculation. Inverse matrices and their calculation.

8. Linear map, linear map matrix.

9. Eigenvalues and eigenvectors of a matrix.

10. Jordan form.

11. Bilinear and quadratic forms.

12. Scalar product, orthogonality. Affine transformation. Translation.

13. Self-correcting codes.

Study Objective:

The aim of the module is to build the foundations of mathematical way of thinking and provide students with basic knowlege of linear algebra necessary to solve systems of linear equations or problems in 2D and 3D analytic geometry.

Study materials:

1. Pták, P. ''Introduction to Linear Algebra''. ČVUT, Praha, 2005.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue	roomTH:A-1030 Milota J. 09:15–10:45 (lecture parallel1) DejviceroomTH:A-1030 Milota J. 14:30–16:00 (lecture parallel1) Dejvice
Fri	roomTH:A-1030 Kalvoda T. 16:15–17:45 (lecture parallel1 parallel nr.101) Dejvice
Thu
Fri

The course is a part of the following study plans:

• Information Systems and Management (Presented in English) (compulsory course in the program)
• Information Technologies (Presented in English) (compulsory course in the program)
• Computer Engineering (Presented in English) (compulsory course in the program)
• Software Engineering (Presented in English) (compulsory course in the program)
• Computer Science (Presented in English) (compulsory course in the program)
• Web and Multimedia (Presented in English) (compulsory course in the program)
• Informatics (Presented in English) (compulsory course in the program)