*Conducted in terms:*2019/20_L, 2020/21_L, 2021/22_L, 2022/23_L

*ECTS credits:*6

*Language:*Polish

*Related to study programmes:*

# Linear algebra WM-I-ALL

Description of the course:

1. Sets. Cartesian products. Relations ( orderings, partitions and equivalence relations ).

2. Relation between two sets, graph, function.

3. Definitions, theorems, proofs.

4. Groups, fields, linear spaces. Group homomorphism, field isomorphism, vector space linear transformation.

5. Structure of linear spaces. Linear combination of vectors, span of a set of vectors. Linear independence, basis, and dimension; linear subspace. Coordinates of a vector relative to a basis , matrix representation of a vector,

6. Linear transformations between finite-dimensional vector spaces. Matrix representations of linear transformations.

7. Algebraic operations on matrices. Composition of linear transformations and matrix multiplication. Linear combination of vectors and matrix multiplication.

8. Algebraic and geometric structure of the field of complex numbers. Complex numbers as ordered pairs of real numbers, an extension of the real numbers by adjoining an imaginary unit, complex numbers as matrices of linear transformations in R2

9. Geometric interpretation of complex numbers. The complex plane: the polar form, Euler’s and de Moivre’s formulas, geometric interpretation of the operations on complex numbers. Exponentiation and root extraction.

10. Matrices and determinants. Definitions, properties and calculating

determinants - Laplace’s expansion, elementary operations.

11. Matrices and determinants. Matrix multiplication non-commutative. Matrix invertibility. Inverce matrix. Oriented volume.

12. Vector and matrix forms of systems of linear equations.

Existence and number of solutions: the Kronecker-Capelli theorem.

13. Solution methods for systems of linear equations. Gauss elimination, Cramer’s rule, inverse matrix method.

14. Kernel and image of a linear transformation, pre-image of a vector and related systems of linear equations.

15. Geometric interpretation of solution sets of homogeneous and non-homogeneous systems of linear equations as linear and affine subspaces in Rn.

## (in Polish) E-Learning

## (in Polish) Grupa przedmiotów ogólnouczenianych

## Subject level

## Learning outcome code/codes

## Type of subject

## Course coordinators

Term 2021/22_L: | Term 2018/19_L: | Term 2020/21_L: | Term 2019/20_L: |

## Bibliography

1. A.I. Kostrikin, Wstęp do algebry, Cz. I: Algebra liniowa, PWN, Warszawa 2004.

2. S. Zakrzewski, Algebra i geometria, Wydawnictwo UKSW, Warszawa 2006.

3. A. Kostrykin, Zbiór zadań z algebry, PWN, Warszawa 2005.

4. T. Jurlewicz, Z. Skoczylas, Algebra liniowa 1 Przykłady i zadania, Oficyna Wydawnicza GiS, Wrocław 2003.

## Additional information

Information on *level* of this course, *year of study* and semester when the course
unit is delivered, types and amount of *class hours* - can be found in course structure
diagrams of apropriate study programmes. This course is related to
the following study programmes:

Additional information (*registration* calendar, class conductors,
*localization and schedules* of classes), might be available in the USOSweb system: