*Conducted in term:*2020Z

*Erasmus code:*14.3

*ISCED code:*0311

*ECTS credits:*6

*Language:*Polish

*Organized by:*Faculty of Economic Sciences

*Related to study programmes:*

# Linear Algebra 2400-PP1AL

1. Systems of linear equations: solutions and general solutions., matrices, elementary matrix operations, solving the system using Gaussian elimination.

2. Linear (or vector) spaces: examples, linear subspaces, linear combination of vectors, linear indepedence, basis and dimension of a linear space, coordinates of a vector in a basis.

3. Linear transformations: examples, matrix of representation of a transformation in a basis. Operations on transformations and matrices, matrix algebra.

4.Determinants: properties and methods of calculations.

5. Matrix inverse and transpose, methods of computing.

6. Rank of a matrix, its relation to elementary operations and determinants.

7. Applications of determinant and rank of a matrix to solving systems of linear equations. Kronecker – Capelli Theorem and Cramer Theorem.

8. Eigenvectors and eigenvalues of linear transformations. Characteristic polynomial of a matrix. Finding of eigenvalues and bases of eigenspaces. Diagonal matrices and diagonalisable matrices.

9. Applications of matrix diagonalisation .

10. Affine subspaces (or layers) of linear spaces. Affine tranformations.

11. Standard scalar product: vector norm, orthogonal systems of vectors , orthogonal and orthonormal bases, Gram – Schmidt procedure. Orthogonal projection and symmetry.

12. Quadratic forms: examples, matrix of a form. Sylvester cryterion of positive definitness. Application of eigenvalues to definitness and semidefinitness.

!3. Basics of linear programming. The symplex method.

## Type of course

## Prerequisites (description)

## Course coordinators

## Mode

## Learning outcomes

The ability to understand and use linear algebra in statistics, econometrics and mathematical models of decision making. Basic techniques of linear algebra including: solving systems of linear equations, finding bases and dimension of linear spaces, determination of rank of a matrix, calculation of determinants and matrix inverse, finding eigenvectors, determination of positive (negative) definitness of a quadratic form. Ability of formulation and solution of a simple linear programming model.

KW01, KU01

## Assessment criteria

The final grade is calculated on the basis of total points earned during the semester in class (up to 20 points depending on results of 3 colloquia, short tests every week and class activity) and the final exam (up to 80 points ).

## Bibliography

Wykłady z Algebry Liniowej I. Tadeusz Koźniewski, MIMUW 2008.

Algebra liniowa w zadaniach. Jerzy Rutkowski. PWN 2008.

Matematyka dla studentów ekonomii. Ryszard Antoniewicz, Andrzej Misztal. PWN 2009.

Algebra dla studentów. Julian Klukowski, Ireneusz Nabiałek. WNT 1999.

Elementy algebry liniowej. Grzegorz Banaszak, Wojciech Gajda. WNT 2002

## 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: