*Conducted in terms:*2022Z, 2023Z

*Erasmus code:*11.1

*ECTS credits:*5

*Language:*Polish

*Organized by:*Faculty of Physics

*Related to study programmes:*

# Algebra I E 1100-1Ind02

The course will present basic concepts of linear algebra together with necessary bacground in abstract algebra. Covered material will serve as basis for development of more advanced linear algebra, analytic geometry and abstract algebra in the second semester.

**Program:**

1. Basics of linear algebra

(concept of a field, the field of complex numbers, polynomial with coeffitients in a field, divisibility and division of polynomials, Euclid's algorithm, Bezout's theorem, roots)

2. Vector spaces

(vector space, linear independence, basis, dimension, subspaces, sums, direct sums)

3. Linear maps

(linear maps, kernel, range, special classes of linear maps (monomorphisms, epimorphisms, isomorphisms, projections), matrix of a linear map, linear maps of kn, systems of linear equations, elementary operations on matrices, column/row reduction of a matrix, different descriptions of subspaces, matrix of an operator - change of basis)

4. Elements of duality theory

(dual space, dual basis, canonical isomorphism with secon dual, dual/adjoint operator)

5. Multilinear algebra and determinants

(mulitlinear maps, tensor products, permutations, determinants, determinant of an operator, the inverse matrix, invertible operators)

There will be written and oral exams. To take the oral exam the student needs to score at least 50% of points in the written part. In order to take the written part 50% of points from mid-term tests must be scored.

May 2008, Piotr Sołtan

## Prerequisites (description)

## Course coordinators

Term 2022Z: | Term 2023Z: |

## Learning outcomes

After having completed the course student should:

a) know the notion of field of complex number and do calculations with complex numbers

b) understand the notions of a vector space, linear independance, basis

c) understand the notions of a linear mapping and a matrix

d) solve systems of linear equations

e) compute determinanta, find inverese matrix

f) understand the notion of the dual space and the dual map

g) understand the notion of the multilinear map

## Assessment criteria

Midterms and written exam -- computational part;

oral exam --theoretical part.

## Bibliography

1. A. Białynicki-Birula "Algebra"

2. A. Mostowski, M. Stark "Algebra liniowa"

## 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: