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Algebra with Geometry

General data

Course ID: 1100-1INZ14
Erasmus code / ISCED: 11.101 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title: Algebra with Geometry
Name in Polish: Algebra z geometrią
Organizational unit: Faculty of Physics
Course groups:
Course homepage: http://www.fuw.edu.pl/~werner/alg
ECTS credit allocation (and other scores): (not available) Basic information on ECTS credits allocation principles:
  • the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS;
  • the student’s weekly hourly workload is 45 h;
  • 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes;
  • weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS;
  • work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load.

view allocation of credits
Language: Polish
Type of course:

obligatory courses

Prerequisites (description):

The lecture is supposed to teach the student basic notions of linear algebra and of geometry in euclidean space.

Mode:

Classroom

Short description:

The purpose of this course is to introduce basic notions of linear algebra and geometry in euclidean space. The concepts are introduced in relation to concrete applications; possible generalizations are also discussed. As a result the students should be able to perform basic operations on vectors and matrices (solving systems of linear equations, linear transformations, matrix diagonalization, using orthonormal basis, describing first and second order manifolds in n-dimensional euclidean space).

Full description:

1. Basic algebraic structures. Real and complex numbers.

2. Systems of linear equations, matrices, Gauss elimination.

3. Operations on matrices.

4. Matrices as an example of algebra, inverse of a matrix.

5. Permutation group, determinants.

6. Properties of determinatns. Kramer's formulae, Laplace expansion.

7. Minors, ranges of matrices, inverting matrices.

8. Vector spaces - linear independence, bases.

9. Linear transformations and connection with matrices.

10. Eigenvalues and eigenvectors. Hamilton-Cayley theorem. Functions of matrices.

11. Change of basis, invariants of endomorphisms.

12. Linerar spaces with product. Gram-Schmidt orthogonalization.

13. Unitary and hermitian operators.

14. Quadratic forms, classification of quadrics.

Expected amount of student's labour: 130 hours including 60 hours of lectures and tutorials, 45 hours of homework and 25 hours for preparations to the examinations and the exam itself.

Bibliography:

1. A. Białynicki-Birula, Algebra liniowa z geometrią

2. J. Klukowski, I.Nabiałek Algebra dla studentów Wydawnictwa Naukowo Techniczne , 2004

3. Jacek Komorowski, Od liczb zespolonych do tensorów, spinorów, algebr Liego i kwadryk.

4. J.A. Mostowski i M. Stark, Algebra liniowa

5. S. Gancarzewicz, Algebra liniowa z elementami geometrii, Wydawnicwo Naukowe UJ, Kraków, 2001.

Learning outcomes:

The student should be able to use vectors, linear transformations, matrices. He/she should know the notion and applications of dot product, determinants, eigenvalues, eigenvectors and eigenspaces of linear transformations, rank-2 manifolds in n-dimensional spaces.

Assessment methods and assessment criteria:

In order to pass the course, the student will have to pass recitations (based on two colloquia and short class tests) and the final exam. The details will be announced at the beginning of the semester after consultations with instructors.

This course is not currently offered.
Course descriptions are protected by copyright.
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