Linear algebra

Enrollment into the program

Vectors in the plane, scalar product, line in plane, linear transformations and matrices in the plane.

Vectors in R3, cross and triple product, areas and volumes, lines and planes, distances between points, lines and planes, linear transformations and matrices.

General matrices and algebraic matrix operations, determinant, systems of linear equations, Gauss elimination, inverse of the matrix.

Real and complex vector spaces, linear independence, basis and dimension, scalar product in Euclidean space.

Linear transformations and matrices of linear transformations, rank, and transition to a new base.

Characteristic polynomial, eigenvalues, eigenvectors, diagonalization.

Real symmetric matrices, conic sections and surfaces of the second order.

T. Košir: Zapiski s predavanj iz Linearne algebre (spletna učilnica)
E. Kramar: Rešene naloge iz linearne algebre, DMFA založništvo, Ljubljana, 1994.
S. I. Grossman: Elementary linear algebra with applications, McGraw-Hill, 1994.
D. C. Lay: Linear algebra and its applications, Reading: Addison-Wesley, 1994.
The linear algebra problem solver : a complete solution guide to any textbook. Piscataway: Research and Education Association, 1993.

Students get familiar with the basic concepts of linear algebra, necessary for further study: basics of two and three-dimensional euclidean geometry, matrix algebra, solving systems of linear equations, calculating with polynomials and basic elements of abstract algebra. They learn a mathematical way of thinking and achieve practical and working knowledge from the field of linear algebra.

Knowledge and understanding:
Knowledge and understanding of the basic concepts and methods of linear algebra. Application of the achieved knowledge.

Application:
Linear algebra is one of the fundamental subjects in the study of natural, technical, social and almost all other science fields.

Reflection:
Integrating theoretical and practical procedures for solving basic practical problems.

Transferable skills:
Mathematically correct formulation of problems, the choice of appropriate methods, capability of acurate solving of problems and analysis of obtained results.

 

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam
Oral exam
5 - 10, a student passes the exam if he is graded from 6 to 10

Tomaž Košir:

1. M. E. Hochstenbach, T. Košir,  B. Plestenjak. Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models. Numer. Linear Algebra Appl. 31 (2024), no. 2, No. e2540, 23 pp.

2. T. Košir, B. Plestenjak. On the singular two-parameter eigenvalue problem II. Linear Algebra Appl. 649 (2022), 433–451.

3. A. Buckley, T. Košir. Simultaneously self-adjoint sets of 3×3 matrices. Rend. Istit. Mat. Univ. Trieste 47 (2015), 81–105.

4. A. Buckley, T. Košir. Plane curves as Pfaffians. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 2, 363–388.

5. T. Košir, P. Oblak. On pairs of commuting nilpotent matrices. Transform. Groups 14 (2009), no. 1, 175–182.

6. T. Košir. The Cayley-Hamilton theorem and inverse problems for multiparameter systems. Linear Algebra Appl. 367 (2003), 155–163.

Jasna Prezelj:

1. J. Prezelj, F. Vlacci: On a class of automorphisms in  H which resemble the property of preserving volume, Math. Nachr. 294 (2021), no. 4, 815--835

2. J. Prezelj, F. Vlacci: An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in ${\mathbb {H}}^2,$ AMPA online (2022), DOI 10.1007/s10231-022-01195-w

3. T. Knific, M. Ocepek, A Kirbiš, B. Krt, J. Prezelj, J. M. Gethmann:  Modeling Paratuberculosis Transmission in a Small Dairy Herd Typical of Slovenia Suggests That Different Models Should Be Used to Study Disease Spread in Herds of Different Sizes, Animals 2022, 12(9), 1150; https://doi.org/10.3390/ani12091150

4. T. Knific, M. Ocepek, A Kirbiš, B. Krt, J. Prezelj, J. M. Gethmann: Quantitative Risk Assessment of Exposure to Mycobacterium avium subsp. paratuberculosis (MAP) via Different Types of Milk for the Slovenian Consumer,  Foods 2022, 11(10), 1472;

5. G. Gentili, J.Prezelj, F. Vlacci: Slice conformality and Riemann manifolds on quaternions and octonions,   Math.Z., 2022, DOI 10.1007/s00209-022-03079-4.

6. G. Gentili, J.Prezelj, F. Vlacci: On a definition of logarithm of quaternionic functions, JCNG,, No 17 (3), DOI: 10.4171/JNCG/514, 1099--1128

7. G. Gentili, J.Prezelj, F. Vlacci,  On a continuation of quaternionic and octonionic logarithm along curves and the winding number, Volume 536, Issue 1, 1 August 2024, 128219, https://doi.org/10.1016/j.jmaa.2024.128219