Linear algebra

2022/2023
Programme:
Practical Mathematics
Year:
1 year
Semester:
first and second
Kind:
mandatory
ECTS:
13
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Content (Syllabus outline)

Plane vectors and three-dimensional space vectors.
Dot product, cross product, box product.
Lines and planes, distances between points, lines and planes.
Matrices, algebraic operations with matrices.
Elementary transformations, row echelon form.
Systems of linear equations, Gauss elimination.
Determinant.
Classical adjoint and inverse of a matrix, Cramer's rule.
Real and complex vector spaces.
Linear independancy, basis and dimension.
Linear transformations.
Matrices of linear transformations.
Rank and change of basis.
Characteristic polynomial, eigenvalues, eigenvectors.
Cayley-Hamilton theorem, minimal polynomial.
Diagonalization, Schur theorem.
Spectral decomposition , functions of matrices.
Scalar product vector spaces.
Orthonormal basis, Gram-Schmidt orthogonalization.
Linear functionals, adjoint transformation.
Self-adjoint transformations, symmetric and hermitian matrices.
Normal, orthogonal and unitary matrices.
Positive definite matrices.
Quadratic forms, Sylvester theorem.
Second order curves and surfaces.

Readings

J. Grasselli, A. Vadnal: Linearna algebra, linearno programiranje, DMFA založništvo, Ljubljana, 1986.
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.

Objectives and competences

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.

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

4 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Tomaž Košir:
BERNIK, Janez, DRNOVŠEK, Roman, KOKOL-BUKOVŠEK, Damjana, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. On semitransitive jordan algebras of matrices. Journal of algebra and its applications, ISSN 0219-4988, 2011, vol. 10, iss. 2, str. 319-333. [COBISS-SI-ID 15908697]
KOŠIR, Tomaž, OBLAK, Polona. On pairs of commuting nilpotent matrices. Transformation groups, ISSN 1083-4362, 2009, vol. 14, no. 1, str. 175-182. [COBISS-SI-ID 15077977]
BERNIK, Janez, DRNOVŠEK, Roman, KOŠIR, Tomaž, LIVSHITS, Leo, MASTNAK, Mitja, OMLADIČ, Matjaž, RADJAVI, Heydar. Approximate permutability of traces on semigroups of matrices. Operators and matrices, ISSN 1846-3886, 2007, vol. 1, no. 4, str. 455-467. [COBISS-SI-ID 14492761]
David Dolžan:
DOLŽAN, David, KOKOL-BUKOVŠEK, Damjana, KUZMA, Bojan. On the lower bound for diameter of commuting graph of prime-square sized matrices. Filomat. 2018, vol. 32, no. 17, str. 5993-6000. ISSN 0354-5180. DOI: 10.2298/FIL1817993D. [COBISS-SI-ID 24985574]
DOLŽAN, David, OBLAK, Polona. Bounds for the completely positive rank of a symmetric matrix over a tropical semiring. The electronic journal of linear algebra. April 2018, vol. 34, str. 152-162. ISSN 1081-3810. DOI: 10.13001/1081-3810. [COBISS-SI-ID 18362201]
DOLŽAN, David. The sums of exceptional units in a finite ring. Archiv der Mathematik. 2019, vol. 112, str. 581-586. ISSN 0003-889X. https://doi.org/10.1007/s00013-019-01304-x, DOI: 10.1007/s00013-019-01304-x. [COBISS-SI-ID 18591065]