Linear algebra

First semester (5 ECTS)
Vectors in Rn. Fundamentals of analytic geometry.
Matrices and determinants. Systems of linear equations.
Basic algebraic structures.
Vector spaces, linear independence, bases.
Linear transformations and their matrices.
Second semester (5 ECTS)
Eigenvalues and eigenvectors of matrices.
Diagonalization and Jordan canonical form.
Inner product spaces, orthonormal bases, adjoint of a linear transformation.
Symmetric, normal and orthogonal matrices.
Quadratic forms. Curves and surfaces of degree two.

Učbeniki in zbirke vaj (Textbooks and problem sets)
J. Grasselli, A. Vadnal: Linearna algebra, linearno programiranje, DMFA založništvo, Ljubljana, 1986.
S. I. Grossman, Elementary linear algebra with applications, McGraw-Hill 1994.
E. Kramar, Rešene naloge iz linearne algebre, DMFA, Ljubljana 1994.
M. Dobovišek, D. Kobal, B. Magajna, Naloge iz algebre I, DMFA, Ljubljana, 2000.
S. Lipschutz: Linear Algebra (Schaum's O.S.), McGraw-Hill, New York 1968.
Elektronski viri (Electronic sourses)
Tomaž Košir, Algebra 1, http://www.fmf.uni-lj.si/~kosir/poucevanje/0910/alg1-fm.html
Bojan Orel, Linearna algebra, http://matematika.fri.uni-lj.si/LA/la1.pdf

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.

Familiarity with basic notions and algorithms of Linear algebra. Ability to apply the knowledge in mathematics and elsewhere.

Lectures, tutorials, homework assignments, consultations.

Written exam, coursework
Oral exam

Jaka Cimprič:
CIMPRIČ, Jaka. Strict positivstellensätze for matrix polynomials with scalar constraints. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2011, vol. 434, iss. 8, str. 1879-1883. [COBISS-SI-ID 15863385]
CIMPRIČ, Jaka. Archimedean operator-theoretic Positivstellensätze. Journal of functional analysis, ISSN 0022-1236, 2011, vol. 260, iss. 10, str. 3132-3145. [COBISS-SI-ID 15997529]
CIMPRIČ, Jaka. Real algebraic geometry for matrices over commutative rings. Journal of algebra, ISSN 0021-8693, 2012, vol. 359, str. 89-103. [COBISS-SI-ID 16315993]
Klemen Šivic:
JELISIEJEW, Joachim, ŠIVIC, Klemen. Components and singularities of Quot schemes and varieties of commuting matrices. Journal für die Reine und Angewandte Mathematik. 2022, vol. 2022, iss. 788, str. 129-187. ISSN 0075-4102. [COBISS-SI-ID 117558275]
BUCKLEY, Anita, ŠIVIC, Klemen. New examples of extremal positive linear maps. Linear algebra and its applications. Aug. 2020, vol. 598, str. 110-144. ISSN 0024-3795. [COBISS-SI-ID 16956675]
KANDIĆ, Marko, ŠIVIC, Klemen. On the dimension of the algebra generated by two positive semi-commuting matrices. Linear algebra and its applications. 2017, vol. 512, str. 136-161. ISSN 0024-3795. [COBISS-SI-ID 17776985]
David Dolžan:
DOLŽAN, David. The probability of zero multiplication in finite rings. Bulletin of the Australian Mathematical Society, ISSN 0004-9727, 2022, vol. 106, no. 1, str. 83-88. [COBISS-SI-ID 94750723]
DOLŽAN, David. Bounds for the number of idempotents in finite rings. Communications in algebra, ISSN 0092-7872, 2021, vol. 49, no. 11, str. 4800-4807, 2021. [COBISS-SI-ID 65073923]
DOLŽAN, David. Zero-divisor graph of direct products of matrices over semirings. Journal of algebra and its applications, ISSN 0219-4988, 2021, vol. 20, no. 5, 10 str. [COBISS-SI-ID 22818819]