Real three-dimensional space. Geometric and algebraic structure of space, vectors. Inner, cross, and triple product. Analytic geometry, planes and lines.

Basic algebraic structures. Relations. Operations and homomorphisms. Groups. Permutation groups. Rings and fields. Vector spaces and linear maps. Algebras.

Finite dimensional spaces. Basis and dimension. Quotient space and direct sum of subspaces. Dual space and dual map.

Linear maps. Space of linear maps and matrices. Change of basis, equivalence and rank. Systems of linear equations.

Endomorphisms. Algebra of endomorphisms and quadratic matrices. Similarity. Determinants. Eigenvalues. Characteristic and minimal polynomial. Jordan form of an endomorphism. Spectral decomposition and functions of matrices.

Inner product spaces. Inner product and norm. Gram-Schmidt orthogonalization. Riesz representation theorem. Hermitian adjoint map.

Normal endomorphisms. Diagonalization. Self-adjoint endomorphisms. Unitary endomorphisms. Unitary similarity of endomorphisms and matrices. Positive definite endomorphisms and matrices.

Quadratic functionals. Bilinear functionals. Congruence and Sylvester's inertia theorem. Second order curves and surfaces.

# Algebra 1

F. Križanič: Linearna algebra in linearna analiza, DZS, Ljubljana, 1993.

J. Grasselli: Linearna algebra, 1. pogl. v I. Vidav: Višja matematika II, DZS, Ljubljana, 1981.

I. Vidav: Algebra, DMFA-založništvo, Ljubljana, 2003.

M. Dobovišek, D. Kobal, B. Magajna: Naloge iz algebre I, DMFA-založništvo, Ljubljana, 2005.

Student gets familiar with the basic concepts of linear algebra that are needed for the further study in mathematics. He learns to think mathematicaly and practices the rigorous mathematical language. At tutorials the student acquires practical applied knowledge of the subject.

Knowledge and understanding: Knowledge and understanding of basic concepts and definitions in linear algebra.

Application: Solving problems using the theory.

Reflection: Understanding of the theory from the applications.

Transferable skills: The skill to transfer the theory into practice.

Lectures, exercises, homework, consultations

4 midterm exams instead of written exam, written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Boris Lavrič:

LAVRIČ, Boris, JERMAN, Marjan. Three-dimensional l-algebras. Communications in algebra, ISSN 0092-7872, 2009, vol. 37, no. 2, str. 684-711. [COBISS-SI-ID 15108697]

LAVRIČ, Boris. Coherent Archimedean f-rings. Communications in algebra, ISSN 0092-7872, 2000, let. 28, št. 2, str. 1091-1096. [COBISS-SI-ID 9502041]

LAVRIČ, Boris. The isometries and the G-invariance of certain seminorms. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2003, vol. 374, str. 31-40. [COBISS-SI-ID 12751193]

LAVRIČ, Boris. Delno urejeni vektorski prostori, (Podiplomski seminar iz matematike, 22). Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, 1995. 152 str. ISBN 961-212-049-8. [COBISS-SI-ID 53745152]

Peter Šemrl:

RODMAN, Leiba, ŠEMRL, Peter. Neutral subspaces of pairs of symmetric/skewsymmetric real matrices. The electronic journal of linear algebra, ISSN 1081-3810, 2011, vol. 22, str. 979-999. [COBISS-SI-ID 16067929]

MBEKHTA, Mostafa, ŠEMRL, Peter. Linear maps preserving semi-Fredholm operators and generalized invertibility. Linear and Multilinear Algebra, ISSN 0308-1087, 2009, vol. 57, no. 1, str. 55-64. [COBISS-SI-ID 15058521]

CHEBOTAR, M. A., ŠEMRL, Peter. Minimal locally linearly dependent spaces of operators. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2008, vol. 429, iss. 4, str. 887-900. [COBISS-SI-ID 14810969]