Algebra 1

2022/2023
Programme:
Financial mathematics, First Cycle
Year:
1 year
Semester:
first and second
Kind:
mandatory
ECTS:
14
Language:
slovenian
Lecturers:

Assoc. Prof. Dr. Karin Cvetko Vah, Prof. Dr. Primož Moravec

Hours per week – 1. semester:
Lectures
4
Seminar
0
Tutorial
4
Lab
0
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

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.

Readings

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-založništv 2bo, Ljubljana, 2003.
M. Dobovišek, D. Kobal, B. Magajna: Naloge iz algebre I, DMFA-založništvo, Ljubljana, 2005.

Objectives and competences

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.

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homework, 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

Karin Cvetko Vah:
CVETKO-VAH, Karin, LEECH, Jonathan. Rings whose idempotents form a multiplicative set. Communications in algebra, ISSN 0092-7872, 2012, vol. 40, no. 9, str. 3288-3307. [COBISS-SI-ID 16432729]
CVETKO-VAH, Karin. On strongly symmetric skew lattices. Algebra universalis, ISSN 0002-5240, 2011, vol. 66, no. 1-2, str. 99-113. [COBISS-SI-ID 16219993]
CVETKO-VAH, Karin, DOLŽAN, David. Indecomposability graphs of rings. Bulletin of the Australian Mathematical Society, ISSN 0004-9727, 2008, vol. 77, iss. 1, str. 151-159. [COBISS-SI-ID 14680409]

Primož Moravec:
DELIZIA, Constantino, MORAVEC, Primož, NICOTERA, Chiara. Groups with all centralizers subnormal of defect at most two. Journal of algebra, ISSN 0021-8693, 2013, vol. 374, str. 132-140. [COBISS-SI-ID 16556889]
MORAVEC, Primož. Unramified Brauer groups of finite and infinite groups. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 6, str. 1679-1704. [COBISS-SI-ID 16521305]
MORAVEC, Primož. Groups of order p [sup] 5 and their unramified Brauer groups. Journal of algebra, ISSN 0021-8693, 2012, vol. 372, str. 420-427. [COBISS-SI-ID 16521049]