Algebra 1

2022/2023
Programme:
Mathematics Education
Year:
1 year
Semester:
first and second
Kind:
mandatory
ECTS:
14
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)

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-založništvo, 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

Type (examination, oral, coursework, project):
4 midterm exams instead of written exam, written exam
oral exam
grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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]

Klemen Šivic:
KLEP, Igor, MCCULLOUGH, Scott, ŠIVIC, Klemen, ZALAR, Aljaž. There are many more positive maps than completely positive maps. International mathematics research notices. June 2019, vol. 2019, iss. 11, str. 3313-3375. ISSN 1073-7928. [COBISS-SI-ID 18670425]
KANDIĆ, Marko, ŠIVIC, Klemen. On the dimension of the algebra generated by two positive semi-commuting matrices. Linear Algebra and its Applications. [Print ed.]. 2017, vol. 512, str. 136-161. ISSN 0024-3795. [COBISS-SI-ID 17776985]
KUZMA, Bojan, OMLADIČ, Matjaž, ŠIVIC, Klemen, TEICHMANN, Josef. Exotic one-parameter semigroups of endomorphisms of a symmetric cone. Linear Algebra and its Applications. [Print ed.]. 2015, vol. 477, str. 42-75. ISSN 0024-3795. [COBISS-SI-ID 17257561]