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1 UČNI NAČRT PREDMETA / COURSE SYLLABUS Predmet: Matematika 2 Course title: Mathematics 2 Študijski program in stopnja Study programme and level Univerzitetni študijski program 1.stopnje Fizika First cycle academic study program Physics Študijska smer Study field Letnik Acade mic year Semester Semester vse 1 drugi all 1 second Vrsta predmeta / Course type Univerzitetna koda predmeta / University course code: obvezni predmet/core course??? Predavan ja Lectures Seminar Seminar Vaje Tutorial Klinične vaje work Druge oblike študija Samost. delo Individ. work ECTS Nosilec predmeta / Lecturer: Prof. dr. Peter Legiša, prof. dr. Bojan Magajna, prof. dr. Janez Mrčun Jeziki / Languages : Predavanja / Slovensko/Slovene Lectures: Vaje / Tutorial: Slovensko/Slovene Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti: Vpis v letnik. Opravljen izpit iz vaj je pogoj za pristop k izpitu iz teorije. Prerequisits: Enrolment into the first year of the program. Positively graded exercises are necessary for the admittion to the theoretical part of the exam. 44

2 Vsebina: Vektorski prostori: linearna odvisnost, baza, razsežnost, podprostori. Skalarni produkt na vektorskem prostoru: pravokotnost, ortonormirane baze, ortogonalni komplementi, Gram-Schmidtova ortogonalizacija. Matrike: operacije med matrikami, transponirana matrika, kvadratne matrike, grupa obrnljivih matrik, Gaussova eliminacijska metoda, vrstična kanonična oblika matrike, rang, sistemi linearnih enačb, inverzna matrika. Determinante: lastnosti, razvoj po vrstici ali stolpcu, determinanta produkta, Cramerjevo pravilo. Linearni operatorji: matrika linearnega operatorja, lastne vrednosti in vektorji, karakteristični in minimalni polinom, diagonalizabilnost, Cayley-Hamiltonov izrek, invariantni podprostori. Operatorji na prostorih s skalarnim produktom: adjungirani operator, unitarni,, sebi adjungirani in pozitivni operatorji, diagonalizacija sebiadjungiranega operatorja. Bilinearne in kvadratne forme: kanonična oblika za linearne in za ortogonalne transformacije, ploskve drugega reda. Content (Syllabus outline): Vector spaces: linear dependence, basis, dimension, subspaces. Inner product spaces: orthogonality, orthonormal basis, orthogonal complements, Gram-Schmidt orthogonalization Matrices: operations with matrices, transpose matrix, square matrices, invertible matrices, Gauss elimination, systems of linear equations, the row canonical form, rang, inverse. Determinants: properties, computation by expansion over rows or columns, determinant of the product of matrices, Cramer s rule. Linear operators: matrix of a linear operator, eigenvalues and eigenvectors, characteristic and minimal polynomial, diagonalizability, Cayley-Hamilton theorem, invariant subspaces. Operators on inner product spaces: the adjoint, unitary, self-adjoint and positive operators, diagonalization of selfadjoint operators. Bilinear and quadratic forms: canonical form for linear and for orthogonal transformations, quadratic surfaces. 45

3 Metrični prostori: odprte in zaprte množice, notranjost in rob, stekališča in limite zaporedij, kompaktnost, kompaktnost v evklidskih prostorih, zvezne preslikave. Vektorske funkcije več spremenljivk: diferenciabilnost in Jacobijeva matrika, verižno pravilo, ekstremi funkcij več spremenljivk, Hessejeva matrika, izrek o inverzni in implicitni funkciji, vezani ekstremi in Lagrangeva metoda množiteljev. Metric spaces: open and closed sets, interior, closure and boundary, limit points, compactness, compactness in euclidean spaces, continuous maps. Vector functions of several variables: differentiability and the Jacobian, the chain rule, extrema, Hessian, the inverse and the implicit function theorem, constrained extrema and Lagrange multipliers. Temeljni literatura in viri / Readings: 1. S. I. Grossman, Elementary linear algebra. Saunders College Publishing, Orlando, P. R. Halmos, Finite-dimensional vector spaces. Springer-Verlag, New York-Heidelberg, F. Križanič, Temelji realne matematične analize. Državna založba Slovenije, Ljubljana, F. Križanič, Linearna algebra in linearna analiza. Državna založba Slovenije, Ljubljana, D. C. Lay, Linear algebra and its applications. Addison-Wesley, Reading, B. Magajna, Linearna algebra, metrični prostor in funkcije več spremenljivk, DMFA, Ljubljana, M. Dobovišek, B. Magajna in D. Kobal, Naloge iz algebre I, DMFA, Ljubljana, M. H. Protter, C. B. Morrey, Intermediate Calculus. Springer-Verlag, New York-Heidelberg,

4 9. I. Vidav, Višja matematika I. Društvo matematikov, fizikov in astronomov Slovenije, Ljubljana, Cilji in kompetence: Študent spozna osnovne pojme linearne algebre ter pojem in uporabo odvoda vektorske funkcije več realnih spremenljivk. Matematika 2 sodi med temeljne predmete pri študiju fizike. Objectives and competences: Objectives: to familiarize students with basic concepts of linear algebra, metric spaces and with the derivative of a vector function in several variables. Competences: students should be able to apply the methods of linear algebra and calculus in several varibles to problems in physics. Predvideni študijski rezultati: Znanje in razumevanje: Poznavanje in razumevanje osnovnih pojmov in definicij. Uporaba: Uporaba teorije pri reševanju problemov. Refleksija: Razumevanje teorije na podlagi uporabe. Prenosljive spretnosti - niso vezane le na en predmet: Spretnosti uporabe domače in tuje literature in drugih virov, identifikacija in reševanje problemov, kritična analiza. Intended learning outcomes: Knowledge and understanding: understanding of basic concepts and definitions. Application: there are numerous applications of linear algebra and several variable calculus to physics (and other disciplines) Reflection: this part of mathematics is basic for understanding classical and modern physics. Transferable skills: ability to apply mathematical methods. Identification of problems and critical analysis. 47

5 Metode poučevanja in učenja: Predavanja in vaje. Opravljen izpit iz vaj je pogoj za pristop k izpitu iz teorije. Learning and teaching methods: Lectures and exercises. Positively graded exercises are the condition for admittance to the theoretical part of the exame. Načini ocenjevanja: 2 kolokvija namesto izpita iz vaj, izpit iz vaj, Delež (v %) / Weight (in %) 50 Assessment: Written exam or two midterm exams. Izpit iz teorije, domače naloge (niso obvezne) (pozitivno), in 1-5 (negativno) (po Statutu UL). 50 Theoretical exam, homework (optional). Reference nosilca / Lecturer's references: prof. dr. P.Legiša: 1. P.Legiša, Adjacency preserving mappings on real symmetric matrices, Math. commun., Croat. Math. Soc., Divis. Osijek, 2011, vol. 16, no. 2, P.Legiša, Automorphisms of Mn, partially ordered by the star order, Linear multilinear algebra, 2006, vol. 54, no. 3, P.Legiša, Automorphisms of Mn, partially ordered by rank subtractivity ordering, Linear algebra appl. 2004, vol. 389, prof. dr. Bojan Magajna: 1. B. Magajna, Linearna algebra, metrični prostor in funkcije več spremenljivk, DMFA, Ljubljana, 2011 (247 strani). 2. B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math. Anal. Appl. 389 (2012) B. Magajna, The Haagerup norm on the tensor product of operator 48

6 modules, J. Funct. Anal 129 (1995) prof. dr. J. Mrčun: 1. I. Moerdijk, J. Mrčun, On the developability of Lie subalgebroid, Adv. Math. 210 (2007), J. Mrčun, On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc. 133 (2005), I. Moerdijk, J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math. 124 (2002),

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