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master's thesis
Primjene linearne algebre u dokazivanju nejednakosti
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:377571
Gornik, Lucija
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Gornik, L. (2022). Primjene linearne algebre u dokazivanju nejednakosti (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:377571

Gornik, Lucija. "Primjene linearne algebre u dokazivanju nejednakosti." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:377571

Gornik, Lucija. "Primjene linearne algebre u dokazivanju nejednakosti." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:377571

Gornik, L. (2022). 'Primjene linearne algebre u dokazivanju nejednakosti', Master's thesis, University of Zagreb, Faculty of Science, accessed 26 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:377571

Gornik L. Primjene linearne algebre u dokazivanju nejednakosti [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2024 November 26] Available at: https://urn.nsk.hr/urn:nbn:hr:217:377571

L. Gornik, "Primjene linearne algebre u dokazivanju nejednakosti", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:377571

Metadata
TitlePrimjene linearne algebre u dokazivanju nejednakosti
Title (english)Applications of linear algebra in proving inequalities
AuthorLucija Gornik
MentorJuraj Šiftar (mentor)
Committee memberJuraj Šiftar (predsjednik povjerenstva)
Committee memberVanja Wagner (član povjerenstva)
Committee memberGoran Muić (član povjerenstva)
Committee memberMaja Resman (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-09-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu je pokazano kako se poznavanjem osnovnih pojmova i činjenica iz linearne algebre mogu dokazati zanimljive, često i važne nejednakosti iz različitih matematičkih disciplina. Primjerima su ilustrirane neke primjene u algebri, kombinatorici, kombinatornoj geometriji te teoriji dizajna. Pri dokazivanju samih nejednakosti korištene su poznate činjenice poput svojstava ranga matrice, dimenzije potprostora vektorskih prostora, kriterij postojanja netrivijalnog rješenja homogenog sustava linearnih jednadžbi te Cauchy-Schwarzove nejednakosti. Zastupljen je i poopćeni oblik Binet-Cauchyjevog teorema o determinanti umnoška matrica, koji je takoder koristan u primjenama. Medu rezultatima dokazanima u radu obuhvaćene su Fisherova nejednakost za kombinatorne dizajne te gornja meda veličine ekviangularnog skupa pravaca u d-dimenzionalnom euklidskom prostoru, što pripada problemima kombinatorne geometrije koji već desetljećima zaokupljaju matematičare i o kojima su posljednjih godina postignuti neki značajni rezultati.
Abstract (english)
In this paper, it is shown how some interesting, often important inequalities from various mathematical disciplines can be proved by using basic concepts and facts from linear algebra. Some applications in algebra, combinatorics, combinatorial geometry and design theory are illustrated by examples. Several well-known statements on the properties of the rank of a matrix, dimensions of subspaces of vector spaces, the criterion of the existence of a non-trivial solution of a homogeneous system of linear equations and the Cauchy-Schwarz inequality were used in the proofs. A generalized form of the Binet-Cauchy theorem on the determinant of the product of matrices is also included, as it is useful in applications. Among the results proved in the paper are Fisher’s inequality for combinatorial designs and the upper limit of the size of the equiangular set of directions in the d-dimensional Euclidean space. The latter result belongs to the topics in combinatorial geometry that have been investigated by mathematicians for decades and about which some significant results have been achieved in recent years.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:377571
Study programmeTitle: Mathematics Education; specializations in: Mathematics Education
Course: Mathematics Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2022-10-26 11:31:13