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master's thesis
Bilinearne i kvadratne forme
Zagreb: University of Zagreb, Faculty of Science, 2016. urn:nbn:hr:217:145897
Halambek, Kristina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Halambek, K. (2016). Bilinearne i kvadratne forme (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:145897

Halambek, Kristina. "Bilinearne i kvadratne forme." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:145897

Halambek, Kristina. "Bilinearne i kvadratne forme." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:145897

Halambek, K. (2016). 'Bilinearne i kvadratne forme', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:145897

Halambek K. Bilinearne i kvadratne forme [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2016 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:145897

K. Halambek, "Bilinearne i kvadratne forme", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2016. Available at: https://urn.nsk.hr/urn:nbn:hr:217:145897

Metadata
TitleBilinearne i kvadratne forme
AuthorKristina Halambek
MentorOzren Perše (mentor)
Committee memberOzren Perše (predsjednik povjerenstva)
Committee memberDražen Adamović (član povjerenstva)
Committee memberVjeran Hari (član povjerenstva)
Committee memberZvonimir Tutek (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2016-11-30, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Tri su glavna tipa bilinearnih formi: simetrične, unitarne i alternirajuće. U ovom diplomskom radu dajemo strukturne teoreme o matričnim prikazima ovih formi s obzirom na odgovarajuće baze. Rad između ostalog, prati standardni postupak rastava objekata na direktnu sumu prostih objekata, koliko je to moguće. Nakon prvog poglavlja, u kojem su definirani osnovni pojmovi potrebni za razumijevanje daljnjeg sadržaja rada, slijedi drugo poglavlje koje je podijeljeno u jedanaest potpoglavlja. Pobliže promatramo bilinearna te kvadratna preslikavanja, a također i bilinearne i kvadratne forme. Dokazujemo također i egzistenciju ortogonalne baze za simetrične bilinearne forme te donosimo dijagonalni prikaz te forme. Zatim promatramo simetrične bilinearne forme definirane nad uređenim poljem te iskazujemo i dokazujemo Sylvesterov teorem te opisujemo induktivni postupak, Gram-Schmidtov postupak ortogonalizacije. Donosimo i dokaz egzistencije ortogonalne baze za vektorski prostor na kojem je definirana hermitska forma, a nakon toga slijede spektralni teoremi, u hermitskom, unitarnom te simetričnom slučaju. Slijede alternirajuće bilinearne forme te uvođenje definicija hiperboličke ravnine te hiperboličkog prostora. Na kraju, iskazujemo te dokazujemo Wittov teorem te definiramo pojam Wittove grupe.
Abstract (english)
There are three major types of bilinear forms: symmetric, unitary, and alternating. In this diploma thesis, we present structural theorems giving normalized expressions for these forms with respect to suitable bases. We follow the standard pattern of decomposing an object into a direct sum of simple objects, insofar as possible. After the first chapter, in which the basic concepts necessary to understand the further contents of the work were defined, follows the second chapter which is divided into eleven subsections. We closely study bilinear and quadratic maps and also bilinear and quadratic forms. We also prove the existence of orthogonal basis for symmetric bilinear forms and present the diagonal representation of this form. Furthermore, we study bilinear symmetric forms defined on an ordered field, we state and prove Sylvester theorem and we describe the inductive process, Gram-Schmidt orthogonalization process. We present the proof of existence of orthogonal basis for a vector space in which the hermitian form is defined, followed by spectral theorems, in hermitian, unitary and symmetric cases. After that we study alternating bilinear forms and define hyperbolic plane and hyperbolic space. Finally, we state the Witt theorem and define the concept of Witt group.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:145897
Study programmeTitle: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education
Course: Mathematics and Computer Science Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2017-03-21 13:38:13