Neke primjene linearne algebre u geometriji

master's thesis

• Author: Manuela Jurić
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prikazane su samo neke od mnogobrojnih primjena linearne algebre u matematici, posebno u geometriji. Iskazana je Cauchy - Bunjakovski - Schwarzova nejednakost na unitarnim prostorima, te su prezentirana četiri različita dokaza. Potom je prikazana njena važnost na nekoliko primjera iz geometrije (na primjer, definicija kuta u realnom unitarnom prostoru, određivanje maksimuma funkcije). Iskazan je i dokazan Mazur - Ulamov teorem koji karakterizira izometrije (realnih) unitarnih prostora, odnosno daje dovoljne uvjete uz koje su izometrije nužno linearna preslikavanja. Primjena opisanih izometrija je pokazana na dva primjera u ravnini pri svođenju krivulja drugog reda na kanonski oblik.

Abstract (english)

In this thesis the focus is on presenting some of many applications of linear algebra in mathematics, particularly in geometry. The Cauchy - Bunjakovski - Schwarz inequality on unitary (inner product) spaces is stated and four different proofs are given. The importance of this inequality is shown on four examples (e.g.~definition of the angle in real unitary spaces, maximisation of functions under certain constraints). The Mazur - Ulam theorem is stated and a proof presented. By this theorem a characterisation of all isometries on (real) unitary spaces is given, i.e.~sufficient conditions on isometries to be linear mappings are provided. The use of above isometries is shown on two examples on the plane transforming second-order curves into their canonical form.

• Keywords: Cauchy - Bunjakovski - Schwarzova nejednakost na unitarnim prostorima; Mazur - Ulamov teorem; izometrija
• Keywords (english): Cauchy - Bunjakovski - Schwarz inequality on unitary spaces; Mazur - Ulam theorem; isometry
• Language: croatian
• URN:NBN: urn:nbn:hr:217:123779
• Study programme: Title: 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 resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-11-29 12:03:57