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Analytic representation of affine transformations in the plane
Horáčková, Blanka ; Beran, Filip (advisor) ; Zamboj, Michal (referee)
The thesis deals with three main topics - isometric, similar, and affine transformations of the plane from the point of view of analytic geometry. In the first chapter, the most basic concepts are recalled, which will be subsequently dealt with throughout the thesis. The second chapter focuses on identical transformations. Here, we find an analytic representation of this transformation in matrix and complex forms. The third chapter focuses on similar transformations. The central point is then the decomposition of similarity into identity and identicalness, where knowledge of both similarities and identities are combined. The last chapter focuses on affinities. This chapter is not so theoretical anymore but focuses mainly on the characteristic elements of affine transformations and examples. This work's important and beneficial factor is the solved examples, supplemented by several figures. The work is intended primarily for mathematics students as a study material. However, it may also be used by secondary school teachers to supplement the secondary school curriculum.
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Mappings in geometry
Trkovská, Dana ; Kubát, Václav (advisor)
This diploma dissertation is dedicated to applications of geometrical mappings. It is intended as a tuitional material specially for students of the third year of the mathematics teachers programm at Mathematical and Physical faculty of Charles University in Prague. The text can be used as a supplementary material for a seminar at secondary school as well. It is based on lectures of the course Geometry II. Students are familiar with the term mapping already during the lessons at elementary and secondary schools. Therefore in the diploma dissertation we at first give only a summary of basic knowledge about mappings in geometry, in the language of mathematics textbooks. Next part of this thesis includes theoretical knowledge about mappings in geometry in the form of definitions and propositions together with their proofs. A great part is dedicated to characterization of affine mappings, specially isometries and similarities. At the end circular inversion is explained as an example of a mapping that is not affine. For better imagination the whole text is complemented with a number of figures. Theoretical part is followed by a collection of exercises. Of course, solutions of all exercises are given.
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Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Zamboj, Michal (advisor) ; Jančařík, Antonín (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
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Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Tůmová, Veronika (advisor) ; Zamboj, Michal (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
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Mappings in geometry
Trkovská, Dana ; Kubát, Václav (advisor)
This diploma dissertation is dedicated to applications of geometrical mappings. It is intended as a tuitional material specially for students of the third year of the mathematics teachers programm at Mathematical and Physical faculty of Charles University in Prague. The text can be used as a supplementary material for a seminar at secondary school as well. It is based on lectures of the course Geometry II. Students are familiar with the term mapping already during the lessons at elementary and secondary schools. Therefore in the diploma dissertation we at first give only a summary of basic knowledge about mappings in geometry, in the language of mathematics textbooks. Next part of this thesis includes theoretical knowledge about mappings in geometry in the form of definitions and propositions together with their proofs. A great part is dedicated to characterization of affine mappings, specially isometries and similarities. At the end circular inversion is explained as an example of a mapping that is not affine. For better imagination the whole text is complemented with a number of figures. Theoretical part is followed by a collection of exercises. Of course, solutions of all exercises are given.
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