# 230-0541/01 – Geometry (G)

 Gurantor department Department of Mathematics Credits 4 Subject guarantor RNDr. Jana Volná, Ph.D. Subject version guarantor RNDr. Jana Volná, Ph.D. Study level undergraduate or graduate Requirement Optional Year 1 Semester winter Study language Czech Year of introduction 2019/2020 Year of cancellation Intended for the faculties FEI Intended for study types Follow-up Master
Instruction secured by
VOL18 RNDr. Jana Volná, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2
Combined Credit and Examination 16+0

### Subject aims expressed by acquired skills and competences

In this course the students can acquaint with properties of affine and euclidean spaces, with affine, isometric and similar tranformations in this spaces.

### Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

### Summary

Vector spaces. Orientation, spaces with scalar multiplication. Vector multiplication, orthogonal transformations. Affine spaces. Affine coordinates, relations between linear subspaces. Linear systems of hyperplanes. Affine transformations, classification of affine transformations in spaces of dimension 2 and 3. Euclidean spaces. Cartesian coordinates, Euclidean metrics, orthogonality. Isometric transformations. Classification of isometries in spaces of dimension 1 and 2. Similarities.

### Compulsory literature:

Holme, A.: Geometry. Springer – Verlag, Berlin, Heidelberg, New York, 2002, ISBN 3-540-41949-7.

### Recommended literature:

Jennings, G.A.: Modern Geometry with Applications, Springer-Verlag, New York, 1994, ISBN 0-387-94222-X.

### Way of continuous check of knowledge in the course of semester

Conditions for credits of the seminar in Geometry Necessary conditions: 1. At least 70% presence in seminar. 2. Due to presence and activity on seminar students can obtain up to 9 points. Supplementary condition: 3. Three tests, each 0 – 7 points. It is necessary to obtain at least 10 points from possible 30 points. Conditions for exam in Geometry 1. At least 10 points in seminar. 2. The exam consists of written (0 - 40 points) and oral (0 – 30 points) examination. The minimum of written part is 15 points, in oral part is 10 points. 3. The result consists of sum of the points from seminar and from exam. 4. Evaluation is based on VSB-TUO statue. Set of questions: 1. Definition of affine space, affine coordinates, arithmetic affine space, linear independent points, transformation of affine coordinates. 2. Subspaces of affine space, analytical expressions of subspaces, symbolic equations, parametric and general equations of affine subspaces. 3. Mutual position of subspaces, collinearity of linear systems of hyperplanes. 4. Affine transformations of affine space, determinateness of analytical expression, module, invariant points, composition of transformations. 5. Classification of transformation classes. 6. Classification for line and plane. 7. Definition of Euclidean space, Cartesian coordinates, distance of points, transformation of coordinates. 8. Orthogonality and distances in Euclidean spaces. 9. Angles in Euclidean spaces for lines, for line and hyperplane and for subspace and hyperplane. 10. Isometries in Euclidean space, analytical expression, group of isometries. 11. Symmetries, classifications of symmetries, particularly in the plane. 12. Similarity in Euclidean space.

### E-learning

Extra requirements are not.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

1. Affine spaces, affine coordinates. 2. Subspaces of the affine space, analytical formulation of the subspaces. 3. Mutual position of the subspaces. 4. Collinearity, linear systems of hyperplanes. 5. Affine transformations, affine transformation of the affine space. 6. Classification of the affine transformations. 7. Classification of the affine transformations for the line and the plane. 8. Vector spaces with scalar multiplication, Euclidean space. 9. Cartesian coordinates, orthogonality. 10. Distances, perturbations. 11. Isometries. 12. Classification of isometries in spaces of dimension 1 and 2. 13. Similarities. 14. Reserve.

### Conditions for subject completion

Full-time form (validity from: 2019/2020 Winter semester)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 30  10
Examination Examination 70 (70) 25
Written examination Written examination 40  15
Oral examination Oral examination 30  10
Mandatory attendence parzicipation: At least 70% attendance at the exercises. Absence, up to a maximum of 30%, must be excused and the apology must be accepted by the teacher (the teacher decides to recognize the reason for the excuse).

Show history

### Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2019/2020 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Optional study plan
2019/2020 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Optional study plan
2019/2020 (N0541A170007) Computational and Applied Mathematics (S01) Applied Mathematics P Czech Ostrava Choice-compulsory type B study plan
2019/2020 (N0541A170007) Computational and Applied Mathematics (S01) Applied Mathematics K Czech Ostrava Choice-compulsory type B study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner