714-0441/02 – Geometry (G)

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

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.

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

Conditions for credits of the seminar in Geometry Necessary conditions: 1. At least 80% presence in seminar. 2. Processing of two projects: I. Examples of partially parallel spaces of the affine space (dimension n≥4 ). II. Examples of the distance of partially parallel, nonparallel and nonintersecting spaces (dimension n≥4 ). 3. If all necessary conditions are fulfilled – 9 points. Supplementary condition: 4. 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.