GEOMETRY I
Academic year and teacher
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- FILIPPO ALFREDO ELLIA
- Credits
- 12
- Didactic period
- Annualità Singola
- SSD
- MAT/03
Training objectives
- The main goal of this course is the study of linear algebra and its applications to affine geometry and euclidean geometry.
Linear algebra is the study of linear phenomenons. These are the simplest ones and so their study is well understood and complete. Linear phenomenon are used to approximate more sophisticated ones (for instance one uses the derivative to study complicated functions). This is why you will find linear algebra anywhere in mathematics. Another goal is to show to the student that this theory can be developed by rigorous proofs starting from some basic facts (logic and set theory). We will show then that affine and Euclidean geometries are just a consequence of linear (and multi linear) algebra. Concerning affine geometry we will stress the correspondence between the geometrical point of view and the algebraic point of view.
At the end of the course the student will be familiar with the main techniques of proof in mathematics (by absurd, induction, contraposition). The student will also be able to solve standard (and less standard) problems in linear algebra and geometry. Prerequisites
- Basic facts of set theory (union, intersection, applications, injectve, surjective, bijective maps). The student must be able to solve, by substitution, a simple linear system of two, three equations in two, three or four unknowns.
Course programme
- I) Preliminaries: basic facts of logic, methods of proof. Set theory, maps. Equivalence relations. Groups. Rings and fields.
II) Linear algebra: vector spaces. Linear maps. Finitely generated spaces. Linear independence, bases. Theorem of the rank, Grassmann’s relation. Ring of endomorphisms. Duality. Omogeneous linear systems and duality. Matrices and linear maps. Change of base. The rank of a matrix. Determinants. How to compute a determinant. Rank and determinants. Diagonalization. Linear systems.
III) Affine geometry: affine subspaces of a vector space. Equations for affine subspaces. Relative position of affine subspaces; complete study in the plane and in a three dimensional space. Independent points, affine basis. Affine maps. General theory.
IV) Euclidean geometry: bilinear forms, symmetric bilinear forms, ortogonality with respect to a bilinear forms. Orthogonal basis. Ortonormal basis. The theorem of Sylvester. Metric spaces and normed vector spaces (sketch). Euclidean spaces. Linear isometries. Isometries. Classification of isometries of the plane. Didactic methods
- Lectures and exercises (homework).
Learning assessment procedures
- During the course some "standard" exercises will be assigned to the students (computation of the rank of a matrix, "standard" exercise on diagonalization of a 3x3 real matrix, diagonalization of a quadratic form, relative position of affine subspaces). To pass the exam the student must be able to solve such exercises. This is the minimal requirement.
More precisely there will be three partial written examinations P1, P2, P3 (mark on 35 points). The first will be on linear algebra. Partial P2 will be on affine geometry (but also with linear algebra questions) and P3 on bilinear forms and euclidean spaces. In each partial examination there will be ‘’standard’’ exercises (but also some more difficult ones).
The final mark, P, for partials is computed as follows. Let Pi >= Pj >=Pk be the three marks. If Pk >= 23 then P = Pi (it is necessary to have done all the three partials), otherwise P = (Pi+Pj)/2.
Then there will be a final written examination, S (mark on 30 points). The student is admissible to the oral examination if A >= 15, where A is computed as follows:
If S < 12, A=S (rejected); if 12<= S < 22, A = max(S, (S+P)/2), if S >=22, A = max(S,P). The final mark is given during the oral examination. The oral examination consists in working some exercises at the blackboard . It is not required to the student to learn the full theory (except for a list of 4-5 theorems which will be given during the course). In general (except if a disaster occurs!) the final mark is >= A. Reference texts
- "Appunti di Geometria I", Ph. Ellia (Pitagora Ed.)
