GEOMETRY
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- ROSSANA CHIAVACCI
- Credits
- 12
- Didactic period
- Annualità Singola
- SSD
- MAT/03
Training objectives
- This is the first course on linear algebra and geometry
The main goal of the course consists in providing the bases to tackle the study of the geometry defined by the particular algebraic structures to be fitted in the finite-dimensional space, with particular reference to the real ones.
The main acquired knowledge will be:
depth knowledge of vector spaces of finite dimension;
knowledge of morphisms between vector spaces ;
knowledge of Euclidean spaces ;
knowledge of isometric and symmetric operators in Euclidean spaces.
The basic acquired abilities (that are the capacity of applying the acquired knowledge) will be:
analysis and use of elements of finite dimensional vector spaces ;
to visualize and recognize geometrically what has been theoretically treated in general in low-dimensional environments;
recognizing of the characteristics of an Euclidean space;
to know how to analyze in depth the isometric transformations;
classification of quadrics. Prerequisites
- Elements of analytical geometry and elementary algebraic calculus, in particular: Cartesian coordinates, polynomials, first and second degree equations and inequalities, systems of linear equations in two or three variables, elementary trigonometry.
Course programme
- The course includes 108 teaching hours during which the following topics will be covered:
homogeneous e non homogeneous linear systems; matrices, determinant, rank of a matrix; system's matrix and its canonical form; row operations. Vector spaces; linear independence , basis; change of basis;linear maps; matrices of linear maps; dimension's theorem, Grassmann's relation.Invariant spaces, eigenvectors. eigenvalues. Diagonalization of matrices. Affine subspaces of a vector space; parallelism, incidences. Bilinear forms, the matrix of a bilinear form; quadratic forms, diagonalization of quadratic forms, Sylvester's theorem. Euclidean geometry: definitions and first properties, orthogonality, angles. Adjoint operators and spectral theorem. Linear isometries: classification in dimension two and three; general structure theorem. Symmetric operators: general structure theorem. Classification of quadrics. The orthogonalization theorem, orthogonal basis; Gram's determinant; volums. Didactic methods
- Frontal lectures on all the topics of the course; written exposition of the theory with performed exercises carried out as illustrative examples of the treated theory.
Learning assessment procedures
- The aim of the exam is to verify at which level the learning objectives previously described have been acquired.
The examination is divided in 2 sections that will be place in different days.
One written test with solutions of numerical exercises based on all topics tackled in the class. The aim of the test is evaluating how deeply the student has studied the subject and how he is able to understanding the basic topics analyzed; the section is selective as the student who does not show a sufficient knowledge of the subjects, cannot be admitted to the oral section. The time allowed for the written test is 2 hours and 30 minutes . To pass the test it is required to get at least 18 points out of 30 ; It is only allowed to consult personally handwritten notes, written on an A4 sheet that can hold all the knowledge deemed useful for passing the test ;
one oral section where the knowledge of the basic concepts and results of the theory and the ability of linking different subjects related to geometry are evaluated, rather than the ability of “repeating” specific topics tackled in the course; demonstrations of some of the main theorems are required;
The final mark takes account of the mark of the written test and the oral presentation that does show a sufficient knowledge of the subjects; if this knowledge is not considered sufficient, it is necessary to repeat only the oral section in a subsequent exam session.
To pass the exam it is necessary to get at least 18 points of 30. Reference texts
- Textbook of the course:
R.Chiavacci - C.Alessi - D.Rebba, "GEOMETRIA" , Pitagora Editrice, Bologna, 2021
