GEOMETRY AND ALGEBRA
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- Versione italiana
- Academic year
- 2019/2020
- Teacher
- GIULIANO MAZZANTI
- Credits
- 9
- Didactic period
- Primo semestre (primi anni)
- SSD
- MAT/03
Training objectives
- The main objective of the course is to provide students with the basics of Linear Algebra and Analytical Geometry , fundamental for other scientific teachings . The main knowledge gained will be related to:
- Basic elements on vector spaces
- Fundamental theorems on linear systems
- Diagonalization of matrices
- Analytical representation of lines , planes, spheres , cylinders , cones
- Tensors , dyads
- Spectral decomposition of a symmetric tensor
- Product tensor and the scalar product of tensors
- Fundamental Theorem of dyads and tensors .
The main skills will be :
- Solve problems related to vector spaces dependent on parameters ,
- Discuss linear systems of various tipologies containing parameters
- Determine whether a matrix is diagonalizable ,
- Solve problems of analytic geometry of space ,
- - Determine the spectral Decomposing of a symmetric tensor ,
- Determine the axial vector of a emisimmetrico tensor in R3 and vice versa ,
- Decompose a tensor in the sum of a symmetric tensor and an emisymmetric tensor. Prerequisites
- Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane. First elements of mathematical logic : concepts , theorem , demonstration , role of examples and counterexamples .
Course programme
- The course includes 90 hours of teaching between lessons and exercises. The topics covered in the course are the following .
Vector spaces ( 15 h ) . Matrices , determinants , linear systems and applications ( 16 h ) . analytic geometry in space ( 13 h ) . Euclidean spaces (8 h ) . Orthogonal matrices ( 5 h ) . Diagonalization of a matrix . Diagonalization of a symmetric matrix with an orthogonal matrix ( 10 h ) . Quadratic forms . Reduction to diagonal form . Square root of a matrix . Applications of quadratic forms ( 8 h ) . Tensor . Tensor remarkables . Matrix of a tensor . Eigenvalues ¿¿and eigenvectors of a tensor ( 5 h ) . Dyads . Spectral decomposition of a symmetric tensor . Emisymmetric tensors in R3 and axial vector ( 5 h ) . Product tensor and the scalar product of tensors . Fundamental Theorem of dyads and tensors ( 5 h ) . Didactic methods
- Lectures to introduce the theoretical concepts . Exercises relating to the application of these concepts. Also receiving students for questions and clarifications.
Learning assessment procedures
- The aim of the examination is to test the level of achievement of knowledge , skills and abilities related to topics previously mentioned .
Examination is divided into two parts , which take place on different days .
The first part consists of a written test on the application of the concepts introduced ( exercises ) .
The second part consists of a written test on the theoretical aspects of the course topics , also customized based on the outcome of the previous trial .
The final grade takes account of both tests .
If the student fails to achieve a minimum of 18 to 30 must repeat both tests .
Passing the exam is proof that he has acquired the knowledge and skills specified in the learning objectives of teaching. ' Reference texts
- Giuliano Mazzanti-Valter Roselli
"Appunti di Algebra Lineare, Geometria Analitica, Tensori: Teoria, Esempi,Esercizi svolti,Esercizi proposti"
Pitagora Editrice, Bologna 2013
Giuliano Mazzanti-Valter Roselli
Esercizi di Algebra Lineare e Geometria Analitica
Pitagora Editrice Bologna 1997
