GEOMETRY
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- ALBERTO CALABRI
- Credits
- 9
- Didactic period
- Secondo Semestre
- 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
- Linear Functions and dimensional theorem.
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 basic elements of a linear function . Prerequisites
- Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane. First elements of mathematical logic: definition, 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 ) . Linear functions . Kernel and image concepts . Dimensional theorem. Fundamentals properties of linear functions . Linear functions and matrices . Eigenvalues and eigenvectors of a linear operator ( 15 h ) . Didactic methods
- Lectures to introduce the theoretical concepts . Exercises relating to the application of these concepts.
Learning assessment procedures
- The goal 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
