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00005 - Algebra

Academic Year 2020/2021

                
                        Docente:
                        Pietro Rigo
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/03
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Pietro Rigo
                            (Modulo 1)
                        
                            Sabrina Mulinacci
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Statistical Sciences (cod. 8873)

                            Teaching resources on Virtuale

Learning outcomes

By the end of the course, the student is expected to know the basic (elementary) theory of linear algebra, and in particular of matrix algebra. Specifically, the student should be able: to work with finite dimensional vectors and matrices, to solve linear systems, to make orthogonal projections in Euclidean spaces, to diagonalize matrices, and to classify real quadratic forms.

Course contents

1. Preliminary notions:

Groups, rings and fields (just a quick mention). Cartesian products. Equivalence relations. Real and complex numbers.

2. Linear spaces and linear transformations:

General definition of a linear space over a field K. Dimension. Isomorphism. Subspaces. Basis. Linear mappings. Dual spaces (just a quick mention).

3. Matrices:

In the remainder, we focus on finite dimensional linear spaces on the real field (namely, K=R). Such spaces are actually isomorphic to R^n. Matrix associated to a linear transformation. Rank. Operations on matrices. Change of the basis. List of some (meaningful) types of matrices.

4. Determinants, inverse matrices, linear systems.

5. Euclidean spaces:

Inner products and norms. Quadratic forms. Projection theorem and some of its consequences.

6. Eigenvalues and eigenvectors:

General definitions. Diagonalizations. The case of symmetric matrices.

Readings/Bibliography

The notes (taken by the students directly) are enough to overcome the exam, obviously provided they are correct and complete. If the notes are not sufficiently clear, and/or to deepen the various topics, the following text-books are suggested:

Abate M. (2000) Algebra lineare, McGraw-Hill

Schlesinger E. (2017) Algebra Lineare e Geometria, Zanichelli

Teaching methods

Lectures and class exercises

Assessment methods

The first part of the exam consists of a written assignment. The second part, subject to overcoming the first, lies in an oral interview

Teaching tools

Notes and the text-books quoted above

Office hours

See the website of Pietro Rigo

See the website of Sabrina Mulinacci