00005 - Algebra

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. Usually, the latter consists in 2 or 3 simple exercises which are obvious versions of those solved by the teacher.

The second part of the exam, subject to overcoming the first, lies in an oral interview. The possible questions may concern each part of the course. Typically, the interview starts with a very general question (such as "The complex numbers" or "The linear functions") and then, as the topic is introduced, they become more specific. In addition to knowledge of the topics discussed in the course, evaluation criteria are the skill to connect different arguments and the adequacy and consistency of the adopted language. A mnemonic exposition, as well as the inability to discuss with the teacher, are penalized. In other terms, it is important to be able to discuss with the teacher, to be interrupted, and possibly to address some simple objections.

The above remarks do not depend on whether the exam is online or in presence. However, for online interviews, it is desirable (even if not mandatory) that the camera is able to frame the sheet where the student is writing.

Teaching tools

Notes and the text-books quoted above

Office hours

See the website of Pietro Rigo

See the website of Sabrina Mulinacci