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:

Cartesian products. Equivalence relations. Real and complex numbers. Polynomials and the Fundamental Theorem of Algebra

 

2. Linear spaces and linear transformations:

General definition of a linear space. Dimension. Isomorphism. Subspaces. Basis. Linear mappings.

 

3. Matrices:

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, integrated with some lecture notes provided by the teacher.

If the notes are not sufficiently clear, and/or to deepen the various topics, the following text-books are suggested:

Abate M.: Algebra lineare, McGraw-Hill

Fioresi R., Morigi M.: Introduzione all'algebra lineare. Seconda edizione, Cea

Schlesinger E.: Algebra Lineare e Geometria, Zanichelli

Teaching methods

Lectures and class exercises

Assessment methods

The exam is of written type with two theoretical questions and three exercises which are obvious versions of those solved by the teacher.

Teaching tools

Notes and the text-books quoted above

Office hours

See the website of Sabrina Mulinacci