09757 - Geometry and Algebra

Learning outcomes

Basic topics of matrix and vector calculus; in particular: computation of determinants, matrices inverses, linear systems, eigenvalues, eigenvectors and eigenspaces of matrices.

Course contents

 Linear systems.
Algebraic structures. Standard operation on K^n. Linear systems.

Matrices.
Basic definitions. Matrix algebra. Linear systems and matrices.

Vector spaces.
Basic definitions. Subspaces. Linear combinations. Sum of subspaces.

Bases.
Linear dependence. Bases and dimension. Rank of matrices. Application to linear systems.

Linear transformations.
Linear transformations. Isomorphisms and endomorphisms. Imagine and Kernel of linear transformations.

Matrix representation of linear transformations.
Matrix representation of linear transformations and endomorphisms. Bases change formula.

Determinants.
Permutations. Determinants. Properties of determinants. Laplace formula. Inverse matrix. Applications to computation of rank. Applications to linear systems.

Rappresentation of subspaces.
Cartesian and parametric representation of subspaces.

Eigenvalues, eigenvectors and eigenspaces.
Eigenvalues, eigenvectors and eigenspaces of endomorphisms and matrices. Matrix similarity. Characteristic polynomial. Diagonalization of matrices.

Bilinear and quadratic forms.
Bilinear forms. Matrix representation of bilinear forms. Matrix congruence. Index. Real quadratic forms. Canonical forms. Euclidean product, orthonomal bases, Gram-Schmidt algorithm.

Readings/Bibliography

M.R. Casali, C. Gagliardi, L. Grasselli, "Geometria", Esculapio, Bologna, 2016.
A. Cattabriga, M. Mulazzani, "Prove d'esame risolte di Geometria e Algebra per i corsi di laurea in Ingegneria", Esculapio, Bologna, 2014.
A. Barani, L. Grasselli, C. Landi, "Algebra lineare e Geometria - Quiz ed esercizi commentati e risolti", Esculapio, Bologna, 2015.
L. Gualandri, "Algebra lineare e Geometria. Esercizi e quiz risolti e d'esame", Esculapio, Bologna, 2007.

Teaching methods

The course consists of 60 hours of frontal teaching during which the topics will be presented through examples, counterexamples and numerous exercises. The solution of exercises of various difficulty levels will be explained to the students and they will be given exercises to be solved independently. The correction of these exercises will be done later in the class. Some hours will be devoted to the discussion of the students' questions: students will be invited to express any doubt in the class and the resolution of these doubts will be discussed collegially. The structure of a demonstration will be explained to the students through some relevant theorems, albeit with elementary content.

Assessment methods

written and oral exams

Teaching tools

blended teaching

Office hours

See the website of Luca Moci