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

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

Lessons on blackboard

## Assessment methods

See
http://www.dm.unibo.it/~mulazza/didattica.htm

## Teaching tools

See
http://www.dm.unibo.it/~mulazza/didattica.htm