09757 - Geometry and Algebra

Learning outcomes

Basic topics of matrix and vectorial calculus; in particular: 
computation of determinants, matrix inverses, linear systems, eigenvalue, 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. Quadratic forms. Real and complex quadratic forms. Canonical forms.

Euclidean vector spaces..
Scalar products. Orthogonality. Orthogonal and orthonormal bases. Orthogonal subspaces.

Readings/Bibliography

M.R. Casali, C. Gagliardi, L. Grasselli, "Geometria", Progetto Leonardo, Bologna, 2002 (testo ufficiale del corso).
G. Parigi, A. Palestini, "Manuale di Geometria - Esercizi", ed. Pitagora, Bologna, 2003.
L. Gualandri, "Esercizi di algebra lineare e geometria", Progetto Leonardo, Bologna, 1995.
S. Lipschutz, M. Lipson, "Algebra lineare", Collana Schaum's, McGraw Hill.

Teaching tools

See at:

http://www.dm.unibo.it/%7Ecattabri/didattica.htm

Links to further information

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

Office hours

See the website of Michele Mulazzani