58416 - Linear Algebra and Geometry

Learning outcomes

Once the course is concluded, the student gets the fundamental notions on vector spaces and linear mappings, and is able to solve simple problems of analytic geometry.

Course contents

Preliminaries: sets and functions, number sets, modular arithmetic, polynomials, complex numbers, groups and fields.

Linear systems: resolution, relation with vector spaces and affine spaces, applications to affine geometry.

Matrices: matrix multiplication, invertibility, similarity, congruence, determinant, canonical Jordan form.

Vector spaces: subspaces, generators, linear independence, bases, coordinates, dimension, Grassmann formula, change of basis.

Linear maps: kernel and image, rank theorem, associated matrices, endomorphisms (determinant, eigenvalues and eigenvectors, digonalisability, nilpotence), dual vector space.

Bilinear and quadratic forms: associated matrices, diagonalisability, rank and signature, Sylvester theorem, conics, positive definite scalar products and norms (Cauchy-Schwartz inequality, orthogonality, angles, isometries, Euclidean spaces), Hermitian products, spectral theorem.

Readings/Bibliography

Main book: B. Martelli - https://people.dm.unipi.it/martelli/alg_lin.html (pdf)

Exercises: S. Francaviglia - "Geometria e Algebra T"

Teaching methods

Traditional lessons, exercise sessions, individual and collective meetings.

Assessment methods

The exam consists of a written test and an oral examination.

The written test may be carried out in the traditional way after the course, or by means of some partial tests during the course. During the test it is forbidden to consult books, notes, or electronic devices, with the only exception of a hand-written A4 paper. The test is 3-hour long, and consists of several exercises, each divided into several questions. To the test a score between 0 and 30+ will be assigned.

One has access to the oral examination in case the written test is passed. It is possible to attend the oral test within the next two available dates. The questions may regard definitions, theorems, proofs, examples, or little exercises. Reasoning skill and the quality of the exposition are subject to evaluation. 

The final score reflects an overall evaluation of the two tests. An approximate gradation follows: preparation on basic topics and analysis skill that emerges only with the help of the teacher, autonomous analysis skill only on purely executive matters, correct language → 18-19; preparation on the main topics (proofs of the main theorems included), knowledge of the main statements (not only theorems), ability to prove secondary propositions at least with the help of the teacher, correct and rigorous language → 20-24; preparation on an ample number of topics, ability to prove statements, mastery of language → 25-29; excellence → 30-30L. The final score may be refused at most twice.

Students with learning disabilities or temporary or permanent disabilities: please contact the relevant University office promptly (link). The office will advise students of possible adjustments, that will be submitted to the professor for approval 15 days in advance. He will evaluate their suitability also in relation to the academic objectives of the course.

Teaching tools

Lecture diary, with detailed bibliography and related exercises for each week, old written exams, etc. on Virtuale; individual and collective meetings, weekly tutoring.

Links to further information

https://www.unibo.it/sitoweb/stefano.riolo/avvisi

Office hours

See the website of Stefano Riolo