58414 - Algebra and Geometry

Learning outcomes

At the end of this course, students master some basic notions in abstract algebra (Euclidean algorithm and congruence relation) and in linear algebra (vector spaces, bases, linear maps, eigenvalues and eigenvectors, Euclidean scalar product). They are able to solve linear systems and study linear maps, with particular regard to diagonalization. Moreover, they can recognize and operate withorthogonal bases.

Course contents

Linear Systems

Matrices

Gaussian elimination

Real vector spaces and subspaces

Linear maps

Diaginalization of matrices

Orthogonality in R^n

Orthonormal bases

Orthogonl bases and spectral theorem

Modular arithmetic, congruence classes modulo n

Readings/Bibliography

Fioresi Rita, Morigi Marta, Introduzione all'algebra lineare, Edizioni CEA (2021)

Notes on https://virtuale.unibo.it/

Teaching methods

The course will alternate between theoretical lectures and practical exercises, with the aim of clarifying the theoretical concepts through examples. Students will be encouraged to participate actively in class by asking questions and providing answers.

Attendance is not mandatory; however, being present in the classroom is highly beneficial, as active participation in the lessons greatly facilitates the learning process.

Assessment methods

Learning assessment consists of an exam divided into two written parts: a first written test, comprising practical exercises, and a second written test, containing theoretical questions (definitions, examples and counterexamples, statements of key theorems, and potentially one proof). Both tests are mandatory, will take place consecutively on the same day, and cover the entire course syllabus. The first test lasts 2 hours, and the second lasts 30 minutes.

A maximum of 24 points can be scored in the first written test, and up to 8 points in the second. The final grade will be the sum of the marks obtained in both parts. To pass the exam, students must achieve a minimum score of 14 points in the first test and 4 points in the second.

The instructor reserves the right to request a short oral examination prior to finalizing and confirming the score of the written tests.

Regardless of the marks obtained in the written tests, students wishing to achieve a final grade higher than 27/30 must undergo a brief oral interview consisting of one theoretical question.

Students have the option to withdraw from either test and may reject the proposed final grade once.

Registration for exam sessions must be completed via AlmaEsami, closing 5 days prior to the date of the written examination.

Students must present their student ID card (tesserino universitario) to take the exam.

No reference materials or electronic devices are permitted, except for a non-programmable scientific calculator. Any use of AI tools during the examination is strictly prohibited and constitutes a violation of academic integrity.

Students with Disabilities and Specific Learning Disabilities (SLD)

Students with disabilities or specific learning disabilities (SLD) are entitled to special accommodation tailored to their needs, subject to prior evaluation by the University Service for Students with Disabilities and SLD. Please do not contact the instructor directly; instead, reach out to the Service to arrange an appointment. The Service will determine the appropriate accommodations. For more information, please visit: https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students

Teaching tools

Exercise sheets and exam samples will be provided.

Occasionally, quizzes will be administered using Wooclap.

Office hours

See the website of Marta Morigi