29228 - Geometry and Algebra T

Learning outcomes

At the end of the course, students will be familiar with the main tools of linear algebra (matrices, vector spaces, linear systems, eigenvalues and eigenvectors) and will know how to apply them in a geometric context. They will understand the links between the different parts of the theory and will be able to apply it operationally.

Course contents

Review of numbers (real, complex, integers mod p) and functions.

Vector spaces. Linear combinations and linear dependence; linear and affine subspaces; systems of generators, bases and dimension; operations on subspaces and Grassmann's formula.

Linear systems: matrix notation; Gaussian elimination; parametric and Cartesian representation of subspaces of R^n; structure theorem for linear systems; Rouché-Capelli theorem; parametric systems and resolution techniques.

Linear functions, kernel and image, rank theorem. Matrix representations, composition of functions and matrix product, invertibility, change of basis. Invariants for conjugation: rank, trace, determinant, characteristic polynomial; calculation techniques. Eigenvalues, eigenvectors, triangulability and diagonalizability. Linear duality, spaces of matrices.

Standard scalar product and vector product in R^3. Metric concepts: orthogonal projection, Gram-Schmidt orthogonalisation. Orthogonal matrices. Symmetric matrices, spectral theorem, quadratic forms and signature.

Readings/Bibliography

Suggested readings:

■ (IT) Abate, de Fabritiis, "Geometria analitica con elementi di algebra lineare", McGraw-Hill

■ (IT) Sernesi, "Geometria I", Bollati Boringhieri (capitoli 1 e 2)

■ (EN) Schilling, Nachtergaele e Lankham, "Linear Algebra", LibreTexts

■ (EN) Boyd e Vandenberghe, "Introduction to Applied Linear Algebra", Cambridge University Press

Problems and solutions:

■ (IT) Abate, de Fabritiis, "Esercizi di geometria", McGraw-Hill

Teaching methods

Lectures with examples and exercises. Tutorials, subject to availability.

Assessment methods

The exam will be written and divided into two parts:

-a one-hour multiple-choice test will verify that students have learned the computational algorithms introduced in the course and are able to independently solve exercises similar to those seen in class;

-a 30-minute open-ended written test will assess students' learning and understanding of the theoretical content of the course and their mastery of the mathematical language.

A maximum of 24 points can be obtained in the test and a maximum of 10 in the written exam. The final mark will be the sum of the partial results. A minimum of 12 points must be obtained in the test in order to be admitted to the written exam, which will take place on the same day as the test after a short break. It is not possible to take the test and the written exam in two different sessions.

In order to be admitted to the written exam, you must register on Almaesami, arrive in the classroom on time, and bring your university badge and ID card with you.

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

The teaching material will be posted on Virtuale.

Office hours

See the website of Luca Battistella