Linear Algebra and Elements of Geometry (2020/2021)

Course code
Francesca Mantese
Teaching is organised as follows:
Unit Credits Academic sector Period Academic staff
ALGEBRA LINEARE 6 MAT/02-ALGEBRA I semestre Francesca Mantese
ELEMENTI DI GEOMETRIA 6 MAT/03-GEOMETRY See the unit page See the unit page

Learning outcomes

First of all, the students are introduced to the language and formal reasoning required for the study of higher mathematics. Furthermore, the main notions and techniques of linear algebra and matrix theory are presented, focussing both on theoretical and computational aspects. Moreover, the course provides an introduction to planar and spatial analytic geometry, within the projective, affine, and euclidean setting. Finally, the main properties of conics will be discussed. Both analytical (coordinates, matrices) and synthetic tools will be employed.

At the end of the course the student must be able to demonstrate an adequate synthesis and abstraction ability, be able to recognize and produce rigorous demonstrations and be able to formalize and solve problems of moderate difficulty, limited to the syllabus of the teaching.


The entire course will be available online. In addition, a number of the lessons (see the course
schedule) will be held in-class.

Groups, fields. The field of complex numbers. Matrices, matrix operations and their properties. Determinant and rank of a matrix. Inverse matrix. Systems of linear equations. The method of Gaussian elimination. Vector spaces, subspaces, bases, dimension. Linear maps.
Eigenvalues and eigenvectors. Canonical form. Affine and Euclidean spaces. Lines, planes, hyperplanes. Vector product and mixed product. Affine and isometric transformations. Projective spaces. Geometry of projective plane. Conics.

The course consists of front lessons and classroom exercises. Moreover optional tutoring activities are offered. In particular, weekly home exercises are given. They are individually corrected by a tutor and discussed during the exercise hours.

Assessment methods and criteria

The exam aims to verify the ability to solve problems on the teaching program, the possession of an appropriate capacity for analysis, synthesis and abstraction, and the ability to recognize and produce rigorous demonstrations.

The exam consists of:
- a joint written examination on the module Linear Algebra and the module Elements of Geometry,
- a joint oral examination on both modules.

Only students who have passed the written examination will be admitted to the oral examination.
The oral examination can also be supported in a subsequent exam session.
Voting obtained in the written test will remain valid until the February 2022 exam session.

Intermediate Testing: for each module there are two partial tests, on dates that will be communicated to the students after the beginning of the lessons.

Bonus exercises: Each week will be assigned exercises to be done at home preparing for the written test. Your works will be corrected individually by a tutor. A good score in the exercises gives rise to a bonus for the exam.

The assessment methods could change according to the academic rules. The remote mode is however guaranteed for all students who will ask for it in the academic year 2020/21.

Reference books
Author Title Publisher Year ISBN Note
E.Gregorio, L.Salce Algebra Lineare Libreria Progetto Padova 2005
Abate, M. Algebra Lineare Mc Graw Hill 2001
Candilera,Bertapelle Algebra lineare e primi elementi di Geometria Mc Graw Hill   9788838661891