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Linear Geometry

Code: 100095
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OB 2

Contact lecturer

Name :
Eduardo Gallego Gómez
Email :
eduardo.gallego@uab.cat

Teaching staff

Alejandro García Sánchez
Eduardo Gallego Gómez

Group languages

You can consult this information at the end of the document.

Prerequisites

To successfully complete this course, students are expected to have a solid command of the basic concepts and techniques of linear algebra covered in the course Àlgebra Lineal. Likewise, it is essential to have consolidated the knowledge and concepts acquired in the course Fonaments de les Matemàtiques.


Objectives

The main objective of this course is to introduce the fundamental concepts of projective, affine, and Euclidean geometry.

On the one hand, students will become familiar with the use of synthetic methods and arguments in geometry, that is, approaches that do not rely on coordinates. This will help them further develop their ability to construct, analyze, and communicate rigorous mathematical reasoning.

On the other hand, particular emphasis will also be placed on the analytical perspective. Students will learn to work with coordinates and perform calculations efficiently, developing the ability to identify the simplest and most appropriate tools and techniques for solving a given problem.


Learning outcomes

  • CM12 (Adapt the tools of geometry to solve problems in other areas of mathematics such as algebra, calculus or statistics.) Adapt the tools of geometry to solve problems in other areas of mathematics such as algebra, calculus or statistics.
  • KM16 (Identify the formal axiomatic systems underlying Euclidean and projective geometries.) Identify the formal axiomatic systems underlying Euclidean and projective geometries.
  • KM17 (List the different types of motion in the plane and in space.) List the different types of motion in the plane and in space.
  • KM18 (Perform elementary geometry proofs.) Perform elementary geometry proofs.
  • SM13 (Classify geometric objects based on their algebraic or analytical properties.) Classify geometric objects based on their algebraic or analytical properties.
  • SM14 (Work with points, vectors, distances and angles in affine and Euclidean spaces, as well as with their corresponding reference systems, subspaces and transformations.) Work with points, vectors, distances and angles in affine and Euclidean spaces, as well as with their corresponding reference systems, subspaces and transformations.

Contents

Affine Geometry

  • Affine spaces.
  • Coordinate systems.
  • Linear varieties (affine subspaces).
  • Simple ratio.
  • Affinities (affine transformations).

Euclidean Geometry

  • Euclidean vector spaces and Euclidean affine spaces.
  • Distances, angles, and volumes.
  • Orthogonal transformations and isometries.
  • Motions of the plane and space.

Projective Geometry

  • Projective spaces and affine charts.
  • Cross-ratio.
  • Duality.

Conics and Quadrics

  • Affine and projective conics and quadrics.
  • Cones and cylinders.
  • Polarity, centers, and tangent varieties.
  • Affine and projective classification of conics and quadrics.

Learning activities and methodology

Title Hours ECTS Learning outcomes
problem solving 41 1.64 CM12, KM16, KM17, KM18, SM13, SM14
seminars 8 0.32 CM12, KM16, KM17, KM18, SM13, SM14
study 30 1.2 CM12, KM16, KM17, KM18, SM13, SM14
test oriented study 10 0.4 CM12, KM16, KM17, KM18, SM13, SM14
exercises 15 0.6 CM12, KM16, KM17, KM18, SM13, SM14
lessons 30 1.2 CM12, KM16, KM17, KM18, SM13, SM14

The course consists of 30 hours of lectures and 15 hours of problem-solving classes. In addition, there will be three seminar/tutorial sessions of two hours each. Attendance at all of these activities is strongly encouraged, as they play a complementary role in the learning process.

Throughout the course, students will regularly receive problem sets to be solved independently. Some of these exercises can be tackled by directly applying the concepts and techniques introduced in the lectures, whereas others require a higher degree of mathematical insight, creativity, and problem-solving ability, providing a stimulating challenge for the student.

During the seminar/tutorial sessions, students will work, typically in small groups, on selected exercises with the guidance and support of the instructor. These sessions are designed to deepen understanding of the course material and to foster the development of effective problem-solving strategies.

However, attendance alone is not sufficient to acquire the competencies associated with the course. Successful learning requires a significant amount of independent study and sustained individual effort throughout the semester.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Handovers and other activities 20% 4 0.16 CM12, KM16, KM17, KM18, SM13, SM14
Exam #2 40% 4 0.16 CM12, KM16, KM17, KM18, SM13, SM14
Second chance exam 80% 4 0.16 CM12, KM16, KM17, KM18, SM13, SM14
Exam #1 40% 4 0.16 CM12, KM16, KM17, KM18, SM13, SM14

Throughout the course, students will complete a number of seminar and/or problem-set assignments. These activities will account for 20% of the final grade and cannot be retaken or recovered.

In addition, there will be two midterm examinations, each contributing 40% of the final grade.

Students will pass the course if they obtain an overall grade of at least 5.0/10. Otherwise, they will be eligible to take a resit examinationcovering all course contents. The grade obtained in the resit examination will replace the grades of the two midterm examinations. The maximum grade that can be awarded through the resit examination is 7.5/10.

Students who choose the single-assessment option will be required to take a comprehensive final examination covering all course contents. The final grade for the course will be the grade obtained in this examination. If this grade is below 5.0/10, the student may take the resit examination, for which the maximum attainable grade is likewise 7.5/10.

A student will receive the grade of Not Assessed if the completed assessment activities account for less than 50% of the total course assessment weight.

Bibliography

Basic Bibliography

  • Euclid, Elements of Geometry.
  • D. Hilbert, Grundlagen der Geometrie.
  • R. Hartshorne, Geometry: Euclid and Beyond.
  • A. Reventós, Projective Geometry.
  • A. Reventós, Affinities, Motions and Quadrics. UAB Manuals, No. 50. Bellaterra, 2008.
  • J. Aguadé, A Course in Linear Geometry. Available at: <http://mat.uab.cat/~aguade/teaching.html>.


Supplementary Bibliography

  • M. Berger, Geometry I. Springer, New York, 1987.
  • A. I. Kostrikin and Yu. I. Manin, Linear Algebra and Geometry. Gordon and Breach Science Publishers, New York, 1989.

Software

Throughout the course, SageMath and GeoGebra may occasionally be employed as complementary tools for visualization, computation, and the exploration of geometric concepts.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan first semester morning-mixed
(SEM) Seminars 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 2 Catalan first semester morning-mixed
(SEM) Seminars 2 Catalan first semester morning-mixed