The student is supposed to acquire basic knowledge on inner product spaces and analytic geometry (vide syllabus) in a deductive and critical perspective

General characterization





Responsible teacher

Ana Cristina Malheiro Casimiro


Weekly - 5

Total - 70

Teaching language



Familiarity eith to the contents of Linear Algebra I.


1. Monteiro, A., Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 2001.

2. Anton, H., and Rorres, C., Elementary Linear Algebra -  Applications Version, 8th Edition, John Wiley & Sons, 2000.

3. Giraldes, E., Fernandes, V. H., and Marques-Smith, M. P., Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 1995.

4. Santana, P., Queiró,  J.F., Introdução à Álgebra linear, Gradiva 2010

5. Lipschutz, S., Linear Algebra - Shaum''''s Outline of Theory and Problems

Teaching method

There are classes in which theory is lectured and illustrated by examples. There are also problem-solving sessions. Some exercises are left to the students to be solved on their own as part of their learning process.

Evaluation method

Students enrolled for the first time in the unit must attend all classes, except up to 3 lectures and up to 3 problem-solving classes.

Students that have already been enrolled in the unit must attend, at least, 2/3 of the lectures and 2/3 of the problem-solving classes.

The students that do not fulfill the above requirements automatically fail "Geometria". 

Continuous evaluation

During the semester two tests will be carried out with a duration of 1 hour 30 minutes and an evaluation of the practical classes (ap). Each test is rated up between 0 and 18 values ​​and the practical classes can be rated 0, 1 or 2 values.

 1st Test (t1): all students enrolled in the course may present themselves to the 1st test.

2nd Test (t2): all the students enrolled in the course that have obtained a frequency or have a special status may submit to the 2nd test.

Evaluation of the practical classes: the teacher of the practical class in which the student is enrolled will provide at the end of the semester the classification between  0 and 2 values. This corresponds to the evaluation made by the teacher, through the student''''s performance in solving the problems proposed in the classes and doing a presentation of a part of the sylabus.

The classification of continuous evaluation (AC) is obtained by the following formula:

AC =min{20, (t1+t2)/2 + ap}

The student is approved in the course if AC is greater than or equal to 9.5 values. 


All the students enrolled in the course that have obtained Frequency or have special status may submit to the Exam.

At the date and time scheduled for the Exam any student enrolled in the course that has obtained Frequency or has special status and who has not obtained approval in the Continuous Evaluation can take the exam for 3 hours or can opt for repeat one of the tests for 1 hour 30 minutes. If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation.

If the student performs the Exam (his or her classification is er) and

AE =min{20, er + ap}

is greater than or equal to 9.5, the student is approved with the classification AE.

Grade improvement

Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they may take the 3-hour Exam or repeat one of the 1-hour 30-minute Tests as described in the previous paragraph. In the case that a student wants to improve his grade, having obtained approval in a previous semester, he can only take the 3-hour Exam.


In tests and in exam, any kind of consultation is not allowed.

In tests and in exam, students are allowed to use the sheets of paper provided by the professor and a ballpoint and nothing else.

More detailed rules are available in the portuguese version. 

The non-portuguese students should address the professor to ask any question that is not in this english version.

Final considerations

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

Subject matter

1. Inner product spaces – Definition of inner product and elementary properties. Euclidean space and unitary space. Matrix of an inner product (relative to a fixed basis). Norm. Schwarz inequality. Triangle inequality. Angle between two non-zero vectors of a euclidean space. Orthogonal and orthonormal (finite) vector systems. Gram-Schmidt orthogonalization process. Orthogonal complement. Cross product and mixed product.

2. Bilinear forms and quadratic forms – Definitions and elementary properties. Polar form.

 3. Affine Geometry.

3.1 Affine spaces – Definition and dimension. Affine euclidean space. Affine subspace. Incidence propositions. Coordinate system of an affine space. Point coordinates. Vectorial, cartesian and parametric  equations of affine subspaces.

3.2 Euclidean or metric geometry in euclidean affine spaces –Orthogonal affine subspaces. Distance and angles. Quadric surfaces.


Programs where the course is taught: