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. Lipschitz, 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". 

There are two mid-term tests. These tests can substitute the final exam if the student has grade, at least 9.5.  CT is the arithmetic mean of the non-rounded grades of the tests.

To be approved in final exam, the student must have a minimum grade of 9.5 in it.  

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.

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: