Double Qualification in Engineering Studies in Audio-visual Systems and Multimedia

Double Degree in Engineering Studies in Audio-visual Systems and Multimedia

La Salle Campus Barcelona offers 5 double degrees in the ICT Engineering and the Business Management field. With the double degrees, you can finish the university studies in 5 academic years with two official degree qualifications

In this course, typical topics of linear algebra are presented. The intention is not only for students to assimilate theoretical concepts, but also their association with technical world problems. That is, a great effort is made to achieve that the student sees algebra as a tool to solve real problems. For this purpose, all along the course, different practical applications of concepts studied in the classroom to different engineering areas are presented. Applications in digital signal processing, digital image processing, digital modulations or in computer graphics world are an example of this. Some of these technical situations are posed and solved by the teacher in the classroom, and others situations are posed for the students to model and solve them using suitable algebraic tools.
Type Subject
Primer - Obligatoria

Titular Professors

Previous Knowledge

Basic knowledge of matrix algebra.


Students will acquire the following capabilities and knowledge:

1. Mathematical background essential for the degree studies.

2. Written communication improvement in their native language in the different reports given to the teacher.

3. Capacity for applying algebraic knowledge to practice.

`Teaching well is helping to discover what you want to transmit´. Student´s achievement of the association between algebraic concepts and technical real situations is the goal. If students realize this association is possible, then they will probably increase their motivation towards the subject. It is necessary to remember that the first element for students´ learning is students´ motivation.


The study plan for this subject is the following:

1.- Determinant and matrix
1.1- Determinant concept and properties.
1.2- Determinant calculus.
1.3- Matrix rank.
1.4- Matrix inverses.

2. Linear equations systems
2.1- Linear systems study.
2.2- Solving linear equations systems: Cramer, Gauss and Gauss-Jordan elimination, inverse matrix method.
2.3- Solving similar systems.

3.- Vector spaces.
3.1- Definition and properties.
3.2- Linear dependence and linear independence.
3.3- Vector subspaces.
3.4- Vector space basis and dimension.
3.5- Coordinates of a vector with respect to a basis.
3.6- Change of basis.

4.- Linear transformations.
4.1- Linear transformation concept. Definition and properties.
4.2- Null space of a linear transformation.
4.3- Image space of a linear transformation.
4.4- Propositions and others definitions.
4.5- Matrix representation of a linear transformation.
4.6- Composition of linear transformations.

5.- Eigensystems.
5.1- Introduction.
5.2- Invariant subspace.
5.3- Eigenvalues and eigenvectors.
5.4- Characteristic polynomial.
5.5- Cayley-Hamilton´s theorem. Application to matrix inversion.
5.6- Conditions to existing diagonal matrix.
5.7- Matrix decomposition.
5.8- Applications.
5.9.- Introduction to the Singular Value Decomposition (SV)

6.- Euclidean vector spaces.
6.1- Inner product.
6.2- Cauchy-Schwarz inequality.
6.3- Norm, angle and distance in vector spaces.
6.4- Orthogonal concept. Orthogonal subspace.
6.5- Orthogonal basis.
6.6- Orthogonal projection. Least square error approximation.
6.7- Gram-Schmidt orthogonalization process.


Through the course, different kind of teaching methods are used:

1. Lectures.

The teacher uses lectures to transmit all the theoretical concepts during the course. The teacher also solves some exercises for direct application of the taught concepts.

2. Personal work and study

3. Exercises to solve at home.

These exercises goal is to establish theoretical concepts in order to apply them to practical situations afterwards. This includes also the online questionnaires that the students must complete after finishing each unit.

4. Introductory sessions to Matlab ®.

During the second term of the course, 1 class is dedicated to introduce students to Matlab simulation environment. The goal is that students use Matlab to validate their reasoning when they try to solve the real problems posed.

5. Exposition of technical situations where the application of algebraic tools helps to solve the problem posed.

The teacher uses some class hours to present different real situations in technical environments where the theoretical concepts studied in classroom help to solve the problems posed to students. These presentations are done using ALGTEC (ALGebra and TEChnology) program. This multimedia application has been designed and generated at La Salle. ALGTEC allows the teacher to show students the different situations modeled, analyzed and solved applying algebraic tools. ALGTEC can be consulted by students on the Internet too through the eStudy platform. In ALGTEC students can find, not only a whole exposition of every technical situation presented in the classroom, but also an experimentation block associated with every situation in order to do different tests to completely understand the presented situation.

6. Teamwork in the lab.

Students work in the lab organized in groups to solve a real world problem using algebra concepts. The solution to the problem must be implemented in Matlab.

7. Personal work in the lab.


In order to evaluate if the student has reached the subject objectives adequately different tests are used to obtain information about each student:

1. Exams

During the course four exams are done: two during each semester.

2. Class tests.

3. Classroom participation

The teacher has an observation checklist where he takes notes about the student´s different attitudes and behaviours during the class.

4. Individual or team reports.

Evaluation Criteria

The final grade for the course is calculated by weighting with 50% each of the final marks of each semester. The final mark of each semester is calculated as follows: 60% exams marks, and 40% continuous assessment marks. Exams mark is calculated by weighting with 30% the midterm exam, and 70% the final exam of the semester. Continuous assessment mark is calculated by weighting with 30% participation and attitude mark, and a 70% knowledge mark. Attitude mark is computed from class attendance and student attitude during the lectures. The knowledge mark is calculated from classroom exercises and test, besides online questionnaires completed by students during the semester. In the second semester, the mark obtained in the practice group has a significant weight (40%) in the continuous assesment mark.

The criteria applied in each of the evaluation systems used are specified:

Each examination (midterm or final semester exam) is evaluated from a list of criteria tailored to each exercise in the exam. These criteria are set by the faculty. The test is evaluated with a numerical score from 0 to 10, allowing the use of decimals.

Tests or exercises in the classroom:
They are assessed on 4 levels, as shown in the rubric 1. Sometimes the teacher complements the letter assigned to the exercise with a "+" or "-" sign.

This category also includes online questionnaires, which are offered to students through eStudy at the end of each unit. These questionnaires, however, are evaluated automatically, assigning an integer mark between 0 and 10 to each questionnaire.

Finally, the mark associated with this evaluation system is computed by weighting the marks of the classroom tests, the online questionnaires and exercises completed at home by students. The percentage allocated to each item may vary slightly in each semester. However, in all cases, the weight assigned to the tests shall be greater than that assigned to the online questionnaires and these, in turn, will have a greater weight than exercises completed at home by the students.

Finally, letter marks are converted into numerical marks by using the following correspondence:

Letter mark Numerical mark
A 10
B 7
C 4
D 1

When the letter mark is complemented by a plus sign `+´, one point is added; it it is complemented by a minus sign `-`, one point is subtracted.

RUBRIC 1: Classroom tests and exercises

A. The student shows a full understanding of the concepts that appear in the test. (10 p)
B. The student, in general, understands the concepts, but makes some non-fatal errors. (7 p)
C. The student demonstrates a poor understanding of concepts that lead to major errors. (4 p)
D. The student leaves much of the test unanswered, or makes serious errors that prove that the concepts have not been understood at all. (1 p)


Class participation is also graded according to four levels, as shown in rubric 2.

RUBRIC 2: Class participation (attitude)

(Between 9 and 10): The student attends class regularly, participating and making contributions of quality.
(Between 7 and 8): The student attends class regularly and their attitude is not negative.
(Between 4 and 6): The student is frequently truant but the attitude is not negative when attending class or attend regularly but their attitude is negative.
(1 to 3): The student rarely attends class, and shows a negative attitude when in class.


The work in group is evaluated following rubric 3.


Quality of the generated video (2p)
A. Technically well recorded. Explanations are complete, detailed, clear and easy to follow. (2p)
B. Acceptable quality recording. Explanations easy to follow, but lacking detail. (1.4 p)
C. Low technical quality recording. Explanations are difficult to follow and to understand. (0.8 p)
D. Low technical quality recording. Explanations are difficult to follow. Incomplete. (0.2 p)

Quality of the report (format) (2 p)
A. The report has the required structure, without spelling mistakes. (2p)
B. The report structure or the number of misspellings deviate slightly from what is considered adequate. (1.4 p)
C. The report structure or the number of misspellings are far from what is considered appropriate. (0.8 p)
D. The report structure or the number of misspellings differs a lot from what is appropriate, yielding a low quality report. (0.2 p)

Quality of the report (contents) (6p)
A. All the questions posed were adequately answered. (6p)
B. Most of the questions posed have been answered adequately. (4.2 p)
C. Some of the posed questions are correctly answered, but they are outnumbered by the unanswered or incorrectly answered questions. (2.4 p)
D. None or almost none of the posed questions has been properly answered. (0.6 p)

Basic Bibliography

- Theory and problems collection, Enginyeria La Salle, [NRG 13]
- Lluís Bermudez, Enrique Pociello, Álgebra Lineal (domina sin dificultad), Ediciones Media
- Juan Flaquer, Javier Olaizola, Juan Olaizola, Curso de Álgebra Lineal, Eunsa, 1996

Additional Material

- Ben Noble, James W. Daniel, Applied linear algebra, Prentice Hall, 1988
- Larson-Edwards, Introducción al álgebra lineal, Ed. Limusa,1995
- Howard Anton, Introducción al álgebra lineal, Ed. Limusa, 1997
- Castellet, M. i Llerena, I., Àlgebra lineal i geomètrica, Universitat Autònoma de Barcelona, 1990
- Queysanne, M., Álgebra básica, Vicens Vives, 1990
- Rojo, A., Álgebra lineal, AC 1991
- Puerta, F., Álgebra Lineal, Marcombo, 1991
- Luzarraga, F., Problemas resueltos de Álgebra Lineal, 1970
- Lipschutz, S., Álgebra lineal, McGraw-Hill, 1991