Universitat Internacional de Catalunya
Algebra
Other languages of instruction: Catalan, English
Teaching staff
The office hours will be agreed at ccardo@uic.es
Introduction
Linear algebra is a fundamental subject for technical and scientific studies. Algebra involves the study of concepts such as vector spaces, matrices and systems of linear equations. It provides the student with the necessary means to solve a wide range of problems that govern physical phenomena. Likewise, it constitutes the solid and necessary mathematical base for multiple subjects of the following courses.
Pre-course requirements
None
Objectives
Competences/Learning outcomes of the degree programme
- CB2 - Students must know how to apply their knowledge to their work or vocation in a professional way and have the competences that are demonstrated through the creation and defence of arguments and the resolution of problems within their field of study.
- CB5 - Students have developed the necessary learning skills to undertake subsequent studies with a high degree of autonomy.
- CE1 - To solve the maths problems that arise in the field of Bioengineering. The ability to apply knowledge of geometry, calculate integrals, use numerical methods and achieve optimisation.
- CE4 - To have spatial vision and know how to apply graphic representations, using traditional methods of metric geometry and descriptive geometry, as well as through the application of computer-assisted design
- CG4 - To resolve problems based on initiative, be good at decision-making, creativity, critical reasoning and communication, as well as the transmission of knowledge, skills and prowess in the field of Bioengineering
- CG5 - To undertake calculations, valuations, appraisals, expert reports, studies, reports, work plans and other similar tasks.
- CT5 - To use information sources in a reliable manner. To manage the acquisition, structuring, analysis and visualisation of data and information in your specialist area and critically evaluate the results of this management.
- CT6 - To detect gaps in your own knowledge and overcome this through critical reflection and choosing better actions to broaden your knowledge.
Learning outcomes of the subject
After passing the subject, students will have acquired the following skills:
- A good command of formal mathematical language
- A good domain of matrix calculation
- Ability to solve systems of linear equations
- Ability to identify and characterize vector spaces and subspaces and manipulate vectors
- Ability to identify diagonalizable endomorphisms
- Ability to analyze and synthesize the information obtained in the course
- Ability to use linear algebra programs on a computer
Syllabus
Topic 0. Preliminaries
0.1 Sets and applications.
0.2 Algebraic structures: groups, rings and fields.
0.3 The field of complex numbers.
Topic 1. Linear systems and matrices
1.1 Systems of linear equations.
1.2 Matrices. Elementary transformations.
1.3 Operations with matrices.
1.4 Regular matrices.
1.5 Determinants
Topic 2. Vector spaces
2.1 Definition, properties.
2.2 Linear dependence.
2.3 Generator systems. Bases.
2.4 Vector subspaces. Dimension of subspaces.
2.5 Sum and intersection of vector subspaces. Direct sum.
2.6 Cross product: definition and properties.
Topic 3. Linear applications
3.1 Definition and examples.
3.2 Nucleus and image.
3.3 Injective and exhaustive applications.
3.4 Operations with linear applications.
3.5 Matrix associated with the linear application.
3.6 Base change.
Topic 4. Diagonalization
4.1 Preliminaries (polynomials, factoring).
4.2 Definition of the problem of diagonalization.
4.3 Eigenvalues and autovectors.
4.4 Characteristic polynomial.
4.5 Endomorphism and diagonalizable matrices.
Teaching and learning activities
In person
Methodology
The subject will be taught face-to-face through theoretical classes and problem solving sessions.
The theory of the subject will be exposed in a rigorous way avoiding, however, an excess of formalization, which could mask the true purpose of the subject: teach the fundamentals of linear algebra to future bioengineers. For this reason, conceptual clarity will be emphasized. In addition, students will learn to deal with linear algebra problems using advanced computer programs.
The concepts must be consolidated by solving the exercises that are proposed to the student throughout each topic. These exercises will be resolved or discussed in class. It is also convenient for the student to solve more exercises in the books recommended in the bibliography.
Formation activities
- Master Class and resolution of exercises and problems: 60 h (Presence: 100%)
- Preparation and performance of evaluable activities: 30 h (Presence: 0%)
- Autonomous study and exercise work: 60 h (Presence: 0%)
Evaluation systems and criteria
In person
First sitting
The evaluation of the subject consists of 4 blocks with the corresponding weights:
- (E): Deliveries of short exercises during the course (5%)
- (T): Multiple choice exams (20%): After each topic, students must take an exam with multiple choice questions corresponding to each topic. They do these exams in class. This note is the arithmetic mean of the four exams.
- (P): Midterm exam (20%): In the middle of the course, students must take a theoretical exam where they will evaluate the first and second topic.
- (F): Final exam (55%): At the end of the course, students must take a final theoretical exam where the whole subject will be evaluated.
An essential requirement to pass the subject is to obtain a grade greater than or equal to 4 in the final exam. Otherwise the final grade of the subject will be the final exam grade and the rest of the blocks will not be taken into account.
If the weighted average of these four blocks is greater than or equal to 5 or the final exam grade is greater than or equal to 5, the student has passed the subject. The final grade of the subject will be the maximum between the weighted average of the 4 blocks and the final grade of the final exam.
Second sitting
Students who have not passed the subject in the first sitting will have the opportunity to return to take a final exam. The midterm exam, deliveries and multiple choice exam marks will remain unchanged. The evaluation criteria will be the same as in the first sitting.
Therefore, an essential requirement to pass the subject is to obtain a grade greater than or equal to 4 in the final exam of the second sitting. Otherwise the final grade of the subject will be the final exam grade of the second sitting and the other blocks will not be taken into account.
Then, if the weighted average between E, T, P and the final exam of the second sitting is greater than or equal to 5 or the mark of the final exam of the second sitting is greater than or equal to 5, the student has passed the subject in the second sitting.
Important considerations:
- Plagiarism, copying or any other action that may be considered cheating will be zero in that evaluation section. Besides, plagiarism during exams will mean the immediate failing of the whole subject.
- In the second-sitting exams, the maximum grade students will be able to obtain is "Excellent" (grade with honors distinction will not be possible).
- Changes of the calendar, exam dates or the evaluation system will not be accepted.
- Exchange students (Erasmus and others) or repeaters will be subjected to the same conditions as the rest of the students.
Bibliography and resources
-Luis Miguel Merino y Evangelina Santos: Álgebra lineal con métodos elementales, Ediciones Paraninfo, 2015.
Further reading
-Ferran Puerta Sales: Álgebra lineal. Edicions UPC
-Manuel Castellet y Irene Llerena: Àlgebra lineal y geometria. Universitat Autonoma de Barcelona, Servei de Publicacions.
Evaluation period
- E1 07/01/2021 P2A03 10:00h
- E2 16/06/2021 P2A02 10:00h