Universitat Internacional de Catalunya

Algebra

Algebra
6
12470
1
First semester
FB
FUNDAMENTALS
MATHEMATICS I
Main language of instruction: Spanish

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

The objectives of this subject are basically two: to achieve a basic mathematical training and obtain own knowledge of Linear Algebra.

Competencies

  • 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


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 1. Logic, set theory and algebraic structures

1.1 Introduction.

1.2 Logic and set theory.

1.3 Algebraic structures: groups, rings and fields.

1.4 The complex field.

Topic 2. System of linear equations and matrices

2.1 System of linear equations.

2.2 Kinds of systems and matrix representation.

2.3 Algebra of matrices. Operations with matrices.

2.4 Gauss method.

2.5 Determinant and rank of matrices.

2.6 Rouché-Frobenius theorem.

Topic 3. Vector spaces

3.1 Definition of vector space. Examples.

3.2 Linear combination of vectors.

3.2 Linear independence of vectors.

3.3 Generator set and base.

3.4 Vector subespaces.

3.5 Space dimension.

Topic 4. Linear maps

4.1 Definition of linear map.

4.2 Kernel and image of a linear map.

4.3 Kinds and operations of linear maps.

4.4 Matrix representation of a linear map.

4.5 Change of base.

Topic 5. Diagonalization of endomorphisms

5.1 Eigen vectors and eigen values.

5.2 Characteristical polynomial and eigen subspaces.

5.3 Diagonalization of endomorphisms.

5.4 Orthonormal bases.

5.5 Orthonormal diagonalization.

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



During the course it will be necessary to present and carry out the following evaluation tasks:

E: four deliveries consisting of works of technical application of the theory or exercises.

P: partial exam of half of the syllabus and which will take place in the middle of the course.

F: final exam of the whole syllabus.

Evaluation: 

C: continuous evaluation is the weighted average consisting of C=20%E + 20%P + 60%F.

FINAL GRADE: the final grade in the first call will result from calculating the maximum between the continuous evaluation mark C and the mark of the final exam F. However, when was less than 4, F will be the final grade. 

In the second call will remain without altering the mark of the deliveries and the mark of the partial exam. Only final exam will be repeated and the same system of evaluation as in the first call will be applied. 

Re-sit:

Students who repeat the subject will have to deliver the 4 "homeworks" again and perform both exams.

Important considerations:

  1. 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.
  2. In the second-sitting exams, the maximum grade students will be able to obtain is "Excellent" (grade with honors distinction will not be possible).
  3. Changes of the calendar, exam dates or the evaluation system will not be accepted.
  4. Exchange students (Erasmus and others) or repeaters will be subjected to the same conditions as the rest of the students.

 

Bibliography and resources

Basic bibliography

-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

E: exam date | R: revision date | 1: first session | 2: second session:
  • E1 03/11/2022 A06 12:00h
  • E1 03/11/2022 P2A03 12:00h
  • E1 11/01/2023 P2A02 10:00h
  • E1 11/01/2023 P2A03 10:00h