Universitat Internacional de Catalunya
Algebra
Other languages of instruction: Catalan, Spanish
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
- CP03 - Apply the techniques of graphical representation, both by traditional methods of metric geometry and descriptive geometry, as well as by computer-aided design applications.
- HB09 - Solve problems that may arise in the field of Bioengineering by applying mathematical knowledge (geometry, integral calculation, numerical methods and optimisation) and the general laws of mechanics and biomechanics.
Learning outcomes of the subject
Upon completion of this course, students will be able to:
• Identify vector spaces and subspaces and work with them, as well as diagonalizable endomorphisms.
• Define multivariable functions while deepening the understanding of concepts and methods in multivariable differential calculus.
• Recognize the fundamentals and principles for the study of mathematics, including its basic and specific terminology.
• Determine how mathematics applies to the geometry of the human body, including calculating moments of inertia and centers of gravity.
• Associate appropriate professional attitudes for future practice.
• Integrate the scientific method, promoting reasoning and problem discussion.
• Recognize primary sources of information.
• Interpret real-life problems that can be solved using linear algebra and numerical calculus methods.
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:
P: (Project) submissions of exercises, tests or works.
M: (Mid term exam) partial exam of half of the syllabus and which will take place in the middle of the course.
F: (Final Exam) Final exam of the whole syllabus.
Evaluation:
C: continuous evaluation is the weighted average consisting of C=20%P + 20%M + 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 F was less than 3.5, 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.
Students who register for the second time or more may present the work upon express request and under the conditions specified and in no case will the grade of the work or any exam grade from the previous year count. If the formal request is not made by mail during the first two weeks of the course, the continuous assessment grade will be C=40%P + 60%F. The final grade will be calculated in the same way as mentioned above.
Important considerations:
- Plagiarism, copying or any other action that may be considered cheating will be zero grade in that evaluation section. Besides, plagiarism during an exam will mean the automatic 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.
Important considerations
- Plagiarism, copying or any other form of academic dishonesty will result in a grade of zero for the corresponding component.
- If academic dishonesty is detected during an exam, it will result in the immediate failure of the course, with no chance of resitting.
- The use of artificial intelligence tools for the completion of assessment activities is strictly prohibited, except where their use is expressly authorized by the lecturer as part of the activity.
- The use or possession of electronic devices (mobile phones, smartwatches, earbuds, etc.) during exams is strictly prohibited.
Mere possession, even if the device is turned off, will be considered an attempt to cheat.
- If this occurs during the first call, it will result in the automatic failure of the exam, and the student will be required to attend the second call.
- If it occurs during the second call, it will result in the definitive failure of the course, and the student must re-enrol in the next academic year.
- No changes to the academic calendar, exam dates or evaluation system will be accepted under any circumstances.
- Exchange students (Erasmus or others) and repeaters are subject to the same evaluation and attendance conditions as all other students.
Bibliography and resources
- David C. Lay et al.: Linear algebra and applications, Pearson, 2016
- 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 06/11/2025 P2A03 12:00h
- E1 16/01/2026 A10 10:00h