The Algebra course provides the foundations of linear algebra necessary for basic training in engineering, introducing mathematical concepts and tools aimed at the analysis and modelling of scientific and technological problems. Through the study of matrices, linear systems, vector spaces and linear transformations, the course develops the student’s abstract reasoning and analytical skills, forming a transversal basis for later subjects in the mathematical, physical and technological fields within the curriculum.
Titular Professors
Professors
Basic knowledge of matrix algebra is recommended.
The objective of the course is to provide the student with a solid foundation in the fundamentals of linear algebra, developing the ability to understand and use mathematical structures applied to engineering problem?solving. It also aims to promote logical and abstract reasoning, mathematical modelling, and the application of algebraic methods to system analysis, contributing to the development of analytical skills needed in later scientific and technological subjects.
1. Determinants and matrices
1.1. Concept of determinant and properties.
1.2. Determinant calculation.
1.3. Rank of a matrix.
1.4. Inverse of a matrix.
2. Linear equation systems
2.1. Study of systems.
2.2. Solution methods: Cramer, Gauss, Gauss?Jordan, Inverse.
2.3. Joint resolution of similar systems.
3. Vector spaces
3.1. Basic algebraic structures and definition of vector space. Properties.
3.2. Linear dependence and independence of vectors.
3.3. Vector subspace.
3.4. Basis and dimension of a vector space.
3.5. Components of a vector with respect to a basis.
3.6. Change of basis.
4. Linear applications
4.1. Concept, definition, and properties of a linear application.
4.2. Kernel subspace.
4.3. Image subspace.
4.4. Propositions and other definitions.
4.5. Matrix of a linear application and matrix associated with the composition of linear applications.
5. Diagonalization of endomorphisms
5.1. Introduction.
5.2. Invariant subspace.
5.3. Eigenvalue and eigenvector.
5.4. Characteristic polynomial.
5.5. Conditions for diagonalizability.
5.6. Cayley?Hamilton theorem. Application to matrix inversion.
5.7. Applications: computing powers, polynomials, and square roots of matrices.
5.8. Introduction to singular value decomposition (SVD).
6. Euclidean and unitary vector spaces
6.1. Inner product. Euclidean and unitary spaces.
6.2. Norm.
6.3. Angle between vectors.
6.4. Orthogonality and orthogonal subspaces.
6.5. Orthogonal projection. Minimum?error approximation.
6.6. Orthogonal projection onto subspaces of dimension greater than 1.
6.7. Gram–Schmidt orthogonalization.
The teaching methodology is based on an active and theory?practice approach aimed at the progressive acquisition of the learning outcomes defined for the course.
The subject is structured through a weekly schedule of three teaching sessions, combining conceptual introduction, practical application, and consolidation of learning.
The usual structure of the sessions (except those specifically practical) is divided into three phases:
1. First third of the class
Students, organised in groups of three, solve a proposed exercise requiring the application of concepts previously studied autonomously using course materials.
This phase promotes prior preparation, autonomous learning, and collaborative work.
2. Second third of the class
The lecturer conducts a whole?group discussion, addressing doubts arising during teamwork and solving the exercise on the board or using digital tools, depending on its nature.
This phase consolidates theoretical foundations and ensures correct understanding of algebraic procedures.
3. Final third of the class
Students solve a new exercise, again in groups of three, aiming to verify their understanding of the concepts studied.
In certain sessions, this activity is carried out individually and handed in as part of the continuous assessment system.
In summary, the methodology integrates:
- prior autonomous work,
- collaborative learning in the classroom,
- continuous formative assessment,
ensuring coherence between learning activities, assessment system, evaluation criteria and the workload associated with the assigned ECTS credits.
With the aim of assessing whether the student has achieved the objectives pursued in the course to an adequate degree, different tests and data are used to obtain information about the student.
The assessment of the course is carried out through a continuous assessment system (based on exercises, questionnaires and class tests), complemented by individual examinations held in the middle and at the end of each semester.
These 2 examinations account for 70% of the grade for each semester (35% + 35%), while the remaining 30% corresponds to the continuous assessment completed.
The minimum average grade of the two examinations must be equal to or greater than 3.5. If this minimum is not achieved, the weighting with the continuous assessment grade is not applied and the final semester grade will be directly the average grade of the two examinations.
The final grade for the course is calculated by weighting the grade of each semester at 50%. To pass the course, both semesters must be passed separately (grade equal to or greater than 5). If either semester is not passed, the final grade for the course will be a fail regardless of the grade obtained in the other semester.
If a semester is failed in the ordinary examination session, it may be retaken through an examination in the extraordinary examination session. The final grade for the extraordinary examination session will be calculated in the same way as the ordinary final grade.
Important:
According to the update of the plagiarism regulations carried out during the 2017-18 academic year, the following categorization of the course assessment activities is defined:
Midterm Exam: Highly significant.
Final Exam: Highly significant.
Continuous assessment tests: Moderately significant.
The classification of the severity of plagiarism and the disciplinary measures are determined by the current regulations:
https://www.salleurl.edu/ca/normativa-de-copies
Use of AI tools:
AI tools may not be used in the assessment activities of this course (AIAS Level 1). If their use is permitted in any continuous assessment test, it will be explicitly indicated in the activity statement.
The following will be assessed:
- The correct application of algebraic methods in problem solving.
- Rigor and coherence in the development of mathematical reasoning.
- Conceptual understanding of the fundamentals of linear algebra.
- The ability to mathematically model basic technical situations.
- Precision in calculations and the correct interpretation of the results obtained.
- Clarity and structure in the presentation of procedures and solutions.
Notes and exercises in linear algebra associated with the course (available on eStudy).
Other books that may be consulted:
- Linear Algebra and its Applications ; David C. Lay, Steven R. Lay, Judi J. Mcdonald; Fifth Edition Pearson 2016
- Elementary Linear Algebra; Howard Anton, Chris Rorres; 11th Edition; Wiley 2014
Further literature can be found in Spanish and Catalan (see Spanish and Catalan versions).