Description

The subject Linear Algebra is taught in the first four-month period of the first course of the Grado en Ingeniería de Tecnologías de Telecomunicación, with the main objective of providing students with a clear understanding of the complex numbers, systems of linear equations and elementary techniques of matrix algebra as well as an introduction to the fundamental concepts of Vector Spaces which will be needed in later subjects. Special attention will be paid to the applications of Linear Algebra.

Instructors

González Rodríguez, Ramón

Martín Méndez, Alberto Lucio

Contents

• Topic 1. Complex Numbers – Operations with complex numbers. Geometric concepts associated with complex numbers. Euler's formula and its consequences.
• Topic 2. Matrices and Determinants – Matrix operations: addition, scalar multiplication and matrix multiplication. Matrix inverse. Block matrices. Determinants.
• Topic 3. Systems of Linear Equations – Systems of linear equations. Elementary row operations and the Gauss method. Numerical methods for systems of linear equations.
• Topic 4. Vector Spaces and Linear Transformations – Linear independence. Subspaces. Basis. Dimension. Rank of a system of vectors. Introduction to linear transformations. Matrix representation of a linear transformation.
• Topic 5. Matrix Diagonalization – Eigenvalues and eigenvectors. Eigenspaces. Matrix diagonalization and diagonalizable matrices.
• Topic 6. Spaces with Inner Product and Applications – Inner product spaces. Orthogonality. Gram–Schmidt method. Orthogonal and unitary diagonalization. Singular value decomposition. Matrix rank reduction. Least squares method. Quadratic forms.

Learning Outcomes

• B3 (CG3) – Knowledge of basic subjects and technologies that enables students to learn new methods and technologies, providing versatility to face and adapt to new situations.
• B4 (CG4) – Ability to solve problems with initiative, make creative decisions, and communicate and transmit knowledge and skills, understanding the ethical and professional responsibility of the Technical Telecommunication Engineer.
• C1 (CE1/FB1) – Ability to solve mathematical problems in engineering, applying knowledge of linear algebra, geometry, differential geometry, differential and integral calculus, ordinary and partial differential equations, numerical methods, numerical algorithms, statistics and optimization.
• D2 (CT2) – Understanding engineering within a framework of sustainable development.
• D3 (CT3) – Awareness of the need for lifelong learning and continuous quality improvement, showing a flexible, open and ethical attitude toward different opinions and situations, particularly regarding non-discrimination based on sex, race or religion, and respect for fundamental rights and accessibility.

Recommended Readings and Tools

Subjects that continue the syllabus

• Physics: Analysis of Linear Circuits — V05G301V01108
• Mathematics: Calculus 2 — V05G301V01106
• Physics: Fields and Waves — V05G301V01202

Subjects recommended to be taken simultaneously

• Mathematics: Calculus 1 — V05G301V01101

Planned Activities

• Laboratory practical – Solving assigned exercises and model problems using the computer tool MATLAB.
  Competences developed: B3, B4, C1, D2, D3.
• Lecturing – Explanation and development by the teacher of the contents of the different topics included in the syllabus.
  Competences developed: B3, C1, D3.
• Problem solving – The professor will solve exercises related to each topic. Students will also participate in solving exercises in order to reinforce their understanding.
  Competences developed: B3, B4, C1, D2, D3.

Assessment Methods and Criteria

Evaluation Activities

• Problem and/or exercise solving – Continuous evaluation consists of three tests conducted during class hours:

  1. Exam covering topics 1 and 2.
  2. Exam covering topics 3 and 4.
  3. Exam covering topics 5 and 6.

  Each test has a weight of 20% in the final grade, for a total continuous assessment weight of 60%.
  The schedule for these intermediate tests will be approved by the Academic Commission of the Degree and published at the beginning of the semester.
  Training and learning results: B3, B4, C1.

• Essay questions exam – Written exam with a maximum duration of three hours, covering topics 1–6. The exam will take place at the end of the semester on the date, time and venue established in the official examination calendar of the School.
  Weight: 40%.
  Training and learning results: B3, B4, C1.

Other Comments on the Evaluation

Ordinary assessment

Continuous assessment

The final grade is calculated using the formula:

M = (2 × (E1 + E2 + E3) + 4 × EF) / 10

Where:

• E1, E2, E3 are the marks (0–10 scale) obtained in the three continuous assessment tests.
• EF is the mark (0–10 scale) obtained in the final exam.

Before each test, the procedure and date for reviewing the grades will be announced. After each test, the marks will be published within a reasonable period. If a student cannot attend a scheduled test, it will not be repeated.

The marks obtained in the continuous assessment tests are valid only for the academic year in which they are obtained.

A student is considered to have chosen continuous assessment if they take the second test (topics 3 and 4).

Global assessment

Students who do not opt for continuous assessment will be evaluated through a single final exam covering all topics of the subject (1–6). This exam is graded on a 0–10 scale, with 5 as the passing mark.

Extraordinary exam

Students who do not obtain a passing grade at the end of the semester may take a second final exam on the date established in the official exam calendar of the School. This exam will cover topics 1–6 and will be graded on a 0–10 scale, with 5 as the passing mark.

During the exam correction period, some students may be contacted by phone or online by the teacher to clarify aspects of their answers; these clarifications may affect the final exam grade.

“No Presentado” (Not Presented)

A student will receive the grade “No Presentado” in the ordinary call if they do not attend either the continuous assessment tests or the final exam.

A student will receive “No Presentado” in the second call only if they obtained “No Presentado” in the first call and did not attend the second final exam.

End-of-program exam

Students taking the end-of-program examination will sit for an exam covering topics 1–6, graded on a 0–10 scale, with 5 as the passing mark.

Ethical behaviour

Students are expected to behave ethically and responsibly in all tests and examinations so that the results accurately reflect their knowledge and skills. Any dishonest behaviour (such as copying or using unauthorized materials) will result in a grade of 0 in the affected test and a report submitted to the School Director’s Office.