6 ECTS; 1º Ano, 1º Semestre, 28,0 PL + 28,0 TP + 5,0 OT , Cód. 81436.
Lecturer
- Maria Isabel Vaz Pitacas (1)(2)
(1) Lead Professor
(2) Teaching Professor
Prerequisites
Not applicable
Objectives
This module aims to provide fundamental knowledge and skills in Linear Algebra and Analytic Geometry, which are essential for a basic grounding in Computer Science and Multimedia Technologies and for learning the content of subsequent modules, enabling students to achieve the following learning outcomes:
1. Identify, describe and explain the fundamental concepts of matrices, determinants, analytical geometry and eigenvalues.
2. Apply the methods of Linear Algebra and Analytical Geometry to solve mathematical and technological problems.
3. Analyse and relate different concepts to model simple situations related to computer science and multimedia technologies.
4. Critically evaluate the results obtained, verifying their correctness and appropriateness.
5. Communicate and justify mathematical reasoning using appropriate notation and language.
6. Develop independent learning and problem-solving skills.
7. Construct simple mathematical models to represent situations related to the field of study.
Program
1. Matrices
1.1. General concepts and notation.
1.2. Matrix algebra.
1.3. Solving systems of linear equations.
1.4. Determinant of a matrix application to the
analysis of a system of linear equations.
1.5. Inversion of a regular matrix the
Gauss-Jordan method.
2. Determinants
2.1. Definition of first- and
second-order determinants.
2.2. nth-order determinants. Laplaces theorem.
2.3. Some properties of determinants.
2.4. Applications of the theory of determinants.
3. Concepts of Analytic Geometry
3.1. Inner product of vectors, outer product and
mixed product applications.
3.2. Analytic representation of a straight line.
3.3. Analytic representation of a plane.
3.4. Distances.
4. Eigenvalues and Eigenvectors
4.1. Brief introduction to real vector spaces
(definition; examples of vector spaces and
subspaces; linear combinations of vectors;
linear independence of vectors).
4.2. Eigenvalues and eigenvectors of a
square matrix; characteristic polynomial.
4.3. Calculation of eigenvalues and eigenvectors.
4.4. Properties of eigenvalues.
4.5. Eigen subspace associated with an eigenvalue.
4.6. Diagonalised matrices and diagonalisation of
matrices.
Evaluation Methodology
i) Continuous Assessment
During the semester, students must complete two closed-book written tests, WT1 and WT2, each graded on a scale from 0 to 20.
The minimum mark required in each Written Test (WT1 and WT2) is 3 out of 20.
The final mark is calculated as follows:
Final Mark = 0.5 × WT1 + 0.5 × WT2
Students pass the Course Unit and are exempt from the final examination in accordance with paragraphs 11 and 12 of Article 11 of the Academic Regulations of PUTomar.
ii) Examination Assessment
Students must sit a final examination.
The examination consists of a closed-book written test graded on a scale from 0 to 20.
Students pass the Course Unit in accordance with paragraphs 11 and 12 of Article 11 of the Academic Regulations of PUTomar.
iii) Remarks
Any student who is not exempt from the examination is eligible to sit the final examination.
Following any assessment component (continuous assessment or examination), students may be required to undertake an oral examination.
Students obtaining a final mark of 17 out of 20 or higher may be required to undertake an additional assessment. Failure to attend this assessment will result in a final mark of 17 out of 20.
Students pass the Course Unit in accordance with paragraphs 11 and 12 of Article 11 of the Academic Regulations of UPTomar.
Bibliography
- Amaral, I. e Ferreira, M. (2017). Álgebra Linear Espaços Vectoriais Geometria Analítica. (Vol. 1). Lisboa: Edições Sílabo
- Amaral, I. e Ferreira, M. (2020). Álgebra Linear - Matrizes e determinantes. . (Vol. 1). Lisboa: Edições Sílabo
- Anton, H. e Rorres, C. (2019). Elementary Linear Algebra: Applications Version. United States of America: John Wiley
- Cabral, I. e Perdigão, C. e Saiago, C. (2021). Álgebra Linear: Teoria, Exercícios Resolvidos e Exercícios Propostos com Soluções. Lisboa: Escolar Editora
- Lay, D. e Lay, S. e McDonald, J. (2024). Linear Algebra and Its Applications. (Vol. 6th ed.). Harlow: Pearson
- Queiró, J. e Santana, A. (2026). Introdução à Álgebra Linear. Lisboa: Gradiva
Teaching Method
Practical classes for the presentation and discussion of fundamental concepts, supplemented with examples. In lecture classes, students apply, analyse and consolidate their knowledge through guided problem-solving.
Software used in class
Productivity tools and e-learning platforms.
