Class Linear Algebra

The Linear Algebra course provides the essential mathematical foundations for Engineering education, enabling the modelling and solution of scientific and technological problems. Its scope covers real and complex numbers, matrices, systems of linear equations, determinants, vector spaces, linear transformations, and eigenvalues and eigenvectors. The course develops students' ability to apply algebraic methods to analyse and solve engineering problems, providing a foundation for areas such as scientific computing, control systems, signal processing, artificial intelligence, and data science. Within the study programme, this course plays a fundamental role by supporting subsequent engineering courses and fostering logical reasoning, abstract thinking, and the application of mathematical methods to engineering problem-solving.

Not applicable

Real numbers and complex numbers. Matrices. Algebraic operations with matrices. Invertible matrices. Matrix transposition. Elementary transformations and elementary matrices. Row-echelon matrix and rank of a matrix. Invertible matrices. Systems of linear equations. Classification. Equivalence between systems of linear equations. Gauss elimination method. Determinants. Determinant function. Properties determinants. Laplace theorem. Determinant of the product. Adjoint matrix. Linear Spaces. Linear subspaces. Linear dependence and independence. Bases and dimension. Change of basis. Rows and column spaces, and nullspace of a matrix. Linear Transformations. Matrix representation of a linear transformation. Algebraic operations with linear transformations. Kernel and image. Invertible linear transformations. Eigen Values and Eigen Vectors. Characteristic polynomial. Eigen subspaces. Algebraic multiplicity and geometric multiplicity. Diagonalization problem.

Series of exercises will be proposed to consolidate knowledge and stimulate problem-solving skills. In each session, students are encouraged to focus on questions related to the topics covered in class and to show the results of their individual work in the following session where questions are resolved on the board, whenever necessary, and details that have caused doubts or difficulties are clarified. The fundamental idea is repeatedly highlighted that solving exercises has the main objective of allowing a deeper understanding of the conceptual body of the curricular unit, mastering which will allow the application of these tools to solve more advanced problems. The progression adopted, which starts from more calculative questions and develops towards the more abstract parts of linear algebra, aims to gradually consolidate mastery of the various tools provided by this discipline.

Cabral, I., Perdigão, C., & Saiago, C. Álgebra Linear: Teoria, Exercícios Resolvidos e Exercícios Propostos com Soluções. 6.ª ed. Lisboa: Escolar Editora, 2021. Barreira, L., & Valls, C. Exercícios de Álgebra Linear. Lisboa: IST Press, 2011. Santana, A. P., & Queiró, J. F. Introdução à Álgebra Linear. 4.ª ed. Lisboa: Gradiva, 2018.