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

Introduction. 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 with the aim of consolidating knowledge and stimulating problem-solving skills. The evaluation of the discipline, expressed on a scale from 0 to 20 points, will be made at different times, including 2 midterms (40% + 50%) and individual work to be developed outside the classroom (10%). If the weighted average of these evaluations is equal to or greater than 9.5, the student will be successful in the subject, otherwise the student will be able to attend a global frequency. In the final exam, the student can improve the grade. The minimum passing grade for these assessments is also 9.5. Assessment criteria are explained at the beginning of the semester.

Cabral, I., Perdigão, C., & Saiago, C. Álgebra Linear: Teoria, Exercícios Resolvidos e Exercícios Propostos com Soluções. 7.ª ed., Forte da Casa: Escolar Editora, 2024.