Linear Algebra

Knowledge and mastery of the subjects taught in Mathematics in secondary school.

This course unit is essentially formative and tries to coordinate the fundamental knowledge with the
necessary developments in the remaining course units in the study plan. At this level, the intuitive
understanding of concepts and the ability to calculate are encouraged. In Theoretical classes, the expository and interrogative method is used, in order to involve the students as much as possible in the functioning of the classes. In some classes, specific support and information software available on the Internet is used. Problem solving lectures are exclusively devoted to solving exercises.

Teaching Mathematics should facilitate mathematical communication, reflective thinking, the application of mathematical techniques to problem solving, critical analysis of the results obtained, that is,
interdisciplinarity. One of the objectives of Linear Algebra is to provide the basic foundations of
mathematical methods, usually applied in the areas of Engineering, used by the various course units of the Degree in Mechanical Engineering.

1. Matrices and Linear Systems.

Introduction; Matrix operations and their properties; Row echelon form and rank; Classification and
geometry of linear systems; Gaussian elimination; Homogeneous systems; Matrix inversion: Gauss-Jordan method; Block matrices.

2. Determinants
Definition and properties; Adjugate matrix and the inverse; Applications to Cryptography.

3. Vectors in R^n
Vectors and lines in the plane; Vectors in 3D-space; Cross product; Lines and planes in 3D-space; Linear
transformations: application to Computer Graphics; Vectors in R^n: Properties; Subspaces; Linear
combinations; Linear expansion; Linear independence; Basis and dimension.

4. Eigenvalues
Eigenvalues, eigenvectors and their properties; Diagonalization; Cayley-Hamilton Theorem.

(Remark: The software MATLAB and other is used throughout the course)

NAO

Cardoso, J. (2023). Apontamentos de Álgebra Linear. Coimbra: ISEC. (available at Moodle and “Secção de
Textos” of ISEC)

Kolman, B. & Hill, D.R. (2004). Introductory Linear Algebra – an applied first course (8ª ed.). Pearson-Prentice Hall. (available in Library of ISEC: 3-1-121)

Leon, S. (1999). Álgebra Linear com Aplicações (4ª ed). Rio de Janeiro: Livros Técnicos e científicos. (two
books available at Library of ISEC: 3-1-89 e 3-1-120)

Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. Philadelphia: SIAM.