The fundamental results of Linear Algebra are presented with detail. All the new ideas and results are followed by examples of application to test whether they are being understood. Independent resolution of problems is greatly encouraged. In class and as homework the students must solve many proposed exercises with various degrees of difficulty The evaluation is done by final examination.
Being a first formal contact with mathematical abstraction, subjects concerning the general concepts of matrix, vector space and linear transformation are motivated by examples well known by students (real and complex numbers endowed by their usual algebraic structures and functions between those structures). The mathematical tools developed in the context of such concepts are to be applied in other areas of mathematics
as well as in engineering (determinants, Gauss Elimination, Least Squares approximation, diagonalization of matrices).
The course aims at developing the following skills: analysis and synthesis, organization and planning, oral and written communication, problem-solving skills. On the personal level it also allows to develop self-learning skills and independent thinking, as well as the capacity to apply theoretical knowledge.
1. Matrices – operations with matrices.
2. Systems of Linear Equations – Gauss Elimination.
3. Inversion of a Matrix - Gauss-Jordan Algorithm.
5. Vector Spaces.
6. Linear Transformations.
7. Inner Vector Spaces – Least Squares Approximation.
8. Diagonalization of Matrices.
9. Geometrical Applications to R2 and to R3.
Ana Paula Santana & João Filipe Queiró, Introdução à Álgebra Linear, Gradiva, 2010
Gilbert Strang, Linear Algebra and its Applications, Harcout Brace Jovanovich, San Diego, 1988