Linear Algebra

Knowledge of the basic mathematics taught at secondary schools

This course unity is essentially formative and attempts to coordinate the theoretical foundations with the developments needed in the subsequent course unities included in the curriculum. At this level, the intuitive understanding of the concepts and calculation skills are promoted. In Theoretical-Practical lessons the expository and interrogative method is used during the explanation of the theoretical subjects and exercises are solved in groups or individually.
Assessment
Continuous Assessment: There are two examination tests, where each one is rated to 10 points. To be successful in this unit course, students need to get a minimal score of 3.25 points in each one and the sum of the classifications has to be greater than or equal to 9.5.
Assessment by Final Examination: There is a final exam, rated for 20 points, where the approval requires a score greater than or equal to 9.5 points.

Perform basic matrix operations.
Compute matrix determinants, eigenvalues and eigenvectors.
Understand and apply concepts related to vector spaces and linear transformations.
Solve and interpret linear systems using matrix theory.
Understand the importance of linear algebra in computer science and informatics engineering.
Recognize the importance of the algorithms in linear algebra.
Solve real problems which are modeled by matrices and systems.
Develop algorithms using a logical and structured reasoning.
Base problem solving on mathematics.
Compare, with criticism, the results obtained by analytical means with the ones obtained by computational means.
Select appropriately the accessible information (from monographs, textbooks, web, …).
Explain the concepts and problems’ solution in an appropriated way.
Solve practical problems with autonomy using, not only the subjects treated in the class, but also other related topics.

0. A brief revision of complex numbers.
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; Matrices and linear systems in MATLAB.
2. Determinants
Definition and properties; Adjunct 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; Diagonalisation; Cayley-Hamilton Theorem.

NAO