Course: 2023/2024

Algebra

(18692)

Skills and learning outcomes

Description of contents: programme

1. Complex numbers.
Number sets. Need for complex numbers. Binomial form of complex numbers. Graphical representation. Operations. Conjugates, modulus and argument. Polar form. Roots of complex numbers. Complex exponentiation. Solution of equations.
2. Systems of linear equations.
Geometrical interpretation. Matrix notation. Gaussian elimination. Row equivalence. Row echelon forms. Resolution of linear systems. Homogeneous systems.
3. Vector spaces.
Vector spaces and subspaces. Linear combination. Dependence and linear independence. Range and dimension. Generator set and bases.
4. Matrix algebra.
Operations with matrices. Transpose of a matrix. Conjugate transpose of a matrix. Inverse of a matrix. Range and determinants. Fundamental subspaces of a matrix. Col A, Row A and Ker A bases.
5. Linear transformations.
Operations. Image and core. Associated subspaces. Types of transformations. Rank-nullity theorem.
6. Bases.
Matrix associated to a base. Systems of coordinates. Change of basis and linear applications.
7. Orthogonality and least squares.
Scalar product and modulus. Orthogonal sets. Unitary matrices. Orthogonal complement. Orthogonal projection. Gram-Schmidt process. Least squares problems.
8. Introduction to spectral theory.
Eigenvalues and eigenvectors. Characteristic equation. Similarity of matrices. Diagonalisation. Applications.
9. Symmetric and normal matrices.
Unit similarity. Orthogonal and unitary diagonalization.

Learning activities and methodology

Learning activities:
Classroom:
Participatory lectures, problem sessions in class, individual and/or small group tutorials, exams. Each mathematical concept will be illustrated with examples and the corresponding theory will be developed. Active student participation will be encouraged and fostered as an important part of the learning process.
Outside classroom:
Individual and independent work and study.
Methodology:
Expositional method: lectures given by teacher supported, if necessary, by computer materials (PowerPoint/Beamer presentations, videos, etc. ). They provide transfer of knowledge and activation of cognitive processes in students.
Problem-based learning: active learning through problem solving, facing students with new situations in which they must find information and apply the newly acquired knowledge to solve these problems.
Face-to-face activities Nº hours
Lectures 24
Problem sessions 24
Office hours (tutorials) 4
Assessment 4

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- J. Arvesú, R. Álvarez, F. Marcellán. Algebra lineal y aplicaciones. Síntesis. 1999
- Strang, G. . Introduction to linear algebra. Wellesley-Cambridge Press. 1993
- Strang, G. . Algebra lineal y sus aplicaciones. Addison-Wesley Iberoamericana.
- Treil, Sergei. Linear algebra done wrong. Disponible en la pagina web del autor.

Additional Bibliography

- E. Hernández. Álgebra y geometría. Addison-Wesley/UAM. 1994
- L. Merino, E. Santos. Álgebra lineal con métodos elementales. Thomson-Paraninfo. 2006
- M. Castellet, I. Llerena. Algebra linear y geometría. Ed. Reverté. 1991
- S. Grossman. Algebra lineal con aplicaciones. McGraw Hill. 1992

The course syllabus may change due academic events or other reasons.