Course: 2020/2021

Linear Algebra

(13870)

Students are expected to have completed

Basic knowledge on vectors and Euclidean space of dimension 2 and 3.
Basic knowledge on matrices and determinants.
Basic knowledge on systems of linear equations.
Basic trigonometry.

ObjectivesFurther information on this link

1. Learning objectives: (PO: a, CGB1)
- To solve systems of linear equations and to interpret the results.
- To understand the concept of algebraic structure.
- To know and understand the notion of vector spaces and their applications.
- To understand the notion of bases and coordinates in a vector space.
- To understand linear transformations and to represent them by matrices.
- To compute the fundamental vector spaces associated to a matrix.
- To understand the concept of eigenvalues and eigenvectors of a matrix, and to know their computation and applications.
- To compute the QR decomposition of a matrix.
- To find an approximate solution to an inconsistent system of linear equation by least-square fitting.
- To obtain the singular value decomposition of a matrix.
2. Specific skills: (PO: a, CGB1)
- To raise the abstraction level.
- To be able to solve real problems using typical linear algebra tools.
3. General skills: (PO: a, CGB1)
- To improve the oral and written communication ability using the language and signs of mathematics properly.
- To be able to model a real situation described with words using mathematical concepts.
- To improve the ability to interpret a mathematical solution and define its limitations and reliability.
- To be able to use mathematical software.

Description of contents: programme

1. Matrices
- Review of definitions and concepts related to matrices.
- Matrix operations.
- Transpose.
- Inverse.
- Determinant.
- Sets induced by a matrix.
2. Systems of linear equations
- Geometric interpretation of linear systems in R^n.
- Existence and uniqueness of solutions.
- Matrix methods to solve systems of linear equations.
3. Vector spaces
- Vector spaces.
- Vector subspaces.
- Operations between subspaces.
4. Basis and dimension
- Spanning sets.
- Basis. Dimension.
- Coordinates.
5. Linear transformations
- Definition and properties.
- Operations between linear transformations.
6. Linear transformation and matrices
- Representation of linear transformations using matrices.
7. Change of basis
- Change of basis.
- Normal form of a linear transformation.
8. Eigenvalues and eigenvectors
- Definitions.
- Characteristic polynomial and characteristic equation.
- Diagonalization.
9. Inner product. Orthogonality
- Inner product.
- Length and angle.
- Orthogonal projection.
- Orthogonal complement.
10. Orthogonal basis
- Orthogonal sets and orthogonal bases.
- Gram-Schmidt process.
- QR factorization.
11. The spectral theorem
- Diagonalization of symmetric matrices.
- Spectral decomposition.
12. Geometry of linear transformations
- Reflections.
- Contractions and dilations.
- Rotations.
- Projections.
13. Least squares
- The least squares problem.
- Geometric interpretation.
- Approximation of functions.
14. Pseudoinverse. Singular value decomposition
- Pseudoinverse.
- Singular value decomposition.
- Applications.

Learning activities and methodology

Synchronous lecture sessions (3 credits) (PO: a, CGB1)
During these sessions we will cover the course topics with the aim of using theory to solve problems.
Practicals, working individually and in groups (3 credits) (PO: a, CGB1)
During these sessions we will solve exercises of different levels of difficulty.

Assessment System

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

Basic Bibliography

- B. Kolman. "Introductory linear algebra: an applied first course". Prentice Hall. 8th ed. - 2005
- D. C. Lay. "Linear algebra and its applications". Addison Wesley. 4th ed. - 2011

Additional Bibliography

- D. POOLE. "Álgebra Lineal. Una introducción moderna". Thomson - Primera edición - 2004.
- O. BRETSCHER. "Linear algebra with applications". Prentice Hall. 4th ed. - 2009