Linear Algebra
Code: 104843 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Applied Statistics | FB | 1 |
Contact
- Name:
- Ramon Antoine Riolobos
- Email:
- ramon.antoine@uab.cat
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
Elementary knowledge of Mathematics corresponding to secondary education and high school.
Objectives and Contextualisation
(from Google Translate)
This subject is a presentation of matrix algebra, with emphasis on solving systems of equations and diagonalization of matrices, in particular symmetric matrices.
The main goal is for the student to reach maturity in matrix manipulation and acquire the theoretical knowledge that should allow him to use matrices in statistical treatments. In particular, the decompositions of matrices such as PAQ-reduction, decomposition into singular values (SVD), diagonalization, ...
Learning Outcomes
- KM02 (Knowledge) Recognise the language and basic tools of linear algebra.
- SM03 (Skill) Solve, using numerical methods, optimisation problems, linear algebra and analysis in general that appear in science and, especially, in statistics.
Content
(from Google Translate)
1. Systems of linear equations and matrices. Operations with matrices. Invertible matrices. Elementary transformations of matrices. Normal form of Gauss - Jordan. Range of an array. Inversibility criteria. Matrix of a system of linear equations. Solving systems of linear equations. Determinant of a square matrix. PAQ-reduction and generalized inverse.
2. Vector Spaces and Linear Applications: Vectors in R ^ n and Linear Applications. Definition of vector space and examples. Vector structure of R ^ n and subspaces. Definition of linear application and examples. Core and image of a linear application. Dependence and linear independence of vectors. Generator systems, bases of vector spaces. Dimension and range. Coordination, base change matrices, matrix associated with a linear application with respect to bases fixed to the departure and arrival spaces.
3. Diagonalization of endomorphisms: Eigenvectors and eigenvalues of an endomorphism. Characteristic polynomial and minimum polynomial. Diagonalization criterion.
4. Vector spaces with scalar product. Bilinear product, definition and properties. Orthogonality. Orthonormal bases. Gram-Schmidt orthonormatization method. Screenings. Orthogonal complement. Orthogonal matrices. Orthogonal diagonalization of symmetric matrices, spectral theorem. Data adjustment. Singular values and decomposition into singular values.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Lesson | 25 | 1 | KM02, SM03, KM02 |
| Problem solving and practical lessons | 24 | 0.96 | KM02, SM03, KM02 |
| Type: Supervised | |||
| Solving exercises | 40 | 1.6 | SM03, SM03 |
| Type: Autonomous | |||
| Learn theoretical concepts | 27 | 1.08 | KM02, SM03, KM02 |
| Prepare avaluations | 26 | 1.04 | KM02, SM03, KM02 |
Time commitment
Considering that this subject is worth 6 credits, the total number of hours (theory classes, problems, seminars, personal work and exams) that an average student should dedicate to it during the semester is 150 hours, adequately distributed over time. It is therefore advisable to allocate an average of 5 hours of personal work each week to assimilating the theory and solving problems.
Methodology
During the semester, the subject has 2 hours of theory classes per week and 2 hours of problem or practical classes per week.
In the theory classes, the contents of the subject will be presented, giving special emphasis to the meaning, motivations and reasoning that lead us to each of the concepts that will be worked on. During the problem classes, lists of exercises will be worked on that the student will receive in advance in which the most practical aspect of the concepts worked on will be emphasized. Finally, in the practical classes, you will learn to use a certain computer program to assist us in solving the problems. As a complement to all this, it is advisable for the student to get used to consulting textbooks, which are well-structured tools where both the mathematical language used in the classroom and the logical reasoning to demonstrate the concepts are clearly reflected.
Periodically there will be small tests in the classroom (type "Quiz") to assess the student's progress in the subject. These tests will be announced in advance, and should help the student to stay up to date with the subject.
Within the computer practice sessions, small evaluative tests will also be done with the corresponding software.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Solving exercises | 10 | 1 | 0.04 | KM02, SM03 |
| Work with Sage Math | 10 | 1 | 0.04 | SM03 |
| Writting exams | 80 | 6 | 0.24 | KM02, SM03 |
(from Google Translate)
Continuous assessment
The assessment of the subject will consist of the following activities:
Exams:
-First part (November) P1 (30%)
-Second part (January) P2 (50%)
Classroom questionnaires:
- Continuous assessment questionnaires Q (10%)
- SageMath tests S (10%)
These activities, scored out of 10, will receive the weight indicated in the final grade. That is, the final grade of the subject will be:
Final grade = 0.1Q+0.1S+0.3P1+0.5P2
In the event of not achieving a pass, the student may opt for a single retake exam, R, which will allow them to recover the grade of the two parts (P1 and P2).
The student will be considered "Not assessable" if they have carried out assessment activities that represent a weight below 50% of the final grade of the course.
Single assessment
If the student opts for the single assessment, he/she will take a single exam coinciding with the date of the second part. The exam will consist of the content of the entire subject including the practical part of SageMath.
As in the case of continuous assessment, the grade of this exam can be recovered in a retake exam.
Bibliography
Basic:
M. Masdeu, A. Ruiz, Apunts d'Àlgebra lineal (https://mmasdeu.github.io/algebralineal/)
Otto Bretscher: Linear Algebra with Applications. Pearson Prentice Hall, 3rd edition.
Complementary:
Ferran Cedó i Agustí Reventós: Geometria plana i àlgebra lineal, Manuals UAB, (2004), UAB.
Stanley I. Grossman, Álgebra lineal, Grupo Editorial Iberoamérica, 1983.
Shayle R. Searle, Matrix Algebra Useful for Statistics, Wiley-Interscience
David A. Harville, Matrix Algebra from a Statistician's Perspective, Springer
Software
We use Sage Math (www.sagemath.org) software during some lessons.
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PLAB) Practical laboratories | 1 | Catalan | first semester | afternoon |
| (PLAB) Practical laboratories | 2 | Catalan | first semester | afternoon |
| (SEM) Seminars | 1 | Catalan | first semester | afternoon |
| (SEM) Seminars | 2 | Catalan | first semester | afternoon |
| (TE) Theory | 1 | Catalan | first semester | afternoon |