Linear Algebra
Code: 104843 ECTS Credits: 6| Degree | Type | Year | Semester |
|---|---|---|---|
| 2503852 Applied Statistics | FB | 1 | 1 |
Contact
- Name:
- Marc Masdeu Sabate
- Email:
- marc.masdeu@uab.cat
Use of Languages
- Principal working language:
- catalan (cat)
- Some groups entirely in English:
- No
- Some groups entirely in Catalan:
- Yes
- Some groups entirely in Spanish:
- No
Teachers
- Ricard Riba Garcia
- Eduard Vilalta Vila
Prerequisites
None.
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, ... will be worked on.
Competences
- Calculate and reproduce certain mathematical routines and processes with agility.
- Critically and rigorously assess one's own work as well as that of others.
- Make efficient use of the literature and digital resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Use quality criteria to critically assess the work done.
Learning Outcomes
- Critically assess the work done on the basis of quality criteria.
- Make effective use of references and electronic resources to obtain information.
- Master the basic language and tools of linear algebra.
- Master the specific algebraic tools that in future will be used for advanced modelling.
- Reappraise one's own ideas and those of others through rigorous, critical reflection.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
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.
Methodology
(from Google translate)
Dedication
Given that this subject is assigned 6 credits, the total number of hours (theory classes, problems, seminars, personal work and exams) that an average student should devote during the semester is 150 hours, appropriately spread over time. It is therefore advisable to devote an average of 5 hours of personal work each week to assimilating the theory, problem solving and writing a paper.
It is clear that according to the abilities of work, assimilation, abstraction, mechanisms, etc. some students may need more dedication and others with fewer hours of work will have enough.
Description of the practices.
Throughout the semester there will be 13 practice sessions or problems, some of which will be held in the computer lab or virtually.
Classroom practices
Classroom practices will consist of the approach and resolution of exercises on the subject already explained atheism. In some of the sessions the students will have to solve and deliver in writing a problem (typology "Quiz"). The score for solving these problems will be scored for continuous assessment. Some of these "Quiz" can also be done during theory hours.
Computer practices
Some practice sessions will consist of working on the subjects already explained in theory using an algebraic manipulator (SageMath). Part of the continuous assessment will consist of the delivery of some of the practices, in which the acquisition of the ability to use algebraic manipulators for problem solving will be scored.
Methodology
The subject has 2 hours a week of theory class and 2 hours a week of problem and practice classes. Attendance at all sessions is recommended. The theory taught is quite contained in the texts recommended in the bibliography, although in each of them its presentation has slightly different characteristics. The student should become accustomed to learning from textbooks, which are well-structured and written tools and where both mathematical language and logical reasoning are clearly reflected. Books, at least one, are a very important complement to classes.
Periodically the student will receive lists of problems that he must try to solve individually or in groups and which will be worked on in the problem classes.
Every 3 or 4 weeks approximately there will be an evaluation test (type "Quiz") that the student will have to answer in class or at a specific time before starting the usual class.
The methodology of the practice sessions is described in detail in the section "Description of the practices".
There will be two partial tests specially designed as a test for the student, and for teachers, which will measure the progress of the student and will have value in the continuous assessment note.
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.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Lesson | 26 | 1.04 | 5, 3, 4 |
| practical lessons with or without Math sources | 26 | 1.04 | 5, 1, 3, 4, 7 |
| Type: Supervised | |||
| Solving exercises | 40 | 1.6 | 5, 1, 7, 6 |
| Type: Autonomous | |||
| Learn theoretical concepts | 24 | 0.96 | 5, 3, 4, 7 |
| Prepare avaluations | 26 | 1.04 | 5, 1, 3, 4, 7, 2 |
Assessment
(from Google Translate)
The evaluation of the subject will consist of:
a) Problem solving, "Quiz" type tests (every three or four weeks, starting with topic 1): 2 points.
b) The use of computer tools, computer exam (date to be determined, after the topic of diagonalization): 1.5 points.
c) A partial exam: 1.5 points
d) A final exam: 5 points
In the examination period there will be a joint recovery test of sections (c) and (d).
To pass the course you must obtain a grade higher than 5, and have taken at least a 4 in the final exam or in the resit test. (In case of having a grade of the subject higher than 5 but not having the minimum grade of 4 in the final exam and recovery test, the student's grade will be 4.5 points).
It will be considered that a student has presented to the asignatura if has realized activities of evaluation that represent a weight equal or upper to 50% of the final note of the course. The award of the qualification of "matrícula d'honor" will be made before the recovery tests.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Solving exercises | 15 | 1 | 0.04 | 5, 3, 4, 7, 6 |
| Work with Sage Math | 15 | 1 | 0.04 | 3, 4, 2 |
| Writting exams | 70 | 6 | 0.24 | 5, 1, 3, 4, 6, 2 |
Bibliography
Basic:
Otto Bretscher: Linear Algebra with Applications. Pearson Prentice Hall, 3rd edition.
Enric Nart,Xavier Xarles: Apunts d'àlgebra lineal, Material UAB, 237 (2016), UAB.
Complementary:
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
Online:
M. Masdeu, A. Ruiz, Apunts d'Àlgebra lineal (http://mat.uab.cat/~masdeu/wp-content/uploads/2022/06/ApuntsAlgebraLineal.pdf)
Software
We use Sage Math (www.sagemath.org) software during some lessons.