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Linear Algebra

Code: 104381
Credits: 6
2026/2027
Degree programme Type Course
Computational Mathematics and Data Analytics FB 1

Contact lecturer

Name :
Carlos Broto Blanco
Email :
carles.broto@uab.cat

Teaching staff

Ángel Lorenzo Martínez

Group languages

You can consult this information at the end of the document.

Prerequisites

Although the course is essentially self-contained, the student will be required to know how to solve systems of linear equations, the basic arithmetic of numbers and polynomials, and how to perform algebraic manipulations.

Objectives

In order to acquire a proper mathematical training, it is essential to understand linear algebra in depth. One needs to learn how to manipulate the objects introduced in such a class and to interpret their meanings. The tools provided in this course are essential not only in all branches of Mathematics, but also in most Sciences and Engineering studies.

Among the many goals we underline the following: to understand and correctly use mathematical language, to appreciate the need for proofs, and to develop a critical approach to mathematical statements.

As more specific goals: the student will learn to manipulate matrices as a basic tool to analyze systems of linear equations, to formalize the necessary language in order to understand the concepts of vector space and linear map, as well as to manipulate bilinear forms. All of this will be reinforced with the introduction of the appropriate software.

Learning outcomes

  • CM02 (Use matrices to solve systems of equations, make changes of base and study linear applications.) Use matrices to solve systems of equations, make changes of base and study linear applications.
  • CM03 (Contrast the use of calculus with the use of abstraction in algebra and analysis to solve a real problem.) Contrast the use of calculus with the use of abstraction in algebra and analysis to solve a real problem.
  • CM04 (Explain ideas and concepts of fundamental mathematics, communicating one's own reasoning to others.) Explain ideas and concepts of fundamental mathematics, communicating one's own reasoning to others.
  • KM01 (Identify the essential ideas of the proofs of some basic algebra and calculus theorems.) Identify the essential ideas of the proofs of some basic algebra and calculus theorems.
  • SM01 (Write small mathematical texts (exercises, solving theoretical questions, etc.) in an orderly and precise manner.) Write small mathematical texts (exercises, solving theoretical questions, etc.) in an orderly and precise manner.
  • SM03 (Classify matrices and linear applications according to diverse criteria (rank, diagonal forms and Jordan form).) Classify matrices and linear applications according to diverse criteria (rank, diagonal forms and Jordan form).

Contents

The course is structured into 6 blocks: a first, instrumental block to introduce the field of complex numbers, and a second, more computational block that focuses on algebraic manipulation of matrices, introducing their basic operations. The third and fourth blocks formalize the concepts of abstract vector space and linear map, relating them to the contents of the second block. The fifth and sixth blocks are devoted to more advanced concepts that build on the structure of vector spaces and linear maps.


Blocks:

  1. Complex numbers
  2. Matrices and linear equations
  3. Vector spaces
  4. Linear maps
  5. Diagonalization
  6. Orthogonality and quadratic forms


Learning activities and methodology

Title Hours ECTS Learning outcomes
Problem sessions 12 0.48
Preparation of problems to deliver 15 0.6
Practice sessions 11 0.44
Problem solving 30 1.2
Lectures 27.5 1.1
Theory study 26 1.04
Use of software 20 0.8

The subject has 4 hours per week during the semester, grouped into 2-hour blocks. Each of these blocks will combine theoretical content and problem solving, which may be on paper or with the use of software.


In order to introduce the software, more time will be dedicated to this part in the sessions at the beginning of the course.


This subject will be taught in person, taking advantage of the resources that the UAB makes available to us. It will also have the corresponding Moodle classroom on the UAB servers to complement the explanations given in class, offer the necessary material, open forums and make deliveries. This tool will also be used to communicate dates, regulations, grades and any other topic related to the subject.

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

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Practice exam 15% 2 0.08 CM02, CM03, SM03
Final exam 50% 4 0.16 CM03, CM04, KM01, SM01, SM03
Problems to turn in 15% 0 0 CM02, CM03, SM01
Regular quizzes 20% 2.5 0.1 CM02, CM03, KM01, SM03

The assessment activities are structured as follows:


Submission of solved problems (15%)

Follow-up test (20%)

Practical exam (15%)

Final theory exam (50%)


Problem assignments are completed at the end of the problem classes designated in advance. There will be two assignments. The practical exam is carried out in one of the practical classes, also designated in advance. There will be 3 follow-up tests that students must take individually, in person in the classroom. The theory exam requires a minimum grade of 3.5. This exam is retaken, but the other assessment activities are not.


The date of the theory exam is set by the Faculty. The dates of the other assessment activities will be announced on the virtual campus one week in advance.


If the minimum grade of 3.5 is obtained in the final exam, the final grade for the course is calculated as the weighted average of the grades obtained in the different activities. Otherwise, the final grade will be limited to 4.5, that is, the lowest value between the grade obtained using the standard weighting and 4.5.


Students who have requested the single assessment and have been accepted by the Faculty will be assessed through a final exam that will count 80% of the grade and a practical exam that will count 20% and both have a minimum grade of 3.5 points out of 10 to be able to pass the subject. These exams will be taken one after the other on the date of the final exam of the subject.


In this subject, the use of Artificial Intelligence (AI) technologies is not allowed in any of its phases.


Anyone who has not completed assessment activities that add up to a minimum of 50% of the final grade will be considered non-assessable.

Bibliography

Basic:

  • Marc Masdeu, Albert Ruiz, Apunts d'Àlgebra Lineal. Available at the Moodle classroom.
  • Otto Bretscher, Linear Algebra with Applications. Pearson, 2013.
  • Enric Nart, Xavier Xarles, Apunts d'àlgebra lineal. Materials UAB, 2016.

Complementary:

  • Sheldon Axler, Linear algebra done right. Springer UTM, 2015.
  • Manuel Castellet i Irene Llerena, Àlgebra lineal i geometria. Manuals UAB, 1991.
  • Ferran Cedó i Agustí Reventós, Geometria plana i àlgebra lineal. Manuals UAB, 2004.
  • Gilbert Strang, Linear Algebra and Learning from Data. Wellesley-Cambridge Press, 2019, pp.446. ISBN:978-06921963-8-0
  • Mike X. Cohen, Practical Linear Algebra for Data Science: From Core Concepts to Applications using Python. O'reilly Media, 300pp (2022). ISBN:978-1098120610
  • Charu C. Aggarwal, Linear algebra and optimization for Machine Learning: a textbook. Springer International Publishing (2020). ISBN: 9783030403430

Software

SageMath (https://www.sagemath.org/)

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PLAB) Practical laboratories 1 Catalan first semester morning-mixed
(SEM) Seminars 1 Catalan first semester morning-mixed