ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Faculty of Physics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
On this subject, students will learn the basic concepts and techniques of linear algebra. The language of sets and applications are introduced in order to become familiarized with the formal aspects present in all branches of mathematics. The theory of vector spaces will enable us to handle vectors from an algorithm point of view. The reference example is Rn, but we may also show other examples of vector spaces. We will also study the linear maps between vector spaces and their relationship with matrices and the resolution of systems of linear equations, as well as the diagonalization criterion for endomorphisms. As a more geometric application we give a brief introduction to analytic geometry in the plane and in th espace, using it to study conics.
Learning results:
After studying the subject, the student will have learned to:
1. Possess knowledge in solving systems of linear equations and relate the solutions with the geometric objects of the plane and space.
2.- Calculate matrices associated with linear applications.
3.- Calculate bases of subspaces from generator systems.
4.- Know when a matrix is diagonalizable; calculate bases where it diagonalizes and the relationship between them.
5.- Handle with ease straight lines and planes in space. He will also know the conics, and their equations, and the passage, in the plane and in space, of any quadratic equation to its reduced equation.
1. Basic Notions of Sets and Applications. (4 theoretical hours).
2. Elements of Group Theory. (2 theoretical hours).
3. Vector Spaces.
Concept of vector space. Examples. Vector subspace. Linear dependence and independence. Generator systems. Bases. Dimension. Coordinates of a vector on a basis. Coordinates and linear dependence. Base changes. Parametric and linear equations of a subspace. Intersection and sum of subspaces. Direct sum. Grassmann formula for dimensions. (8 theoretical hours).
4. Linear Maps (Homomorphisms) and Matrices.
Definition. Examples. Kernel and image. Injective and surjective linear maps, and isomorphisms. Operations with linear maps. Classification of vector spaces. Matrix associated to a linear map in a given basis. Computation of the kernel and image from the associated matrix. Matrices and operations with linear maps. Associated matrices and base changes. (6 theoretical hours).
5. Determinants.
Determinant of a square matrix. Properties of determinants. Inverse matrix. Rank. (3 theoretical hours).
6. Systems of Linear Equations.
Systems of linear equations. Discussion of a system. Rouché-Frobenius theorem. Cramer's rule. (3 theoretical hours).
7. Diagonalization of endomorphisms.
Eigenvectors and eigenvalues. Geometric interpretation. The characteristic polynomial. Diagonalization. Cayley-Hamilton theorem. (3 theoretical hours).
8. Plane and space analytic geometry.
Equations of lines and planes. Relative position. Euclidean space. Conics. Classification. (3 theoretical hours).
Básica.
M. Castellet, I. Llerena. Álgebra Lineal y Geometría. Editorial Reverté.
G. Jerónimo, J. Sabia, S. Tesauri. Álgebra Lineal. http://mate.dm.uba.ar/~jeronimo/algebra_lineal/AlgebraLineal.pdf
L. Merino, E. Santos: Álgebra Lineal con Métodos Elementales. Editorial Thomson.
Complementaria.
E. Hernández. Álgebra y Geometría. Addison-Wesley
J. Arvesú Carballo, F. Marcellán Español, J. Sánchez Ruiz. Problemas Resueltos de Álgebra Lineal. Editorial Thomson.
BASIC AND GENERAL
CB1 - That the students have demonstrated to possess and understand knowledge in an area of study that starts from the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that imply knowledge coming from the vanguard of their field of study.
CB2 - That students know how to apply their knowledge to their work or vocation in a professional manner and possess the competences that are usually demonstrated through the elaboration and defense of arguments and the resolution of problems within their area of study.
CB5 - That the students have developed those learning skills necessary to undertake further studies with a high degree of autonomy.
CG3 - Apply both the acquired theoretical and practical knowledge and the capacity for analysis and abstraction in the
definition and approach of problems and in the search for their solutions both in academic and professional contexts.
TRANSVERSALS
CT1 - Acquire analysis and synthesis capacity.
CT2 - Have the capacity for organization and planning.
CT5 - Develop critical reasoning.
SPECIFIC
CE5 - Be able to perform the essentials of a process or situation and establish a work model of it, as well as perform the required approaches in order to reduce the problem to a manageable level. Demonstrate critical thinking to build physical models.
CE6 - Understand and master the use of mathematical and numerical methods most commonly used in Physics
CE8 - Be able to manage, search and use bibliography, as well as any source of relevant information and apply it to research and technical development of projects
The Expository classes will be used for the presentation of the basic contents that compose this subject.
The interactive seminar classes, which will serve to illustrate the theoretical contents, will be dedicated to the resolution of questions and problems by the teacher with the participation of students.
Continuous assessment combined with a final test are planned as evaluation criterion. The final test will be held on the date set by the Faculty of Physics for that purpose.
Continuous assessment will consist of tests.
Calculation of the final qualification:
The final test, which will be compulsory, will be face-to-face. The qualification of both the first and the second chance will be max {F; 0.25xC + 0.75xF} where C is the grade of the continuous assessment and F the grade of the final test.
For cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of students' academic performance and the review of qualifications will apply.
It will be considered as Not Presented that student that does not show to the final test, both in the first and in the second chance.
Besides the lectures, interactive classes and the reduced group tutorial sessions, the students will have to dedicate 90 hours of personal work to study the theory and handle the exercises.
Study daily with the help of bibliographic material. Read the theoretical part carefully until you assimilate it and try to answer the questions, exercises or problems presented in the bulletins.
José Javier Majadas Soto
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813168
- j.majadas [at] usc.es
- Category
- Professor: University Professor
Samuel Alvite Pazo
- Department
- Mathematics
- Area
- Algebra
- samuel.alvite.pazo [at] usc.es
- Category
- USC Pre-doctoral Contract
Ana Peon Nieto
- Department
- Mathematics
- Area
- Algebra
- ana.peon [at] usc.es
- Category
- PROFESOR/A PERMANENTE LABORAL
| Monday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 6 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 130 |
| Tuesday | |||
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 6 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 130 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 6 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 130 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 6 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 130 |
| 01.14.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 0 |
| 01.14.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 6 |
| 01.14.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 830 |
| 01.14.2025 16:00-20:00 | Grupo /CLE_01 | Main Hall |
| 06.12.2025 09:00-13:00 | Grupo /CLE_01 | Classroom 0 |
| 06.12.2025 09:00-13:00 | Grupo /CLE_01 | Classroom 6 |
| 06.12.2025 09:00-13:00 | Grupo /CLE_01 | Classroom 830 |