ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Hours of tutorials: 1 Expository Class: 30 Interactive Classroom: 20 Total: 51
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Higher Technical Engineering School
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
In this course, students should develop habits related to the capacity for abstraction and rigor necessary for a professional in the field of Artificial Intelligence.
The specific objectives include:
-Algebraic manipulation of matrices to solve and analyze systems of linear equations.
-Knowledge of matrix decomposition algorithms and their application in problem-solving.
-Handling basic concepts of vector spaces: linear dependence and independence, bases, dimension, subspaces, and linear transformations.
-Identification of linear transformations using matrices and systems of linear equations.
-Understanding and applying procedures for the diagonalization of square matrices.
- Solving matrix problems using diagonalization
-Using the standard inner product, norm, and the Gram-Schmidt method in the Euclidean real space. Explaining their usefulness in problem-solving.
1.-Systems of Linear Equations
Methods for solving linear equations systems. Elementary Operations. Elementary Operations: Matrix Version. Gauss Elimination Method. Solving Systems of Linear Equations.
In-Person Teaching:
Lecture/Practice Hours: 2/2
2.-Matrix Algebra
Addition, Scalar Multiplication, Transpose. Matrix Multiplication. Square, Invertible, and Triangular Matrices. Systems of Equations and Matrices. Elementary Matrices. Invertibility Criteria. Calculating the Inverse. LU Decomposition: Algorithm. Determinant.
In-Person Teaching:
Lecture/Practice Hours: 6/4
3.-Vector Spaces
Properties. Vector Subspaces. Linear Dependence and Independence. Bases and Dimension. Row and Column Rank. Coordinates of a Vector with Respect to a Basis.
In-Person Teaching:
Lecture/Practice Hours: 6/5
4.-Linear transformations and Matrices
General Concepts. Kernel and Image of a Linear transformation. Operations. Matrix Representation of a linear transformation. Row and Column Rank: reformulation. Basis of the Image and Kernel: reformulatation. Change of Basis Matrices. Similar Matrices.
In-Person Teaching:
Lecture/Practice Hours: 6/4
5.-Eigenvalues, Eigenvectors, Diagonalization
Method for finding Eigenvalues. EigenSpaces. Diagonalizable Matrices and Transformations. Algebraic and Geometric Multiplicity. Diagonalization Criterion. Examples.
In-Person Teaching:
Lecture/Practice Hours: 5/3
6.-Orthogonality
Inner Products and Euclidean Spaces. Orthogonality, Orthogonal and Orthonormal Bases. Gram-Schmidt Method: Applications.
In-Person Teaching:
Lecture/Practice Hours: 5/2
Basic bibliography:
- David C. Lay, Álgebra lineal y sus aplicaciones, 4a edición, Pearson 2012.
- Ron Larson, Bruce H. Edwards, David C. Falvo: Álgebra Lineal, 5a edición, Ediciones Pirámide.
- Gilbert Strang, Linear Algebra y sus aplicaciones, Addison-Wesley Iberoamericana, 1989.
- Howard Anton and Chris Rorres, Elementary Linear Algebra: applications version, 9th edition. John Willey and Sons, 2005.
- Kenneth Hoffman and Ray Kunze, Linear Algebra (Second Edition), Prentice Hall, 1961.
Further reading:
- Merino, L. y Santos. E. Álgebra Lineal con métodos elementales. Thomson, 2006.
- Burgos, J., Álgebra Finita y Lineal; García-Maroto Editores, 2010.
- Hernández, E., Álgebra Lineal y Geometría; Addison-Wesley/Universidad
This course contributes to the development of both general and specific competencies outlined in the USC Degree in Artificial Intelligence (CB2, CB3, CB5, CG2, CG4, TR3, CE1). It provides the essential mathematical foundation for techniques and algorithms in AI, covering everything from data representation to model optimization. Specific competencies addressed include:
- Understanding basic concepts of Linear Algebra, such as linear dependence and independence, bases, and linear applications.
- Mastering algorithms for matrix reduction and applying them to calculate rank and solve systems.
- Analyzing the close relationship between matrices, linear applications, and systems of equations.
- Evaluating whether a matrix is diagonalizable and, if so, diagonalizing it.
- Applying concepts of Euclidean spaces, scalar product, and orthogonal projection to solve problems in n-dimensional spaces.
The general methodological guidelines established in the Degree in Artificial Intelligence program at USC will be followed.
In the lectures, the professor will present theoretical concepts, provide examples, and demonstrate results that are most useful for understanding the subject matter (addressing competencies CB2, CB5, CG2, and TR3).
The interactive laboratory classes will illustrate the theoretical content and focus on solving questions and problems, with active student participation (addressing competencies CB3 and TR3). These sessions will also help students acquire practical skills (addressing competencies CB3, CB5, CG4, and CE1).
In the small group tutorials, there will be personalized monitoring of students' learning and their work outside of class. Problems will also be proposed for students to solve in the presence of the professor (addressing competencies CB2, CB5, and TR3).
In the tutorials in very small groups, there will be a personalized follow-up of the students learning and their work outside the class. Problems will also be proposed, to be done in class (competences TR1, TR2 and CG8).
A course on the Virtual Campus will be made available to students to support the teaching of this subject, providing materials related to the lecture content and exercise booklets for use in the laboratory sessions.
The students' grades, including those of students retaking the course, will be based on the assessment of a final theoretical-practical exam (F) and continuous assessment of the work done throughout the semester (C).
The final exam will be held on the date set by the department. All enrolled students may take this exam, which will be conducted at the end of the first semester (January/February). If they do not pass the subject, they can take the exam at the end of the second semester (June/July).
The final theoretical-practical exam will be in-person and written. The final exam will assess competencies CB2, CB3, CB5, and CG2.
For the continuous assessment, two tests conducted in class (in laboratory sessions) will be considered, as well as the student's participation in classes and tutorials. The grade obtained in the continuous assessment (C) will be valid for both exam opportunities. Competencies CB3, CB5, CG4, CE1, and TR3 will be assessed.
The final grade, in each of the opportunities, will be calculated using the formula: Final Grade = 70% F + 30% C. Students who do not attend any of the final exams, both in the first and second opportunities, will be considered "Not Presented."
In cases of fraudulent execution of exercises or tests, the provisions of the regulations on the assessment of students' academic performance and grade review will apply.
IN-CLASS WORK: 60 hours distributed as follows:
Lectures: 30 hours.
Problem-based learning in laboratory classes: 20 hours.
Small group tutorials: 1 hour.
Assessment activities: 9 hours.
STUDENT'S INDIVIDUAL WORK (NON-PRESENTIAL): 90 hours distributed as follows:
Hours of autonomous study related to classes: 45.
Work on proposed problem booklets: 45.
TOTAL: 150 hours (6 ECTS credits)
-Continuous attendance to classes.
-Daily study of class contents.
-It is essential for the student to attend laboratory classes having previously worked on the proposed exercises for each session. To do so, they must acquire sufficient knowledge of the theory to tackle these problems.
-Active participation in tutorials for personalized monitoring.
-The books in the bibliography complement the classes and provide an important source of examples and exercises.
Rosa Mª Fernandez Rodriguez
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813158
- rosam.fernandez [at] usc.es
- Category
- Professor: University Lecturer
Maria Cristina Costoya Ramos
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- cristina.costoya [at] usc.es
- Category
- Professor: University Lecturer
Samuel Alvite Pazo
- Department
- Mathematics
- Area
- Algebra
- samuel.alvite.pazo [at] usc.es
- Category
- USC Pre-doctoral Contract
Alex Pazos Moure
- Department
- Mathematics
- Area
- Algebra
- alex.pazos.moure [at] usc.es
- Category
- Xunta Pre-doctoral Contract
| Monday | |||
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| 09:00-10:00 | Grupo /CLE_01 | Galician, Spanish | IA.11 |
| Tuesday | |||
| 12:00-14:00 | Grupo /CLIL_03 | Spanish | IA.13 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLE_01 | Galician, Spanish | IA.11 |
| 12:00-14:00 | Grupo /CLIL_01 | Galician, Spanish | IA.11 |
| Friday | |||
| 09:00-10:00 | Grupo /CLE_01 | Galician, Spanish | IA.11 |
| 12:00-14:00 | Grupo /CLIL_02 | Galician | IA.11 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_02 | IA.01 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_03 | IA.01 |
| 12.20.2024 09:00-14:00 | Grupo /CLE_01 | IA.01 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_01 | IA.01 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_01 | IA.04 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_02 | IA.04 |
| 12.20.2024 09:00-14:00 | Grupo /CLE_01 | IA.04 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_03 | IA.04 |
| 12.20.2024 09:00-14:00 | Grupo /CLE_01 | IA.11 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_02 | IA.11 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_03 | IA.11 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_01 | IA.11 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_01 | IA.12 |
| 12.20.2024 09:00-14:00 | Grupo /CLE_01 | IA.12 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_02 | IA.12 |
| 12.20.2024 09:00-14:00 | Grupo /CLIL_03 | IA.12 |
| 07.03.2025 09:00-14:00 | Grupo /CLIL_03 | IA.11 |
| 07.03.2025 09:00-14:00 | Grupo /CLE_01 | IA.11 |
| 07.03.2025 09:00-14:00 | Grupo /CLIL_01 | IA.11 |
| 07.03.2025 09:00-14:00 | Grupo /CLIL_02 | IA.11 |