In this subject the classic topics regarding linear algebra will be taught. These are: Matrices, systems of linear equations, vector spaces, linear applications, endomorphisms and vector spaces with a defined scalar product. The intention of the course is a twofold, on the one hand that the students get to know the main theoretical concepts of linear algebra and on the other to associate this theory with real problems. During the course, different applications of the concepts studied are proposed and it is shown how these concepts correctly applied can solve determined situations.
Titular Professors
Basic calculus of matrices
The course aims to provide the student with a solid foundation in the fundamentals of linear algebra, developing the ability to understand and use mathematical structures applied to solving engineering problems. Likewise, it seeks to promote logical and abstract reasoning, mathematical modeling, and the application of algebraic methods to system analysis, contributing to the development of analytical skills necessary for subsequent scientific and technological subjects.
1. Determinants and Matrices
1.1. Concept of determinant and properties.
1.2. Calculation of determinants.
1.3. Rank of a matrix.
1.4. Inverse of a matrix.
2. Systems of Linear Equations
2.1. Study of systems.
2.2. Resolution methods: Cramer, Gauss, Gauss-Jordan, Inverse.
2.3. Joint resolution of similar systems.
3. Vector Spaces
3.1. Basic algebraic structures and definition of vector space. Properties.
3.2. Linear dependence and independence of vectors.
3.3. Vector subspace.
3.4. Basis and dimension of a vector space.
3.5. Components of a vector relative to a basis.
3.6. Change of basis.
4. Linear Maps
4.1. Concept, definition, and properties of a linear map.
4.2. Kernel (null space).
4.3. Image (range).
4.4. Propositions and other definitions.
4.5. Matrix of a linear map and matrix associated with the composition of linear maps.
5. Diagonalization of Endomorphisms
5.1. Introduction.
5.2. Invariant subspace.
5.3. Eigenvector and eigenvalue.
5.4. Characteristic polynomial.
5.5. Diagonalizability conditions.
5.6. Cayley-Hamilton Theorem. Application to matrix inversion.
5.7. Applications: calculation of powers, polynomials, and square roots of matrices.
5.8. Introduction to Singular Value Decomposition (SVD).
6. Euclidean and Unitary Vector Spaces
6.1. Scalar product. Euclidean space and unitary space.
6.2. Norm.
6.3. Angle between vectors.
6.4. Orthogonality and orthogonal subspaces.
6.5. Orthogonal projection. Least squares approximation.
6.6. Orthogonal projection onto subspaces of dimension greater than 1.
6.7. Gram-Schmidt orthogonalization.
The subject is taught weekly in sessions. These sessions will combine formal lectures with exercise/problem lectures and practical sessions focused on self-learning and doubt solving: 1. During teaching sessions, examples will be developed for the acquirement of the explained knowledge 2. The students will solve, individually or by groups, an exercise proposed in class, where the acquirement of knowledge will be determined. These exercises can be solved through the explanations in class or through teaching material given via eStudy. Applied methodologies: Flipped classroom (MD7), Peer instruction (MD09) and sessions for problems and exercises solving (MD1). 3. During the session, the lecturer will solve general questions that the group has and will explain how to solve the proposed exercise. Applied methodologies: Formal lecture (MD0) 4. The lecturer assigns individual exercises for students to complete to confirm their understanding of the concepts covered in the session. These exercises must be submitted at the end of the teaching unit (via the corresponding folder in eStudy) and will count toward the continuous assessment grade. Applied methodologies: Peer instruction (MD9) and lecture of problems and exercises.
The final grade for the course in the ordinary session (Final_Grade) is calculated by combining the exam grade (N_Ex), which accounts for 70% of the qualification, and the continuous assessment grade (N_AC), which accounts for 30%, provided that N_Ex is equal to or higher than 4.
The exam grade (N_Ex) corresponds to the average of two parts —the November test and the January test—, with a minimum score of 4 required in each; if any part is lower than 4, the exam grade will be the lower of the two.
Continuous assessment (N_AC) is obtained by weighting various elements: Practicals: 40%, Homework assignments: 20%, In-class or eStudy tests/controls: 20%, Attendance, attitude, and participation: 20%
In the extraordinary session in July, the student must take a comprehensive exam covering the entire syllabus; partial grades obtained during the semester will not be carried over.
Assessment will be based on whether the student:
· Correctly identifies the type of system and discusses its consistency.
· Selects and applies appropriate resolution methods (Gauss, Gauss–Jordan, Cramer, inverse) justifying each step.
· Interprets the meaning of the obtained solutions with clarity and mathematical foundation.
· Correctly handles the properties of vector spaces and subspaces, applying them coherently.
· Works with bases, dimensions, and change of basis with precision and conceptual understanding.
· Performs and understands linear combinations and the superposition principle in algebraic and geometric contexts.
· Determines conditions of linear dependence or independence using valid procedures (ranks, combinations, associated matrices).
· Explains the algebraic and geometric implications of these concepts in the problems addressed.
· Interprets the behavior of a linear map and correctly characterizes its kernel and image.
· Constructs and uses the matrix associated with a linear transformation to solve problems.
· Applies properties of linear maps to justify results and argue solutions.
Linear algebra notes and exercises (available in eStudy)
- Linear Algebra and its Applications; David C. Lay, Steven R. Lay, Judi J. Mcdonald; Fifth Edition Pearson 2016 - Elementary Linear Algebra; Howard Anton, Chris Rorres; 11th Edition; Wiley 2014 - Álgebra lineal; Stanley l. Grossman, José Job Flores Godoy; Séptima edición; McGrawHill 2012 - Álgebra Lineal para estudiantes de ingeniería y ciencias, Juan Carlos Del Valle Sotelo, McGraw-Hill, 2012 - Howard Anton, Introducción al álgebra lineal, Ed. Limusa 1997 - Castellet, M. i Llerena, I., Àlgebra lineal i geomètrica, Universitat Autònoma de Barcelona, 1990 - Queysanne, M., Álgebra básica, Vicens Vives, 1990 - Rojo, A., Álgebra lineal, AC 1991 - Puerta, F., Álgebra Lineal, Marcombo, 1991 - Luzarraga, F., Problemas resueltos de Álgebra Lineal, 1970