Titular Professors
Professors
Basic knowledge of matrix algebra is recommended.
The Learning Outcomes of this subject are the following:
RA.1 Mathematical knowledge to face the Bachelor Degree
This is a general learning outcome shared with other subjects. In particular for the subject of linear algebra this general outcome is shared with these other two:
RA1.1: Linear algebra knowledge to face the Bachelor Degree
RA1.2: Practical application of the main concepts of linear algebra in the different engineering fields.
1. Determinants and Matrices
1.1 Determinants and its properties.
1.2 Determinant calculation.
1.3 Matrix rank.
1.4 Inverse of a matrix.
2. System of Linear Equations
2.1 System analysis.
2.2 Resolution methods: Cramer, Gauss, Gauss-Jordan, Inverse.
2.3 Resolution of similar systems.
3. Vector Spaces
3.1 Definition and properties.
3.2 Linear dependence and independence vectors.
3.3 Vector subspaces.
3.4 Bases and dimension.
3.5 Vector components referred to a basis.
3.6 Change of basis.
4. Linear applications
4.1 Concept, definition and properties of a linear application.
4.2 Nucleus subspace / Kernel.
4.3 Image subspace.
4.4 Propositions and other definitions.
4.5 Matrix of linear applications and matrix associated to the composition of linear applications.
5. Diagonalization of Endomorsphisms
5.1 Introduction.
5.2 Invariant subspace.
5.3 Eigenvectors and eigenvalues.
5.4 Characteristic polynomial.
5.5 Diagonalization conditions.
5.6 Cayley Hamilton theorem. Applied to matrix inversion.
5.7 Matrix diagonalization.
5.8 Applications: power calculation, polynomial and square matrices.
5.9 Introduction to Singular Value Decomposition.
6. Euclidean and Unitary Vector Spaces
6.1 Scalar product. Euclidean and unitary spaces.
6.2 Norm, dot product and distance.
6.3 Angle in between two vectors.
6.4 Orthogonality and orthogonal subspaces.
6.5 Orthogonal projection. Minimum error approximation.
6.6 Orthogonal projection in subspaces with a dimension greater than 1.
6.7 Gram-Schmidt orthogonalization process.
The subject is taught weekly in 3 sessions of 50 minutes each. These sessions will combine, formal lectures, exercise/problem lectures and practical sessions.
The way to approach these lectures (an exception applies to practical sessions) will be the following:
- First third of the lecture: Then students in groups of 3 will solve an exercise proposed with the content that they have studied at home previously (with the material of the subject they have).
- Second third of the lecture: The teacher will explain to the whole group the doubts that they have found while solving the proposed problem and will solve it in the blackboard/computer (depending on the kind of exercise).
- Last third of the lecture: The students will solve another exercise in groups of three, after all the doubts of the initial exercise have been solved. The aim of this second exercise is to confirm that the group has understood correctly the content of the session.
In order to evaluate if the student has reached the subject objectives adequately different tests are used to obtain information about each student:
1. Exams
During the course four exams are done: two during each semester.
2. Class tests.
3. Classroom participation
The teacher has an observation checklist where he takes notes about the student´s different attitudes and behaviours during the class.
4. Individual or team reports.
The final grade will be calculated taking the average of the grades obtained in the two semesters if both are passed. Otherwise, it will be the minimum of both.
The grades of each semester will be calculated considering the marks obtained in the exams (Ex_Mark) and the mark obtained in the continuous assessment (CA_Mark) according to the following formula:
Semester_Mark = 0,7 · Ex_Mark + 0,3 · CA_Mark
as long as Ex_Mark is greater or equal than 3,5, if not Semester_Mark = Ex_Mark
On the other hand the mark of the exams in each semester will be calculated with an average of the marks of the exam taken in the first part of the semester (Ex_First_Part) and the exam taken in the second part of the semester (Ex_Second_part) as long as the score of Ex_First_Part is at least a 5 and the score of Ex_Second_part is at least a 3:
Ex_Mark = 0,5 · Ex_First_Part + 0,5 Ex_Second_Part
In the case that the student does not pass the first part, she/he will have to do a final exam (Ex_Final_Semester) with all the content of the subject, in this case the final mark will be the following:
Ex_Mark = Ex_Final_Semester
For those students who have not passed in June there will be retake exam in July, where they have to sit the failed semesters. Like on the ordinary exam, to pass the retake both semesters must be passed in order to average them. In the retake the final mark will be the highest grade based on:
a) 70% of the recovery exam and 30% of the continuous assessment based on the corresponding semester (as long as the mark obtained in the exam is at least 3.5).
b) 100% of the recovery exam.
For those students who do not show up in one of the recovery exams with no subject to release, the final mark of the subject will be Not Presented (NP) in the extraordinary announcement.
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
Further literature can be found in Spanish and Catalan (see Spanish and Catalan versions).