Algebra

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Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
In this subject, the concepts of linear algebra necessary to develop the analysis of data and their visualization throughout the Bachelor's degree will be introduced. We will study algebraic concepts from the point of view of matrix algebra, but also from the geometrical and numerical. Emphasis will be placed on examples from the field of computing, data modelling, and image processing.

Teachers

Person in charge

  • Anna Rio Doval ( )

Others

  • Josep Elgueta Monto ( )

Weekly hours

Theory
3
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
7.5

Competences

Technical Competences

Technical competencies

  • CE1 - Skillfully use mathematical concepts and methods that underlie the problems of science and data engineering.

Transversal Competences

Transversals

  • CT5 - Solvent use of information resources. Manage the acquisition, structuring, analysis and visualization of data and information in the field of specialty and critically evaluate the results of such management.
  • CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

Basic

  • CB1 - That 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 from the vanguard of their field of study.

Generic Technical Competences

Generic

  • CG2 - Choose and apply the most appropriate methods and techniques to a problem defined by data that represents a challenge for its volume, speed, variety or heterogeneity, including computer, mathematical, statistical and signal processing methods.

Objectives

  1. Acquisition of the basic knowledge of linear algebra (vector spaces, matrices, linear systems)
    Related competences: CB1,
  2. Recognize concepts of linear algebra within interdisciplinary problems.
    Related competences: CT5,
  3. Learn how to use linear algebra in solving problems of data analysis and modeling.
    Related competences: CT5, CG2,
  4. Using linear algebra tools in mathematical problems
    Related competences: CE1,
  5. Using software to solve exercises related to linear algebra
    Related competences: CT6, CE1,
  6. Understanding of the notions of matrix decomposition, its geometric interpretation and its application in exercise solving
    Related competences: CE1,

Contents

  1. The real coordinate space
    Vectors. Dot product (scalar product). Norm. Angle. Linear independence. Bases. Gram-Schmidt. Coordinate system. Points. Distance. Angle.
  2. Linear Maps
    Linear maps. Matrices. Kernel and image. Systems of linear equations. Gaussian elimination. Subspaces. Invertible matrices. Change of basis. Endomorphisms and automorphisms
  3. Vector subpaces
    Vector subspaces. Bases. Intersection and sum. Orthogonal complement. Orthogonal Projection.
  4. Diagonalization
    Eigenvalues and eigenvectors; characteristic polynomial; algebraic and geometric multiplicity, diagonalization criteria, application to the computation of power of matrices and functions of matrices. Special case of Markov matrices and symmetric matrices. Spectral Theorem.
  5. Projections. Isometries.
    Matriu d'una projecció. Classificació d'isometries en dimensions 2 i 3.
  6. Linear discrete dynamical systems
    Madelling of problems via linear discrete dynamical systems, resolution and analysis of particular and generic solutions; long term behaviour of the solutions; numerical methods for the computation of eigenvalues and eigenvectors. Power iteration. Perron-Frobenius Theorem. Recurrencies.
  7. Applications
    Singular value decomposition; matrix norms; application to rank approximation and dimensionality reduction in data and image analysis. Pseudoinvers and least squares. Errors.

Activities

Activity Evaluation act


Development of topic 1

Classes de teoria i de problemes corresponents al tema 1
Objectives: 1 5 2
Contents:
Theory
4h
Problems
3h
Laboratory
0h
Guided learning
0h
Autonomous learning
10h

Development of topic 2

Classes de teoria i problemesi corresponents al tema 2
Objectives: 1 4 5 2 3
Contents:
Theory
4h
Problems
3h
Laboratory
0h
Guided learning
0h
Autonomous learning
10h

Development of topic 3

Classes de teoria i problemes corresponents al tema 3
Objectives: 1 4 5 2 3
Contents:
Theory
9h
Problems
6h
Laboratory
0h
Guided learning
0h
Autonomous learning
18h

Development of topic 4

Classes de teoria i problemesi corresponents al tema 4
Objectives: 1 4 5 2 3
Theory
4h
Problems
3h
Laboratory
0h
Guided learning
0h
Autonomous learning
10h

Development of topic 5

Classes de teoria i problemes corresponents al tema 5
Objectives: 1 4 5 2 3 6
Contents:
Theory
9h
Problems
6h
Laboratory
0h
Guided learning
0h
Autonomous learning
17h

Development of topic 6

Classes de teoria i problemes corresponents al tema 6
Objectives: 1 4 5 2 3 6
Contents:
Theory
5h
Problems
3h
Laboratory
0h
Guided learning
2h
Autonomous learning
10h

Development of topic 7

Classes de teoria i problemes corresponents al tema 7
Objectives: 1 4 5 2 3 6
Contents:
Theory
10h
Problems
6h
Laboratory
0h
Guided learning
0h
Autonomous learning
17.5h

Final exam

Final exam
Objectives: 1 4 2 3 6
Week: 1 (Outside class hours)
Type: theory exam
Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
5h

Partial exam

Partial exam
Objectives: 1 4 2 3 6
Week: 1 (Outside class hours)
Type: problems exam
Theory
0h
Problems
1.5h
Laboratory
0h
Guided learning
0h
Autonomous learning
5h

Avaluation of problem resolution using Python or another software


Objectives: 4 5 3
Contents:
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
1h
Autonomous learning
2.5h

Teaching methodology

Different methodologies will be considered for lectures and exercises classes.
The lectures will consist mainly of master classes, based on presentations and explanations on the slate; the problem classes will be to solve exercises and practice concepts learned in the theory sessions.
Both of them may incorporate examples o short projects using python or similar software.

Evaluation methodology

The assessment of the subject will consist of the marks: P, F, L
The mark P will be obtained from the partial exam.
The mark F will be obtained from the final exam.
The mark L will be obtained by evaluation of problem resolution using python or another software.
The final mark will be computed as follows:

Note = maximum (60% F + 30% P + 10% L, F)

The re-evaluation grade will be the mark of the reavaluation exam.

Bibliography

Basic:

Complementary:

Web links

Previous capacities

L'alumne ha de dominar els coneixements de matemàtiques de batxillerat i tenir destresa en la resolució de problemes de matemàtiques de nivell de batxillerat.