Algebra

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Credits
6
Types
Compulsory
Requirements
This subject has not requirements
Department
MAT
In this subject we introduce the concepts of linear algebra and geometry necessary to understand the algebraic methods and models used throughout the Bachelor's degree. Special emphasis is placed on examples from bioinformatics, biostatistics, and biomathematics.

Teachers

Person in charge

  • Marta Casanellas Rius ( )

Others

  • Iria Posé Lagoa ( )
  • Xavier Povill Clarós ( )

Weekly hours

Theory
2
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
6

Learning Outcomes

Learning Outcomes

Knowledge

  • K2 - Identify mathematical models and statistical and computational methods that allow for solving problems in the fields of molecular biology, genomics, medical research, and population genetics.
  • K3 - Identify the mathematical foundations, computational theories, algorithmic schemes and information organization principles applicable to the modeling of biological systems and to the efficient solution of bioinformatics problems through the design of computational tools.

Skills

  • S3 - Solve problems in the fields of molecular biology, genomics, medical research and population genetics by applying statistical and computational methods and mathematical models.

Competences

  • C3 - Communicate orally and in writing with others in the English language about learning, thinking and decision making outcomes.
  • C6 - Detect deficiencies in the own knowledge and overcome them through critical reflection and the choice of the best action to expand this knowledge.

Objectives

  1. Acquisition of the basic knowledge of linear algebra (vector spaces, matrices, linear systems, linear maps, diagonalization).
    Related competences: C3, K3, C6,
  2. Using linear algebra for solving mathematical problems and interdisciplinary problems, specially in the field of bioinformatics
    Related competences: K2, K3, S3,
  3. Learning how to use software to solve linear algebra problems
    Related competences: K2, S3, C6,

Contents

  1. Matrices and linear systems
    Matrices: Operations, elementary transformations, rank, determinant, inverse of a matrix. Linear systems: gaussian elimination, solutions
  2. Vector spaces.
    Vectors, linear combinations, dependency. VEctor spaces, systems of generators, basis. Coordinates and change of basis. Subspaces; intersection and sum,
  3. Linear maps
    Linear maps. Matrices. Kernel and image. Composition and inverse map. Change of basis.
  4. Diagonalization
    Eigenvalues and eigenvectors; characteristic polynomial; algebraic and geometric multiplicity, diagonalization criteria. Special case of Markov matrices. Applications.
  5. Linear discrete dynamical systems
    Definition and Computation of solutions. Applications to biology.
  6. Orthogonality
    Inner product, norm, distance. Orthogonal projection, Quadratic least squares. Spectral theorem. Singular value decomposition and rank approximation.

Activities

Activity Evaluation act


Theoretical expository lectures


Objectives: 1
Contents:
Theory
26h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
40h

Problem solving sessions


Objectives: 2 3
Contents:
Theory
0h
Problems
28h
Laboratory
0h
Guided learning
0h
Autonomous learning
40h

Mid term exam


Objectives: 1 2
Week: 7
Theory
0h
Problems
2h
Laboratory
0h
Guided learning
0h
Autonomous learning
0h

Final exam


Objectives: 1 2
Week: 15 (Outside class hours)
Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
0h

Python delivery


Objectives: 3
Week: 12 (Outside class hours)
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
5h

Python assessment



Week: 1
Theory
1h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
5h

Teaching methodology

Lectures will be mainly of expository type. There will be also problem-based sessions and exercise sessions using Python.

Evaluation methodology

For the evaluation of the subject, the grade of the partial exam (P), the grade of the final exam (F), the grade of the Python delivery (Py), and the grade of the Python Assessment (EPy) will be taken into account and will be combined with the following formula:

Grade=max(0.3*P+0.05*Py+0.05EPy+0.6*F;0.05*Py+0.05EPy+0.9*F;F)

A student is considered to have taken the subject if he/she takes the final exam. If the student has taken the subject but has failed, then the student may take the re-evaluation exam (R) and in this case the grade for the subject will be the maximum between R and 0.9*R+0.05*Py+0.05EPy.

Bibliography

Basic:

Complementary:

Web links