Algebra

Credits
6
Types
Compulsory
Requirements
This subject has not requirements , but it has got previous capacities
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

  • Roser Homs Pons (roser.homs@upc.edu)

Others

  • Marta Casanellas Rius (marta.casanellas@upc.edu)

Weekly hours

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

Competences

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, C6, K3,
    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.
    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
    Week: 8
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Final exam


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

    Python project

    Development of a group project applying linear algebra tools to solve a biological problem using the Python programming language. Assessment will be based on the oral presentation of the completed work.
    Objectives: 2 3
    Week: 15 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Python test


    Objectives: 2 3
    Week: 14
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Teaching methodology

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

    Evaluation methodology

    The final grade for the course will take into account the midterm exam grade (P), the final exam grade (F), the Python project grade (Py), and the Python test grade (TPy). These will be combined using the following formula:

    Grade = max(0.25P + 0.1Py + 0.05TPy + 0.6F; 0.1Py + 0.05TPy + 0.85*F).

    Students who do not take the final exam will be considered "Absent". Students who take the final exam but fail the course will be eligible for a retake exam (R), and in this case the final course grade will be 0.85R + 0.1Py + 0.05*TPy.

    If considered necessary by the professors, an oral examination may be conducted to verify the authorship of any of the assessment tasks.

    Bibliography

    Basic

    Complementary

    Web links