# Algebra

## You are here

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: