Public University of Navarre

Academic year: 2020/2021 | Previous academic years:  2019/2020  |  2018/2019  |  2017/2018 
Bachelor's degree in Computer Science at the Universidad Pública de Navarra
Course code: 250101 Subject title: MATHEMATICS I
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Estadística, Informática y Matemáticas

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Module/Subject matter

Module: Basic training

Subject matter: Mathematics


General proficiencies

  • G8 Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations
  • G9 Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in computer engineering
  • T1 Analysis and synthesis ability
  • T3 Oral and written communication
  • T4 Problem solving
  • T8 Self-learning


Specific proficiencies

  • FB1 Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential and integral calculus, numerical methods, numerical algorithms, statistics and optimization
  • FB3 Ability to understand and master the basic concepts of discrete matemathics, logic, algorithmics and computational complexity, and their applications to problem solving in engineering


Learning outcomes

At the end of this course students will be able to

  • Solve systems of linear equations using different methods based on Matrix decompositions (LU, QR, generalized inverse).
  • Make matrix representations of plane and space transformations, proyections, and orthogonal proyections onto subspaces
  • Diagonalize matrices through fundamental subspaces.
  • Use matrix diagonalization to study stochastic processes (Markov chains).
  • Find the singular value descomposition of a given matrix and use it to approximate systems of linear equations by the pseudo-inverse matrix.
  • Find the least-squares polynomial approximation to a function.
  • Use some symbolic computation software such as Matlab or Mathematica.



Methodology - Activity Attendance  Self-study
A-1 Lectures 46  
A-2 Practical clases 14  
A-3 Self-study    75
A-4 Exam, evaluation tests 5  
A-5 Office hours 10  
Total 75 75


Relationship between formative activities and proficiencies/learning outcomes

Proficiency Activities
G8 A-1, A-2, A-3, A-5
G9 A-1, A-2, A-3, A-5
FB1 A-1, A-2, A-3, A-4, A-5
FB3 A-1, A-2, A-3, A-5
T1 A-1, A-2, A-3, A-5
T3   A-1, A-2, A-4
T4   A-2, A-4, A-5
T8 A-2, A-3






Learning outcome
Weight (%)  Recoverable              
 All  Various Exams  80%  Yes
 All  In-class work  20%  Yes



Vector Spaces
Bases and dimension.
Linear Transformations. Matrix associated to a linear transformation.
Diagonalization of matrices.
Orthogonal matrices.

Functions of a real variable.
Approximation of functions by polynomials.

Integration of functions of one real variable.



Lesson 1. Matrices and linear systems

Systems of linear equations. Matrix notation.

Matrix multiplication: block matrices and block multiplication.

The inverse of a matrix.

Row reduction and echelon forms: The Gaussian row reduction algorithm.

Gaussian elimination.

Elementary matrices.

Solution sets of linear systems.



Lesson 2. Vector Spaces
Definitions and examples.
Linear combinations and linear independence.
Bases and dimension.

The  change of basis matrix in a vector space.
Row space and column space of a matrix.
Solutions of linear systems.


Lesson 3. Linear Transformations.
Definitions and examples. Matrix associated to a linear transformation.
Operations with linear transformations.

Change of bases and linear transformations.

The null space of a linear transformation.

Kernel, image and isomorphisms. Rank+Nullity theorem.


Lesson 4. Diagonalization of linear transformations and matrices.

Linear transformations from a vector space into itself.

Eigenvalues and eigenvectors.
Diagonalization of matrices.


Lesson 5. Orthogonality.
Euclidean spaces. Scalar product. Norm of a vector and distance between vectors.   Orthogonal projections. Gram-Schmidt method.
Orthogonal diagonalization of symmetric matrices.


Lesson 6. Least-squares approximations.

Approximate solutions of inconsistent systems.

Optimal least-squares solutions.

Normal equations.



Access the bibliography that your professor has requested from the Library.

  • D. R. Guichard and others, Calculus,
  • J. Hefferon, Linear Algebra,
  • R. Larson and B. Edwards, Calculus of a Single Variable, Brooks/Cole Cengage Learning 2014
  • D. C. Lay, Linear Algebra and its applications, Addison Wesley 2012
  • G. Strang, Linear Algebra and its applications, Thomson 2006
  • G. Strang, Calculus, Wellesley-Cambridge Press