## Public University of Navarre

Academic year: 2021/2022 | Previous academic years:  2020/2021  |  2019/2020  |  2018/2019  |  2017/2018
 Course code: 250101 Subject title: MATHEMATICS I Credits: 6 Type of subject: Basic Year: 1 Period: 1º S Department: Estadística, Informática y Matemáticas Lecturers: ASIAIN OLLO, MARÍA JOSÉ (Resp)   [Mentoring ] MILLOR MURUZABAL, NORA   [Mentoring ]

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### Module/Subject matter

Module: Basic training

Subject matter: Mathematics

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### 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

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### 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

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### 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.

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### Methodology

 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

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### 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

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English

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### Evaluation

 Learning outcome Evaluation procedure Weight (%) Recoverable All in-class work 20 Yes All exams 80 Yes

 Contents Criteria Evaluation procedures Weight (%) Theoretical and practical contents Key concepts identification and understanding of theoretical and operational knowledge of the subject. Theoretical-practical exams 80% Competence for analysis and synthesis. Practical application of knowledge. Proper response in time, form and content suitability. Theoretical-practical exams Practical application of knowledge. Individual tests performed during the course 20% Creativity, ability to analyse and synthesise

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### Contents

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.
Applications.

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### Agenda

Unit 1.- Sets, applications and relationships. Operations. Congruences. Definition of ring and body.

Unit 2.- Vector spaces. Linear combinations. Linear dependence and freedom. Vector subspaces. Supports and dimensions.

Unit 3.- Linear applications. Kernel and image. Construction of linear applications.

Unit 4. Subject- Matrices. Hermit basic operations, rank and form. Matrix equivalence. Systems of linear equations: Rouché-Frobenius theorem. Generalized reverse.

Unit 5.- Eigenvalues and vectors. Diagonalization of square matrices.

Unit 6.- Scalar product. Vector norm. The angle between two vectors. Euclidean vector space. Orthogonal and orthonormal bases. Orthogonal projection. Orthogonal matrices. Symmetric matrix diagonalization.

Unit 7.- Approximate solutions of the system of equations. Applications.

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### Bibliography

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

• D. R. Guichard and others, Calculus, http://www.whitman.edu/mathematics/california_calculus/
• J. Hefferon, Linear Algebra,http://joshua.smcvt.edu/linearalgebra
• 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

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