Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Module: Basic training

Subject matter: Mathematics

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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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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

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Languages

English

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Evaluation

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Agenda

Important warning: Since the English group of this course is common to other Engineering degrees, its contents may differ from the contents of the groups taught in Spanish.

Lesson 1. Vectors and matrices
Vector spaces. Linear combinations and linear independence. Subspaces. Bases and dimension.
Matrices. Rank. Inverse.
Determinants.
Solutions of linear systems. Gauss method.

Lesson 2. Scalar product.
Euclidean spaces. Scalar product.
Orthogonal projections. Gram-Schmidt method.
Least squares method.

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

Lesson 4. Diagonalization and quadratic forms.
Diagonalization of matrices.
Eigenvalues and eigenvectors.
Orthogonal diagonalization of symmetric matrices.
Quadratic forms.

Lesson 5. Real and complex numbers. Sequences and series.
Natural, integer, rational, real and complex numbers.
Sequences and series. Convergence.

Lesson 6. Functions of a real variable.
Limits and continuity.
Weierstrass and Bolzano Theorems.

Lesson 7. Differential Calculus.
Derivatives of functions of one real variable.
Mean Value Theorems. Extrema.
Power series. 

Lesson 8. Integral Calculus.
The Riemann integral.
Mean value theorem for integrals.
Fundamental Theorems of Calculus.
Integration techniques. Integration by parts. Change of variable.
Improper integrals.

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Bibliography

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

• J. Hefferon, Linear Algebra, Virginia Commonwealth University Mathematics 2009
• D. C. Lay, Linear Algebra and its applications, Pearson Education 2006
• D. J. S. Robinson, A course in Linear Algebra with applications, World Scientific
• R. A. Adams: Calculus. A complete course. Addison Wesley
• Larson, Calculus of a Single Variable, 10th Edition, Brooks/Cole, Cengage Learning 2014
• S.L. Salas, E. Hille and Etgen, Calculus, Reverté

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