Course code: 250101 |
Subject title: MATHEMATICS I |
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Credits: 6 |
Type of subject: Basic |
Year: 1 |
Period: 1º S |
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Department: Estadística, Informática y Matemáticas |
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Lecturers: |
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ASIAIN OLLO, MARÍA JOSÉ (Resp) [Mentoring ] | MILLOR MURUZABAL, NORA [Mentoring ] |

**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

**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

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 |

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 | 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 |

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.

**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.

**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