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- Linear Algebra and Numerical Analysis

Code | 12500 |

Year | 1 |

Semester | S2 |

ECTS Credits | 6 |

Workload | TP(60H) |

Scientific area | Mathematics |

Entry requirements | Mathematics of Secondary Education and Calculus I. |

Mode of delivery | Face to Face |

Work placements | Not applicable. |

Learning outcomes |
o familiarize students with the key tools of Linear Algebra (Matrix Calculation, Determinants, Eigenvalues and vectors) and Numerical Analysis (Approximation of roots of algebraic equations, interpolation functions, Numerical integration, Numerical solution of differential equations). Students should be able to: - Solve a system of linear equations - Find the Inverse of a matrix using the Gauss-Jordan Method - Find the Null space and image space of a matrix - Apply the Theory of Eigenvalues and Eigenvectors to solve differential equations - Numerical approximation of the roots of an equation by Newton's method and bisection method. - Numerical approximation of a function using Lagrange interpolation - Numerically integration of a real-valued function |

Syllabus |
1. Matrices and Gaussian Elimination 1.1. The Geometry of Linear Equations 1.2. Matrix notation and matrix operations 1.3. Triangular factors and row exchanges 1. 4. Applications of matrices in solving linear systems 1.5. Special matrices and applications 2. Vector Spaces 2.1. Vector spaces and subspaces 2.2. Linear independence, basis, and dimension 2.3. The four fundamental subspaces 2.4. Linear transformations 3. Determinants 3.1. Formulas for the determinant 3.2. Applications of determinants 4. Eigenvalues and Eigenvectors 4.1. Diagonalization of a matrix 4.2. Eigenvalues and eigenvectors in differential equations 5. Numerical Analysis 5.1. Approximation of the roots by Newton's method and bisection method. 5.2. Interpolation of a function using Lagrange interpolation method 5.3. Numerical Integration |

Main Bibliography |
Isabel Cabral, Cecília Perdigão, Carlos Saiago, Álgebra linear : teoria, exercícios resolvidos e exercícios propostos com soluções, Escolar Editora, 2009. ISBN: 978-972-592-239-2. Gilbert Strang, Linear Algebra And Its Applications, 4th Edition, 2006. ISBN: 0-03-010567-6. Knop, Larry E., Linear algebra. A first course with applications. Textbooks in Mathematics. CRC Press, Boca Raton, ISBN: 978-1-58488-782-9. Kress, R., "Numerical Analysis", Graduate Texts in Mathematics, Vol. 181, Springer-Verlag, NY, 1998. ISBN: 0- 387-98408-9. Ricardo, Henry, A modern introduction to linear algebra. CRC Press, Boca Raton, FL, 2010. ISBN: 978-1-4398- 0040-9 15-01. |

Teaching Methodologies and Assessment Criteria |
The curricular unit is divided into two parts: - The Linear Algebra part will work during the first 7 weeks, being evaluated by two minitestes (March 12 and April 5, with 4 and 6 values, respectively); - The numerical part will work in the second half of the semester, being evaluated by two mini-tests (May 9 and June 11, with 4 and 6 values, respectively). The student, if you wish, can choose to work in Python by implementing Numeric algorithms. In this case, in each of the two minitestes, one of the questions will be replaced by another one focusing on the tasks performed in Python and will be worth 1 value. If the sum of the evaluations of the four minitestes is equal to or greater than 9.5 values, the student is exempt from the final exam. If the sum is less than 5.5 values, the student will be Not Admitted. The student who has chosen Python may choose to keep the grade obtained in the evaluation of these two values, taking the exam for 18 values. |

Language | Portuguese. Tutorial support is available in English. |