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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.
Planned learning activities and teaching methods The course consists of theoretical-practical classes – TP – allowing the connection between theory, examples, and application problems. Are used various teaching methods, including: exposure of the syllabus, also involving the presentation of small
problems and solving practical problems, reading / discussion of readings and solving exercises, individually or in small groups, by the students, developing their autonomy but also mutual aid. The teaching methodology is student-centered; during the semester, the student will learn and apply the acquired concepts with his autonomous work and with the help of the teaching team. Thus, particular importance is given to the continuous evaluation that allows the student, during the semester, to demonstrate the competences gradually acquired.
Metodologias de Ensino e Critérios de Avaliação A avaliação é realizada em duas fases:
- avaliação contínua: testes teórico-práticos ao longo do semestre letivo,
- exame final (com parte teórica e parte prática) para os alunos admitidos.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2016-06-09

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