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Linear Algebra and Numerical Analysis

Code 9093
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements Knowledge acquired in secondary education in Mathematics A.
Mode of delivery Face-to-face with the use of an e-learning platform
Work placements Not Applicable
Learning outcomes The main goal of this curricular unit (CU) is to provide an introduction to the basic concepts and techniques in Linear Algebra and Numerical Analysis.
At the end of this CU, the student will be able to
- understand fundamental properties of matrices, including determinants and inverse matrices and solve systems of linear equations using direct methods;
- describe and analyse numerical methods for solving non-linear equations, systems of linear equations and of non-linear equations, methods for polynomail interpolation, methods for approximating integrals and methods for approximating solutions to simple ordinary differential equations (initial value problems);
- apply the methods discussed to solve mathematical problems in bioengenhary.
Syllabus Chapter I: Matrices
Types of matrices, matrix and vector operations, elementary row operations, row-echelon form, rank of a matrix, Gauss and Gauss-Jordan methods for solving systems of linear equations, inverse matrix, calculation of matrix inverse by the Gaussian elimination method.
Chapter II: Determinants
Definition, properties, Cramer's rule, calculation of matrix inverse using determinants.
Chapter III: Preliminaries on computing
Elementary concepts, errors and convergence.
Chapter IV: Nonlinear equations
Bisection, false position, Newton-Raphson, secant and fixed-point methods.
Chapter V: Systems of linear and of nonlinear equations
Jacobi and Gauss-Seidel methods, Newton-Raphson method.
Chapter VI: Interpolation
Lagrange polynomial and Newton polynomial
Chapter VII: Numerical differentiation and integration
Trapezium rule and Simpson's rule
Chapter VIII: ODEs: initial value problems
Euler, Taylor and Runge-Kutta methods
Main Bibliography • Cabral, I., Perdigão, C., Saiago, C. , Álgebra Linear, Escolar Editora, 2018.
• Lipschutz, S., Álgebra Linear, Schaum's Outline Series. McGraw-Hill, 1994.
• Magalhães, L.T., Álgebra Linear como Introdução à Matemática Aplicada, Texto Editora, 1993.
• Burden, R.L. & Faires, J.D., Numerical Analysis, 9th Ed., Brooks/Cole, Cengage Learning, 2011.
• Pina, H., Métodos Numéricos, Mc Graw-Hill, 2010.
• Valença, M.R., Métodos Numéricos, INIC, 1988.
Teaching Methodologies and Assessment Criteria The classes combine theory with practice. The teacher introduces the concepts, states and proves the fundamental results, provides examples and applications. The student is encouraged to participate in the classes, to interact with the teacher and colleagues and to work independently, by solving exercises, guided reading, problem formulation and problem solving.
The evaluation carried out during the teaching-learning process consists of five tests, four of them using the Maple T.A. software (homework assignments), which are worth 30% of the final grade, and a written in-person test which account for 70% of the final grade. Each exam, consisting of a written and in-person test, has a maximum rating of 20 values.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2022-03-31

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