IPLUSO 1

Mathematics I

• ApresentaçãoPresentation

  In this study cycle, curricular units in the scientific area of  Mathematics play an important role. They are essential for students to acquire solid basic knowledge needed in other curricular units of the study cycle that require mathematical knowledge or new skills and competences acquired through gradual and persistent work in curricular units in this area. The curricular unit of Mathematics I assumes a fundamental and relevant role in the beginning of students' mathematical training. Mathematical contents are essential in the training of qualified staff, either in the understanding and consolidation of the different concepts, or in the specific knowledge of their applicability and in the development of new skills and competences acquired with the work in the curricular unit. Much of the content explored in some topics of the mentioned syllabus has a wide application in other areas of the study cycle.
• ProgramaProgramme

  CP1. Algebraic structures. Fields R and C CP2. Vector spaces. Linear combination and independence. Generating set. Basis and dimension CP3. Vector subspace. Intersection and direct sum CP4. Linear systems. Matrix algebra. Inverse CP5. Gaussian characteristic and condensation. Rouché's theorem and dependence of variables CP6. Elementary matrices. Permutations. Determinant and properties CP7. Complementary minors and adjoint. Laplace's formula. Cramer's rule CP8. Operators and linear transformations (LTs). Image and kernel. Similarity. Change of basis CP9. LTs in computer graphics: composite and geometric transformations CP10. Eigenvectors and eigenvalues (EVVs). Invariants. Characteristic polynomial CP11. Diagonalization of matrices. Jordan block and canonical form. Minimal polynomial CP12. EVVs in system stability linear dynamics: difference and power equations of a matrix, differential and exponential matrix equations, Markov processes, input-output and Von Neumann models
• ObjectivosObjectives

  LO1. Understand the concepts of real vector space and vector subspace; LO2. Master the language of vectors and matrices and perform operations; LO3. Classify sets of vectors according to linear independence; LO4. Obtain systems of generators, bases, and the dimension of vector spaces; LO5. Obtain the coordinates of a vector in different bases; LO6. Calculate determinants, interpret their value, and apply properties; LO7. Solve linear systems using matrices and identify dependent variables; LO8. Calculate eigenvalues and eigenvectors; LO9. Understand the definition of the product of complex numbers as the operation between vectors that allows the structure of a field and a vector space over R in C; LO10. Obtain the matrix of a linear transformation in different bases and determine the kernel and image subspaces; LO11. Use Python (or Octave) as an exploratory work tool; LO12. Apply theory to contextual problems and acquire the skills and reasoning for their formulation.
• BibliografiaBibliography

  Strang, G. (2009). Introduction to Linear Algebra, Wellesley-Cambridge Press. Almada, T. (2007). Álgebra Linear, Edições Universitárias Lusófonas. Magalhães, L. T. (2001). Álgebra Linear como introdução à Matemática Aplicada, Texto Editora. Blyth, T.S.; Robertson (1998). Basic Linear Algebra, Springer. Monteiro, A.; Pinto, G. (1997). Álgebra Linear e Geometria Analítica. Problemas e exercícios, McGraw-Hill.    
• MetodologiaMethodology

  Teaching methodologies are based on two strands:   (1) Theoretical sessions - where fundamental concepts are conveyed; (2) Theoretical-practical sessions, in which teaching is practically oriented and students are invited to analyze and solve problems involving the concepts presented in the theoretical classes. Students are also encouraged to experiment with various problem-solving strategies.
• LínguaLanguage

  Português
• TipoType

  Semestral
• ECTS

  6
• NaturezaNature

  Mandatory
• EstágioInternship

  Não