ISLA IPGT 2

Linear Algebra and Analytical Geometry

• ApresentaçãoPresentation

  Linear Algebra and Analytic Geometry introduces fundamental concepts and methods for representing and solving problems involving linear and geometric structures. It connects the study of vector spaces, points, vectors, lines, planes and curves with matrices, systems of linear equations, linear transformations and complex numbers. The curricular unit contributes to the development of mathematical reasoning, modelling skills, and the selection and use of quantitative tools in science and engineering problems.  
• ProgramaProgramme

  CP1 - Vector spaces: subspaces, linear dependence and bases. CP2 - Points and vectors: distances, plane sections and spheres. CP3 - Vector operations: dot product, parallelism, perpendicularity and relative positions. CP4 - Lines: direction vectors and equations. CP5 - Cross product: normal vectors and plane equations. CP6 - Curves: parametrisation, tangent/normal vectors, intersections and polar coordinates. CP7 - Matrices: operations, inverses and determinants. CP8 - Linear systems: matrix form, solutions and linear transformations. CP9 - Quadratic equations and complex numbers: algebraic/polar forms and Euler's formula. CP10 - Complex vector space: De Moivre's formulas and complex roots.
• ObjectivosObjectives

  By the end of the curricular unit, students should be able to: 1. Identify vector spaces, subspaces, bases and linear dependence. 2. Represent points and vectors in 2D and 3D. 3. Apply dot and cross products to geometric problems. 4. Determine and analyse lines and planes. 5. Parametrise curves and work with tangent and normal vectors. 6. Perform matrix operations, determinants and inverses. 7. Solve and interpret linear systems and transformations. 8. Solve quadratic equations and work with complex numbers. 9. Apply Euler's and De Moivre's formulas to complex numbers.
• BibliografiaBibliography

  Marcos, M. G., Oliveira, M. J. G. P., & Barreiras, A. M. S. (2017). Álgebra linear e geometria analítica. Sílabas & Desafios. Stroud, K. A., & Booth, D. J. (2020). Engineering mathematics (8th ed.). Red Globe Press.  
• MetodologiaMethodology

  Teaching combines brief conceptual explanations with problem solving, worked examples, guided exercises and independent practice. Where appropriate, topics are introduced through a problem that establishes the need for a mathematical tool, followed by intuitive exploration, formalisation of the concept, a commented solution and application to new situations. Problem-based learning, pair or small-group discussion, board work and collective analysis of different strategies are used. Tasks focus on mathematically natural contexts, such as trajectories, distances, intersections, simultaneous constraints and transformations, and ask students to justify their choice of method and interpret the results. Geometric representations and digital visualisation tools may be used when they support understanding.
• LínguaLanguage

  Português
• TipoType

  Semestral
• ECTS

  6
• NaturezaNature

  Mandatory
• EstágioInternship

  Não
• AvaliaçãoEvaluation

  Descrição dos instrumentos de avaliação (individuais e de grupo) ¿ testes, trabalhos práticos, relatórios, projetos... respetivas datas de entrega/apresentação... e ponderação na nota final.

  Exemplo:

  Descrição	Data limite	Ponderação
  Teste de avaliação	10-2026	20%
  Teste de avaliação	12-2026	20%
  Exame Final	01-2027	60% (com nota mínima de 8)

   

  Adicionalmente poderão ser incluídas informações gerais, como por exemplo, referência ao tipo de acompanhamento a prestar ao estudante na realização dos trabalhos; referências bibliográficas e websites úteis; indicações para a redação de trabalho escrito...