Algebra and Geometry - LMATE

T - Theoretical; TP - Theory and practice; PL - Laboratory; S - Seminar; OT - Tutorial. (*) - Variable.

• Intended learning outcomes (knowledge, skills and competences to be developed by the students):

  After the student is approved, he should be able to:

       Compute with matrices and determinants and solve systems of linear equations;

       Recognize vector and affine spaces;

    Master the key concepts of vector calculus (inner, cross and triple product) and its applications to coordinate geometry;

       Identify basic geometric transformations and know how to compute with them using matrices;

       Compute eigenvalues and eigenvectors and diagonalize a matrix/linear transformation;

       Factorize matrices;

      Use computational tools to solve problems in algebra, geometry and their applications

• Syllabus:

  1.Matrices: matrix operations; systems of linear equation; inverse of a matrix.

  2.Determinants: definition and properties; methods of evaluating determinants.

  3.Vector spaces. definition and examples; linear combinations and linear dependence; subspaces; basis and dimension; change of basis.

  4.Vectorial calculus: inner product, norm, angles; cross product, scalar triple product and geometrical applications.

  5.Affine and euclidean spaces: definition and examples; applications of vector calculus to coordinate geometry.

   6.Linear and affine transformations: definition and examples; matrix representations; isometries and similarities in plane and tridimensional geometry.

   7.Matrix decompositions: eigenvalues, eigenvectors and diagonalization; classical decompositions;  applications to conics, quadrics and geometric tranformations.

• Evidence of the syllabus coherence with the curricular unit’s intended learning outcomes:

  Tools from Linear Algebra and Coordinate Geometry are widely used in modelling throughout science and engineering. The curricular unit aims to provide  basic knowledge in linear algebra (learning outcomes 1, 2, 4 and 5 are covered by sections 1, 2, 3, 4, 6 and 7 of the syllabus), coordinate geometry  (learning outcomes 3 and 4 are covered by sections 5, 6 and 7 of the syllabus). Special emphasis will be given to matrix theory (learning outcomes 1, 4, 5, 6 are covered by sections 5, 6, 7 of the syllabus) and computational tools (learning outcome 7, which is common to the whole program).

• Teaching methodologies (including assessment):

  There will be both theoretical and practical components in the teaching. A total of 90 hours of classes will be scheduled, consisting of 67.5 hours of mixed lectures/recitations (TP) and 22.5 hours of laboratory classes (PL).  The total student work time is 160 hours.

  The theory will be presented together with examples and exercises involving concrete applications in the lectures/recitations. The laboratory classes will be devoted to the solution of exercises applying the theory learned in class. Individual or group work on problems directly related to applications will be emphasized.

  The course assessment will have two components. The first is the average grade (NP) obtained in small projects to be completed in the laboratory classes. The second component is the grade in a final exam (NT) which can be taken either in class, or during the exam periods. The student's final grade, NF, will be computed via the formula

  NF=0.7NT+0.3NP .

  In order to pass this course, the student should obtain a minimum grade of 9.5 in both NT and NP.

• Evidence of the teaching methodologies coherence with the curricular unit’s intended learning outcomes:

  The lecture/recitations present the theory and illustrate the solution of diverse types of problem with varying degree of difficulty. This combination will help the student follow the material presented in class. The presentation of applications to engineering and "real life" problems will increase motivation and give students an introduction to the applications of the theory described in goals 1-6 above.The laboratory classes will allow the students to consolidate their knowledge and develop their autonomy in problem modeling and solving (goals 1-7). The inclusion of simple appropriate problems for beginning students, requiring the use of computational tools, will serve to familiarize them with these tools (goal 7).The assessment will have two components: a final exam (assessing the achievement of goals 1 through 6) and in class projects (assessing the achievement of goals 1 through 7). The projects will help the student follow the material presented in class and develop their analytical and critical thinking skills.

• Main Bibliography:

  Santana, A. P., Queiró, J. P., “Introdução à Álgebra Linear”, Gradiva, 2010.

  Anton, H., Rorres, C., “Elementary Linear Algebra: Applications Version”, Wiley, 10th edition, 2010.

  Farin, G., Hansford, D., “Practical Linear Algebra – A Geometry Toolbox”, 3rd edition, CRC Press, 2014.

  Lay, D., “Linear Algebra and its Applications”, Pearson, 4th edition, 2011.

  Poole, D., “Linear Algebra: a modern introduction”, Brooks/Cole, 4th edition, 2014.

  Strang, G., “Linear Algebra and its Applications”, Brooks/Cole, 4th edition, 2005.