Geometry
Code
10974
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Herberto de Jesus da Silva
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
The student is supposed to acquire basic knowledge on inner product spaces and analytic geometry (vide syllabus) in a deductive and critical perspective.
Prerequisites
Knowledge corresponding to the contents of Linear Algebra I.
Subject matter
1. Inner product spaces – Definition of inner product and elementary properties. Euclidean space and unitary space. Matrix of an inner product (relative to a fixed basis). Norm. Schwarz inequality. Triangle inequality. Angle between two non-zero vectors of a euclidean space. Orthogonal and orthonormal (finite) vector systems. Gram-Schmidt orthogonalization process. Orthogonal complement. Cross product and mixed product.
2. Bilinear forms and quadratic forms – Definitions and elementary properties. Polar form.
3. Affine Geometry.
3.1 Affine spaces – Definition and dimension. Affine euclidean space. Affine subspace. Incidence propositions. Coordinate system of an affine space. Point coordinates. Vectorial, cartesian and parametric equations of affine subspaces.
3.2 Euclidean or metric geometry in euclidean affine spaces – Orthogonal affine subspaces. Distance and angles. Quadric surfaces.
Bibliography
1. Agudo, F. R. D., Introdução à Álgebra Linear e Geometria Analítica, Escolar Editora, 1996.
2. Anton, H., and Rorres, C., Elementary Linear Algebra - Applications Version, 8th Edition, John Wiley & Sons, 2000.
3. Giraldes, E., Fernandes, V. H., and Marques-Smith, M. P., Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 1995.
4. Monteiro, A., Álgebra Linear e Geometria Analítica, McGraw-Hill de Portugal, 2001.
Teaching method
There are classes in which theory is lectured and illustrated by examples. There are also problem-solving sessions. Some exercises are left to the students to be solved on their own as part of their learning process.
Evaluation method
Students enrolled for the first time in the unit must attend, at least, 85% of the lectures and 85% of the problem-solving classes.
Students that have already been enrolled in the unit must attend, at least, 2/3 of the lectures and 2/3 of the problem-solving classes.
There are three mid-term tests. These tests can substitute the final exam if the student has grade, at least, 7.5 in the third one and CT is, at least, 9.5. CT is the arithmetic mean of the non-rounded grades of the tests.
If the student satisfies the conditions above with CT (rounded to units) greater than 16, he may choose between having 16 as final grade or undertake a complementary assessment.
To be approved in final exam, the student must have a minimum grade of 9.5 in it. Again, for grades (rounded to units) greater than 16, the student must undertake a complementary assessment, otherwise his final grade will be 16.
More detailed rules are available in the portuguese version.