Calculus I

Objectives

1. Perform matrix operations: multiplication by a scalar, addition and multiplication of matrices and inversion using the Gauss-Jordan and adjoint methods;
2. Solve systems of linear equations using the Gaussian elimination method and Cramer's rule; classify systems according to the type of solution;
3. Calculate determinants using Sarrus' Rule and Laplace's Theorem and recognize some of the properties of determinants;
4. Determine eigenvalues ​​and eigenvectors of matrices, proceeding to their diagonalization;
5. Identify groups, rings and fields, and use their properties;
6. Solve problems using arithmetic and geometric progressions;
7. Apply the mathematical induction method to demonstrate properties.

Program

1. Basic matrix operations: addition, multiplication and multiplication by a scalar.
2. Systems of linear equations: classification; resolution by the Gaussian elimination method and Cramer's rule.
3. Determinants: Sarrus` rule, Laplace's Theorem and properties.
4. Matrix inversion using the Gauss-Jordan method and the adjoint method. 
5. Eigenvalues ​​and eigenvectors of matrices: characteristic polynomial, matrix diagonalization and applications.
6. Groups, rings and fields.
7. Sequences defined by recurrence: arithmetic and geometric progressions.
8. Mathematical induction.

Teaching Methodologies

In theoretical classes themes are presented, illustrated with examples, using computer resources combined with the use of the blackboard.
Theoretical-practical classes work in unification with theoretical classes and are completed by solving exercises, encouraging teacher/student interaction.
The UAc Moodle e-Learning platform (at http://moodle.uac.pt) is used as a repository of pedagogical and didactic material to support learning, as well as a platform for scheduling, disseminating and promoting complementary activities and management of evaluation elements.

EVALUATION:

• In each edition of the curricular unit, evaluationis periodic.
• Students are evaluated through two or three written tests carried out during the semester. 
• The final grade is the weighted average of the results obtained in the written tests. 
• In evaluation oral tests are not carried out. 
• In Exam, the final grade in the subject is the grade obtained in the exam. There are no oral exams.

 

Bibliography

- Leon, Steven J., Álgebra Linear com aplicações, LTC - Livros Técnicos e Científicos S.A., 4a Edição, 1999.
- Monteiro, António, Pinto, Gonçalo e Marques, Catarina, Álgebra Linear e Geometria Analítica, Problemas e Exercícios, McGraw-Hill, 1997.
- António J. Monteiro, Isabel Teixeira Matos, Álgebra: Um primeiro Curso, Escolar Editora, 1995.