Mathematics II

Recommended Prerequisites

Mathematics 12. Knowledge of limits, derivatives and integral of functions studied in Mathematics I.

Teaching Methods

The classes are essentially of expository nature and should include examples that lead the students to understanding and applying the material being taught. The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving. Some tutorial support will be available to help the students with the proposed tasks outside the classroom.

Learning Outcomes

Develop the ability to calculate limits of sequences. Determining the nature and the summation of a series. Study the development of a function series. Use elementary operations with matrices. Discuss and solve linear systems and calculate determinants. Apply the concepts of linear dependence and independence in Rn and the notion of base and dimension of a subspace of Rn.

Application of the concept of inner product and the method of least squares.

Applications of eigenvalues and eigenvectors of a matrix.

Work Placement(s)

No

Syllabus

1. Sequences: properties; limit of a sequence; the squeeze theorem; calculating limits using the limit laws; indeterminate forms.

2. Series: general properties; positive-term series; some convergence tests; numerical approximation; absolute convergence.

3. Function series: absolute and uniform convergence; the Weierstrass test; power and Taylor series.

4. Matrices and systems of linear algebraic equations: operations with matrices); Gaussian elimination; LU decomposition; inverse of a matrix; the  Gauss-Jordan algorithm; determinants.

5. The vector space Rn: vector subspaces of Rn; linear independence; basis and dimension; rank of a matrix; linear transformation

6. Inner product in Rn: projection and orthogonality; the Gram-Schmidt algorithm; method of least squares.

7. Eigenvalues and eigenvectors: diagonalization of a matrix.

Head Lecturer(s)

Olga Maria da Silva Azenhas

Assessment Methods

Assessment
Exam (100%) or Midterm (0-100%) + Problem resolving report (0-25%).: 100.0%

Bibliography

- Cálculo, Volume II  - James Stewart

- Introdução à Álgebra linear - Ana Paula Santana  e João Queiró

- Princípios de análise Matemática Aplicada - Jaime Carvalho e Silva

- Álgebra Linear como Introdução a Matemática Aplicada - Luís T. Magalhães

- Análise Matemática Aplicada - Jaime Carvalho e Silva e Carlos Leal

- Linear Algebra  Pure and Applied First Course - Edgar Goodaire