# Elements of Analysis and Algebra I

## Objectives

To provide the basis for some working knowledge of mathematical techniques (in the Linear Algebra and Analysis domains).

## General characterization

##### Code

10707

##### Credits

6.0

##### Responsible teacher

Cláudio António Raínha Aires Fernandes

##### Hours

Weekly - 6

Total - 78

##### Teaching language

Português

### Prerequisites

Basic knowledge of Mathematics Secondary Education

### Bibliography

[1] Cabral, I. & Perdigão, C. & Saiago, C. - Álgebra Linear, Escolar Editora, 2008.

[2] Anton, H. & Rorres, C. - Elementary Linear Algebra, Applications version, 9th Edition, John Wiley & Sons, Inc., 2005.

[3] Lang, S. - A first course in Calculus. Springer-Verlag, 1986.

[4] Sá, A. & Louro, B. – Análise Matemática, Teoria e Exercícios (Departamento de Matemática da F.C.T-U.N.L).

### Teaching method

The lectures consist on the theoretical exposition of the syllabus, illustrated with examples.The practical classes consist on the resolution of exercises on all the contents.

The students have previously access to lectures-notes and to practical-notes consisting on two different lists of exercises and problems.

Any doubts are clarified during classes, in weekly scheduled sessions, or in extra sessions accorded directly between student and teacher.

The students must attend lectures and solving-problem classes, having a maximum of 3 unexcused absences during the semester.

There are three mid-term tests that can substitute the final exam in case of approval. Otherwise the student must pass the final exam.

### Evaluation method

Students must attend classes and take the 2 proposed tests. The grade will be given by the formula

NAC=0,5*T1+0,5*T2

If NAC < 9,5 or T2 < 7, student need to make new exam.

## Subject matter

**Part I - Linear Algebra**

1- Matrices: algebra of matrices, inverse of a matrix.

2- Systems of linear equations: resolution and discussion.

3- Determinants: definition and properties of determinant, inverse matrix calculation from the adjoint matrix, Cramer''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s rule.

4- Eigenvalues and eigenvectors: linear function, eigenvalues and eigenvectors of a matrix.

**Part II - Analysis**

1-Limits and continuity: limits and continuity of functions of one variable.

2-Differential calculus: derivative of a function, Taylor''''''''''''''''''''''''''''''''s theorem and applications of Taylor''''''''''''''''''''''''''''''''s formula.

3-Integral Calculus: antiderivatives, Riemann integral.

## Programs

Programs where the course is taught: