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# Linear Algebra 2017/2018

• 6 ECTS
• Taught in Portuguese
• Both continuous and final Assessment

### faculty body

• Introduction
• Syllabus
• relevant skillset

### Objectives

It is intended that the student is able to:
(i) operate with matrices (addition, multiplication and multiplication by a scalar);
(ii) define, calculate and use properties of the inverse and transpose of a matrix;
(iii) calculate determinants, using their properties and applications;
(iv) discuss and solve systems of linear equations (Gauss-Jordan elimination and Cramer Rule);
(v) define space and linear subspace;
(vi) identify the relationships between elements of a space / linear subspace: dependency / linear independence, generators and bases systems;
(vii) define and determine eigenvalues and vectors of a matrix;
(viii) operate with vectores/matrices and solve systems, with R software.

### Recommended Prerequisites

Knowledge acquired in Mathematics of Elementary and Secondary Education.

### Teaching Metodology

In practical classes are presented the basic concepts accompanied by the resolution of exercises and study of practical applications. Whenever possible will be done some demonstrations using the software R. Expository, demonstrative and interrogative to introduce the basics methods are used.
They are provided handouts and leaves practical exercises for all the contents.

### Body of Work

1. CALCULATION MATRIX
1.1 Definitions and examples;
1.2 Operations and properties;
1.3 Inverse and transposed matrix and its properties;
1.4 conditions of existence of the inverse matrix;
1.5 Obtaining the inverse matrix by condensation method.
2. SYSTEMS OF LINEAR EQUATIONS
2.1 Discussion and resolution of systems of linear equations.
2.2 Applications with software R.
3. Determinants
3.1 Definition and properties;
3.1 Laplace expansion;
3.4 Inverse of a matrix using the adjoint matrix.
3.5 Cramer's rule.
3.6 Applications with software R.
4. VALUES AND VECTORS OWN
4.1 Concepts and obtaining eigenvalues and eigenvectors.
4.2 Applications with software R.
5. SPACES VECTOR
5.1 Definition of space and linear subspace;
5.2 Linear combinations, linear dependence and independence;
5.3 generator set, base and dimension of a (sub) vector space;
5.4 vector subspace spanned by a set of vectors;

### Recommended Bibliography

- Anton, H. e Rorres, C. (2012). Álgebra Linear com aplicações (10ª edição). Bookman.
- Cabral, I., Perdigão, C. e Saiago, C. (2010). Álgebra Linear. Teoria, exercícios resolvidos e exercícios propostos com soluções (2ª edição). Escolar Editora.

### Complementary Bibliography

- Anton, H. e Rorres, C. (2010). Elementary Linear Algebra with applications (tenth edition). John Wiley
- Santana, A. e Queiró, J. (2010). Introdução à Álgebra Linear. Gradiva.

### Weekly Planning

1. Definition of matrix operations and properties
2. Definition of transposed matrix and inverse and properties
3. Matrix Equations
4. Elementary Transformations and feature an array. Practical appplications with software R.
5. Systems of linear equations. Gaussian elimination method
6. Resolution systems. Practical appplications with software R.
7. Discussion of systems and calculating the inverse of a matrix by Gauss-Jordan elimination
8. Determinants: definition, properties and calculation
9. Laplace expansion
10. Adjoint matrix and matrix inverse
11. Resolution systems by Cramer's rule. Practical appplications with software R.
12. Values and eigenvectors. Practical appplications with software R.
13. Definition of space / vector subspace, linear combinations and generator sets
14. Linear dependence and independence subspace spanned by a set of vectors, base and dimension of a vector space
15. Practical applications using R.

### Demonstration of the syllabus coherence with the curricular unit's objectives

The selected program content, meet consistently the learning objectives. For the purposes of (i) and (ii) the UC contributes directly point 1 of the program. The contents of 1, 2.3 and 4 contribute to create basic skills matrix calculation allowing achieve the objectives (i) to (iv). For the purposes set out in (v) and (vi) contributes directly point 5 of the program and for the purpose of (vii) point 4. The syllabus 4 and 5 are the basis the algebraic calculation reinforced by matrix calculation introduced in points 1, 2 and 3. For the purpose (viii)contributes directly points 1.6, 2.2, 3.6 and 4.2 of the program.
The contents of this unit allow the development of a reasoned logical reasoning scientifically and the acquisition of scientific knowledge of algebraic nature for use in the field to develop other courses.

### Demonstration of the teaching methodologies coherence with the curricular unit's objectives

The union between the theoretical exposition of the matter (expository method, demo, trial and interrogative), the participation of students, the presentation of examples and solving practical problems on the treated materials (guided practice and debate) enables students to familiarize themselves with concepts and mathematical methods that are the foundation of algebra and matrix calculus, thus being able to achieve the objectives of the course. The use of the R software will give students the opportunity to explore computationally some contents of this course.

relevant generic skillimproved?assessed?
Achieving practical application of theoretical knowledgeYesYes