Algebra

Code:

EMM0003

Acronym:

ALGE

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Web Page:	http://consultoriodigitalmatematica.pt.vu/
Responsible unit:	Department of Metallurgical and Materials Engineering
Course/CS Responsible:	Master in Metallurgical and Materials Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LCEEMG	26	Plano de estudos oficial a partir de 2008/09	1	-	6	56	162
MIEA	50	Syllabus since 2006/07	1	-	6	56	162
MIEMM	33	Syllabus since 2006/2007	1	-	6	56	162

Teaching language

Portuguese

Objectives

The promotion of logical reasoning, methods of analysis and the theoretical development of mathematical concepts is fundamental to support the study of the majority of course units along this programme of studies.

This course unit aims to introduce the basic fundamental concepts of Linear Algebra, Vector Algebra and Analytic Geometry.

The student must be acquainted with basic notions on trigonometry, real functions, plane analytic geometry, systems of linear equations and logic operations.

The scientific component is100%.

Learning outcomes and competences

At the end of this, students should be capable of: 
- Knowing basic matrix operations, properties and operations; 
- Define and calculate the rank of a matrix; Define nonsingular matrix, properties of the inverse of a matrix and calculate the inverse of a matrix; 
- Define determinant of a matrix, properties and calculate it; 
- Analyse and solve linear systems of equations; 
- Define linear transformations, define and calculate kernel and algebraic operations; 
- Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations; 
- Calculate eigenvalues and eigenvectors of linear transformations and knowing properties. 
- Knowing vector algebraic operations, their properties and how to apply them; 
- Define vector space, vector subspace and Euclidian subspace; 
- Define linear combination of vectors, linear independence and subspace spanned by a set of vectors; 
- Define a basis and dimension of vector space; obtain the coordinates of a vector with respect to a given basis; 
- Define line and plane, properties and represent lines and planes; 
- Solve problems with lines and planes, such as distances, angles and relative positions.

Working method

Presencial

Program

1. Determinants - Definition and properties. Minors and cofactors. The Laplace theorem. Computation of determinants.
2. Matrices - Algebraic operations. Transpose of a matrix. Square matrices: definitions and special properties. Rank of a matrix. Inverse of a square matrix. The determinant of the inverse of a non-singular matrix. Evaluation of the rank of a matrix with determinants.
3. Systems of Linear Equations. Cramer´s rule. Gauss and Gauss-Jordan methods. Method of determinant.
4. Linear Spaces - Definition and properties. Subspaces of a linear space. Dependent and independent sets in a linear space. Bases and dimension. Inner products.
5. Linear Transformations and Matrices - Definition. Null space and range. Nullity and rank. Algebraic operations. Inverses. One-to-one linear transformations. Matrix representation of linear transformations. Matrices representing the same linear transformation. Similar matrices.
6. Eigenvalues and Eigenvectors - Definition and properties. Linear transformations with similar diagonal matrix representations.
7. Vector Algebra - The vector space of n-uples of real numbers. The dot product. Norm of a vector. Orthogonality and angle between two vectors. Applications of Vector Algebra to Analytic Geometry - Lines in n-space. Properties of straight lines. Lines and vector valued functions. Linear Cartesian equations for straight lines. Planes in n-space. Properties of planes. Normal vectors to planes. Planes and vector valued functions. Linear Cartesian equations for planes. Geometric applications to three-dimensional space. Norms and orthogonality. The cross product. The scalar triple product.

Mandatory literature

A bibliografia de referência básica e obrigatória é disponibilizada pelo docente nos conteúdos da unidade curricular
Luís, Gregório; Álgebra linear
Steinbruch; Álgebra Linear e Geometria Analítica, McGraw-Hill

Complementary Bibliography

Frank Ayres; Álgebra Moderna, McGraw-Hill
Lipschutz, Seymour; Álgebra linear. ISBN: 85-3460197-6
ULL; Álgebra linear e geometria analítica
Agudo, F. R. Dias; Introdução à álgebra linear e geometria analítica. ISBN: 972-592-050-3
J. Sebastião e Silva; Compêndio de Matemática (vols.1,2 e 3), Edição Gep, Ministério da Educação, Lisboa
J. Santos Guerreiro; Curso de Matemáticas Gerais, Livraria Escolar Editora, Lisboa
Lipschutz, Seymour; Álgebra linear. ISBN: 85-3460197-6
Apostol, Tom M.; Calculus
Frank Ayres; Álgebra Moderna, McGraw-Hill

Teaching methods and learning activities

In class concepts are presented and important results associated with an emphasis on geometric interpretations and practical applications. In order to clarify the definitions and theorems presented, several exercises are solved and illustrative applications are presented. The aim is to, whenever possible, the participation of students, not only in solving the exercises, but also in introducing new concepts. It remains to enhance the resolution of individual exercises and the guidance should be in the study of discipline and clarify questions that may arise in proposal exercises.

keywords

Physical sciences > Mathematics > Algebra
Physical sciences > Mathematics > Geometry

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	112,00
Frequência das aulas	50,00
Total:	162,00

Eligibility for exams

Gets frequency to this UC, in the current school year, every student that:
- Regularly registered in UC and does not exceed the maximum number of absences.

All students duly registered in UC, can perform the tests and examinations that are proposed and used for evaluation.

Regarding the assessment there are four different times, and they were the following:
1) First Test (T1) (Item 1, 2 and 3 of the program);
2) Second Test (T2) (4, 5, 6 and 7 of the program);
Note: Only students with grade greater than or equal to 6, in the first test (T1) can go to T2.
4) Exam resource (E) (all matter) - to be marked.

Remarks:
a) classifications of tests from previous years are not considered.
b) if a student misses one test, the rating assigned to this test, to calculate the final average, is zero values.

Calculation formula of final grade

For the final grade of the UC student must have or be exempt from such frequency. In these conditions, any student can choose to get approved by tests or final exam resource (E). If a student does not obtain approval for tests, he can still do the exam resource.

The final grade of UC is (scale 0-20):
- In the case of the arithmetic average ot T1 and T2, be greater than or equal to 9.5, the final grade will be the arithmetic average;
- The grade of the exam recourse (E).

Special assessment (TE, DA, ...)

Students who are under special statutes or having the or who have had previous years frequency, are exempt frequency. The approval can be obtained by performing the tests (T1 and T2) or a exam resource (E), the final classification is done according to the previous point.

Classification improvement

Students who wish to undertake improvement of classification shall be subject to an exam.

Observations

Language of instruction: Portuguese