Algebra

Code:

EC0002

Acronym:

ALGE

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=664
E-learning page:	http://moodle.up.pt/
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	234	Syllabus since 2006/2007	1	-	7	75	187

Teaching language

Portuguese

Objectives

JUSTIFICATION
Essentially, two reasons justify the existence of this unit: the need to develop a scientifically based logical reasoning, the capacity of reasoning and communication in scientific and technical approaches of the branches of civil engineering; the need for acquiring scientific knowledge of algebraic nature for use in the subjects that will be studied in the remaining semesters of the course.

GOALS
Acquisition of fundamental concepts of Linear Algebra and Matrices. Developing the capacity of formal algebraic calculus, formulation and resolution of explicit algebraic problems, including issues of Analytical Geometry.

 

Learning outcomes and competences

RESPONSIBILITIES AND OUTCOMES OF LEARNING
To define vector spaces, bases of spaces (finite or infinite dimensional) arrays, linear applications, quadratic forms, linear systems, eigenvalues and eigenvectors. To compute these entities in specific problems, solve problems that use them as a tool and apply these concepts and properties that are involved in operational aspects. To discuss the validity of solutions, to distinguish problems with one or more solutions. To formulate problems with algebraic components in mathematical terms. To draw conclusions from calculations performed on the basis of mathematical properties or known theories .

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

The student should have basic knowledge of trigonometry, calculus of roots of polynomials and factorization, real functions of one real variable, analytic geometry in the plane, systems of linear equations and logics.

Program

Vector spaces: definition and examples, subspaces, generated subspaces, linear dependence, bases and dimensions, sum of subspaces. Matrices: definition and examples, operations and properties, changes of basis, special classes of matrices. Linear transformations: definition and examples, kernel and image, injectivity and sobrejectivity, inverse transformation, projections, symmetries, vector space of linear transformations, matrix representation, inversions and rotations. Determinants. Systems of Linear Equations: homogeneous and inhomogeneous, discussion of systems. Brief review of polynomials: irreducible in R and C, factorization, calculation of roots. Eigenvalues and eigenvectors: invariant subspaces, calculation of proper elements, diagonalization of matrices, canonical forms (illustrated). Euclidean spaces: scalar product and norm in spaces of dimension n, projections, vector product, mixed product. Analytic geometry: the n-dimensional affine space, a brief review of intersections, parallelism and perpendicularity, distances and angles, relative positions. Quadratic forms. Introduction to algebraic surfaces.

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:
This curricular unit is within the group of curricular units in the scientific area of mathematics, mainly focusing on providing students a solid education in the concepts and calculation in linear algebra and matrix. The syllabus includes the definition of vector spaces, bases of spaces (finite or infinite dimension), arrays, linear applications, quadratic forms, linear systems, eigenvalues and eigenvectors. These issues are the basis of algebraic calculation and matrix, being presented in lectures. In practical classes, students are guided in solving problems with several examples of other curricular units related to the area of physics and mechanics.

Mandatory literature

Emília Giraldes, Vitor Hugo Fernandes, Maria Helena Santos; Curso de álgebra linear e geometria analítica. ISBN: 972-9241-73-2
Isabel Cabral, Cecíilia Perdigão, Carlos Saiago; Álgebra Linear, Escolar Editora, 2009. ISBN: 9789725922392

Complementary Bibliography

António Monteiro; Álgebra Linear e geometria analitica, McGraw-Hill. ISBN: 972-8298-66-8
Anton Rorres; Elementary Linear Algebra with Applicattions, John Wiley, 2005. ISBN: 0471449024
Anton Rorres; Álgebra Linear com aplicações, Bookman, 2000. ISBN: 85-7307-847-2
Elon Lages Lima ; Álgebra Linear e geometria analitica, Instituto de Matemática Pura e Aplicada, 1996. ISBN: 852440102-8
Sheldon Axler; Linear Algebra done right, Springer, 1997. ISBN: 0-387-98258-2

Comments from the literature

Notes edited by the chair of the UC for this subject that are available through the web pages of the UC in SIGARRA system and in MOODLE. Other documentation available in the Moodle pages.

Teaching methods and learning activities

Essentially lecturing subjects, coordinating basic theoretical knowledge necessary to develop subsequent subjects in the syllabus. At this level intuitive understanding of the concepts is encouraged as well as computational ability. The materials are presented in a clear and objective form in the classroom, making frequent use of examples taken from other disciplines such as Physics, Mechanics I, Mechanics and Theory of Structures II. In practical classes, students are guided in solving problems as applications of materials taught in theoretical classes.

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:
The focus is on coordination between the fundamental theoretical knowledge and the developments required in the following curricular units, being promoted the intuitive understanding of the concepts and calculation capabilities. It is intended to apply these subjects in specific problems that use them as a tool and apply these concepts and properties that are involved in operational aspects. To discuss the validity of solutions, to distinguish problems with one or more solutions. To formulate problems with algebraic components in mathematical terms. To draw conclusions from calculations performed on the basis of mathematical properties or known theories.

keywords

Physical sciences > Mathematics > Algebra

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Teste	50,00
Trabalho escrito	50,00
Total:	100,00

Eligibility for exams

Achieving final classification requires compliance with attendance at the course unit, according to the MIEC assessment rules. It is considered that students meet the attendance requirements if, having been regularly enrolled, the number of absences of 25% for each of the classes’ types is not exceeded.

Calculation formula of final grade

Evaluation consists on 2 written assessments (see 1st year calendar) and 2 questions answered during lecture classes, in the group for which they registered. All assessments are compulsory. Not being present in one assessment  implies a grade of "0" in the correspondent grading.

CF= final mark
CT1= first writtent test' mark
CT2= second writtent test' mark
CQ1= question 1' mark (from lecture class)
CQ2= question 2' mark (from lecture class)

Result of the evaluation at the end of the semester:

CF = 0,450*CT1+0,450*CT2+0,05*(CQ1+CQ2)

Students that were admited for evaluation at the end of the semester but did not succeed are admited and can regiter for an exam of appeal and must choose to repeat one of the 3 options:

(1) part 1 of the syllabus, replacing their mark CT1 in the previous formula;

(2) part 2 of the syllabus obatining a new  mark CE2 resulting in a final mark given by the formula

CF = 0,450*CT1+0,550*CE2;

(3) all syllabus, resulting in a single final mark CF.

The dates of occurence of the 2 assessments in lecture classes, Q1 and Q2, will be announced to the respective groups of students during the week before that, followed by an email announcement.


NOTE 1: All students enrolled in the course are classified according to this method.
NOTE 2: Students who have attended the course in the previous academic year are not required to attend this year and can request that their mark of the component (CQ1+CQ2) be kept, but not for the other parts of the assessment. This will require the student to inform the administration (smf@fe.up.pt) before the deadline that will be anounced.
NOTE 3: Students who have attended the course in academic years 2011/2012 and before can not keep the classification of parts of the assessment which then held.
NOTE 4: For the exam of appeal, students must announce their choice of assessment when registering to smf@fe.up.pt with a deadline to be announced.

 

Special assessment (TE, DA, ...)

Evaluation of students that register to special exames is made by means of a single written exam which replaces all other marks.

Classification improvement

In accordance to Article 10th of the General Standards of Evaluation (Normas Gerais de Avaliação),  students who were approved at the curricular unit and wish to improve their grade can do so by participating in the correpondant appeal exam. They will follow the same grading rules as the students that did not succeed previously.

Observations

PRIOR KNOWLEDGE: The student should have basic knowledge of trigonometry, calculus of roots of polynomials and factorization, real functions of one real variable, analytic geometry in the plane, systems of linear equations and logics.

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Working time estimated out of classes: 4 hours