Algebra

Code:

EC0002

Acronym:

ALGE

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2010/2011 - 1S

Active?	Yes
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	265	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.

OBJECTIVES
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.

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 .

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.

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

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.

keywords

Physical sciences > Mathematics > Algebra

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	70,00
Test	Exame	4,00
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Individual study	Estudo autónomo	66
	Total:	66,00

Eligibility for exams

According to General Evaluation Rules of FEUP

Calculation formula of final grade

The assessment will consist of two written tests corresponding to two periods.
m1 - the result of an evaluation of time (rounded to one decimal)
m2 - the result of 2. periods of assessment (rounded to one decimal)

Students can get the first test or the second test or a global test, if 0,5*m1+0,5*m2 < 9,5. To do so they will have to send an email to smf@fe.up.pt stating their option, eight days prior to the recovery test.

Whereas
nf - final mark (rounded to the units)
Mg1-note of the recovery of the first moment of evaluation (rounded to one decimal)
Mg2-note the recovery of the second moment of assessment (rounded to one decimal)
For students who do not get the recovery test the note is nf = 0.5 * m1 * m2 +0.5.
For those that recover the first component the final grade is nf = 0.5 * max (m1, Mg1) +0.5*m2.
For those who recover the second component the final score is nf = 0.5 * m1 +0.5 * max (m2, Mg2).
For those who recover the global test, nf = 0.5 * max (m1, Mg1)+0.5 * max (m2, Mg2).

Special assessment (TE, DA, ...)

SPECIAL RULES FOR MOBILITY STUDENTS:
Proficiency in Portuguese and/or English; Previous attendance of introductory graduate courses in the scientific field addressed in this module;

Classification improvement

The assessment will consist of a global writing test

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