Algebra

Code:

L.EC001

Acronym:

ALG

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

Active?	Yes
Web Page:	http://https://moodle.up.pt/course/view.php?id=1441
Responsible unit:	Mathematics Division
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	279	Syllabus	1	-	6	58,5	162

Teaching language

Portuguese

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.

Learning outcomes and competences

To define vector spaces, bases of spaces (finite or infinite dimensional) matrices, 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

Program

-Complements on algebraic structures.

Matrices over fields. Matrix operations: addition and multiplication by a scalar.
Multiplication of matrices. Transpose and conjugate matrix. Transconjugate of a matrix.  Special matrices: symmetric, rectangular and square of order n. Triangular, Diagonal and identity matrices. Normal, hermitian and hemi-hermitian matrices. Unitary and orthogonal matrices. Invertible matrix and its properties. Integer power of a square matrix.

-Determinant.

Definition and properties. Calculation of the determinant of a matrix using the Gauss method of condensation. Laplace theorem for the calculation of a determinant. Minors of a matrix and cofactor of an element of a matrix. Adjugate matrix. Calculation of the inverse of a matrix using determinants. 

-Linear systems of equations and their classification. Gauss and Gauss-Jordan methods. Echelon and reduced echelon forms of a matrix. Cramer’s rule.

-Vector Spaces.

Definition and properties. Subspace of a vector space, definition. Linear dependence and independence of vectors. System of generators of a subspace. Concept of basis and dimension. Change of basis matrices

-Linear transformations.

Definition and properties. Examples of linear transformations, rotations, symmetries, contractions and expansions. Definition of a Monomorphism, endomorphism and isomorphism. Kernel and Image subspaces of a linear transformation. Injective and surjective linear transformations. Definition of Invertible linear transformation. Composition of linear transformations. Rank of a linear transformation. Other relevant theorems regarding linear transformations. Classification of a linear system.

 

-Eigenvalues and eigenvectors.

Definition of eigenvalue and eigenvector of an endomorphism of a finite dimensional vector space over a field. Characteristic equation and characteristic polynomial of a linear transformation over a field. Caley-Hamilton theorem. Definition of diagonalizable endomorphism. Relevant theorems about endomorphism diagonalization.

- Euclidean spaces
Real and complex (unitary) Euclidean spaces. Norm of a vector. Cauchy Schwarz inequality. Orthogonal and orthonormal set. Orthonormed basis in euclidean spaces of finite dimension. Orthogonal projection and orthonormed basis on finite dimensional Euclidean spaces. Orthogonal projection of a vector over a subspace in a finite dimensional Euclidean space. The Gramm-Schmidt orthogonalization process. Dot product and vectorial product in 3 dimensions and its properties.

 -Analytic geometry

Affine spaces, an introduction. Lines and planes in in 3 dimensions. Non-metric problems: incidence and paralelism. Metric problems: distances and angles. Quadric surfaces.

Mandatory literature

Luís Almeida Vieira, Rui Soares Gonçalves; Álgebra linear, Volume 1- Cálculo matricial, sistemas lineares e espaços vetoriais, Efeitos Gráficos, 2021. ISBN: 978-989-53004-5-7
Luís Almeida Vieira, Rui Soares Gonçalves; Álgebra Linear - Volume 2 - Aplicações Lineares, determinantes e espaços euclidianos, Efeitos Gráficos, 2021. ISBN: 978-989-53004-6-4
Emília Giraldes; Curso de álgebra linear e geometria analítica. ISBN: 972-8298-02-1
Isabel Cabral, Cecíilia Perdigão, Carlos Saiago; Álgebra Linear, Escolar Editora, 2009. ISBN: 9789725922392

Complementary Bibliography

Rorres, Chris e Anton, Howard; Álgebra Linear com Aplicações, Bookman. ISBN: 9788540701694
António Monteiro; Álgebra linear e geometria analítica. ISBN: 972-8298-66-8

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.

Type of evaluation: two tests without final exam

Terms of frequency: 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% of presencial classroom lessons.

Software

maxima

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	145,00
Frequência das aulas	55,00
Total:	200,00

Eligibility for exams

Terms of frequency: 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% of presencial classroom lessons.

Calculation formula of final grade

Formula Evaluation: Evaluation consists on 2 written assessments (see 1st year calendar) plus a weekly quizz taking place at the first 4 weeks theoric lessons on moodle. The quizzes are optional.
The written asessments 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 written test' and CQ - quizzes mark. Formula for the final grade:
CF = max{ 0,4*CT1+0,6*CT2 ; 0,35*CT1+0,55*CT2+0,10*CQ}.
It is required a minimum mark of 6 in each component CT1 and CT2.
The appeal exam is mandatory for the students with a component grade lower than 6.  
 Students that were admited for assessment at the end of the semester but did not succeed are admited and automatically registered  for the exam of appeal.
NOTE 1: All students enrolled in the course are classified according to this method, including examinations for special regulation students.
NOTE 2: Students who attended the course unit in previous academic years are exempt from obtaining frequency in this academic year.
NOTE 3: Students who, at the end of a full set of examination, have only been evaluated at one component will receive the mention RFE.

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 correspondant 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.

...........................................................

Working time estimated out of classes: 4 hours