Linear Algebra and Analytical Geometry

Code:

EM0005

Acronym:

ALGA

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEM	204	Syllabus since 2006/2007	1	-	6	78	162

Teaching language

Portuguese

Objectives

This course unit aims 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.

Learning outcomes and competences

At the end of this, students should be capable of:

a) Knowing vector algebraic operations, their properties and how to apply them;

b) Define vector space, vector subspace and Euclidian subspace;

c) Define linear combination of vectors, linear independence and subspace spanned by a set of vectors;

d) Define a basis and dimension of vector space; obtain the coordinates of a vector with respect to a given basis;

e) Define line and plane, properties and represent lines and planes;

f) Solve problems with lines and planes, such as distances, angles and relative positions;

g) Knowing basic matrix operations, properties and operations;

h) Define and calculate the rank of a matrix;

i) Define nonsingular matrix, properties of the inverse of a matrix and calculate the inverse of a matrix;

j) Define determinant of a matrix, properties and calculate it;

k) Analyse and solve linear systems of equations;

l) Define linear transformations, define and calculate kernel and algebraic operations;

m) Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations;

n) Define similar matrices and knowing properties;

o) Calculate eigenvalues and eigenvectors of linear transformations and knowing properties.

Working method

Presencial

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

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

Program

The vector space of n-uples of real numbers. The dot product. Norm of a vector. Orthogonality and angle between two vectors. The linear span of a finite set of vectors. Linear independence and dependence. Bases and dimension in vector spaces.

The cross product. The scalar triple product. 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.

Matrices; algebraic operations. Transpose of a matrix. Square matrices: definitions and special properties. Rank of a matrix. Inverse of a square matrix.

Determinants; definition and properties. Minors and cofactors. The Laplace theorem. Computation of determinants. The determinant of the inverse of a non-singular matrix. Evaluation of the rank of a matrix with determinants.

Systems of linear equations; Gauss and Gauss-Jordan methods. Cramer´s rule.

Linear Spaces; definition and properties. Subspaces of a linear space. Dependent and independent sets in a linear space. Bases and dimension.

Inner products. Euclidean spaces. Norms and orthogonality.

Linear transformations; 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.

Eigenvalues and eigenvectors; definition and properties. Linear transformations with similar diagonal matrix representations.

Mandatory literature

José Augusto Trigo Barbosa; Noções sobre álgebra linear. ISBN: 978-972-752-142-5
Barbosa José Augusto Trigo; Noções sobre matrizes e sistemas de equações lineares. ISBN: 972-752-069-3 972-752-065-0
Anton, Howard; Elementary linear algebra. ISBN: 0-471-44902-4
Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3
Cristina Faria M. Guedes; Apontamentos da disciplina (Aulas Teóricas), 2015
José Augusto Trigo Barbosa; apontamentos sobre "Produto Vetorial e Produto Misto" e "Geometria Analítica"

Complementary Bibliography

Ribeiro, Carlos Alberto Silva; Álgebra linear. ISBN: 972-8298-82-X
Monteiro, António; Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
Luís, Gregório; Álgebra linear. ISBN: 972-9241-05-8
J.A. Trigo Barbosa, J.M.A. César de Sá e A.J. Mendes Ferreira; ALGA - Exercícios Práticos

Teaching methods and learning activities

Theoretical classes: detailed exposition of the program of the discipline illustrated by application examples. Theoretical-practical classes: application of the theoretical concepts in the resolution of several exercises that can be found in the proposed literature.

keywords

Physical sciences > Mathematics > Algebra
Physical sciences > Mathematics > Applied mathematics > Engineering mathematics
Physical sciences > Mathematics > Geometry

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	78,00
Frequência das aulas	84,00
Total:	162,00

Eligibility for exams

Admission to exam:

a) Attend to Article 7 of the General Evaluation Rules of FEUP;

b) Attend to at least one of the two tests with minimal classification of 5,5 out of 20.

Calculation formula of final grade

The student must attend to two written exams, with the duration of 2 hours each. Each exam comprises two different parts: a theoretical part, which worth 20% of the final mark and a theoretical-practical part which worth 80% of the final mark. Final mark will be based on the average grade of the two exams.

Exams are scheduled for these dates: 1st written exam: November 2016; 2nd written exam: January 2017.

According to Article 4 of General Evaluation Rules of FEUP, a student in order to pass the course must earn a grade of five point five out of twenty or better in each of the exams.

At the end of the semester students will be able to attend a new exam in order to improve their final grade. This exam may either only test a part of the program, or the whole program. Only students who got admission to exam can attend this exam.

Students who have passed the end of the evaluation process
Distributed and intend to carry out, to improve the classification obtained, one of the revaluation evidence should make their registration in Academic Services of FEUP.

Date of the exam: February 2017.

Examinations or Special Assignments

Not applicable.

Special assessment (TE, DA, ...)

According to items 1b), 1c), 2 to 4 in Article 9 of General Evaluation Rules of FEUP.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to Article 10 of General Evaluation Rules of FEUP.