Linear Algebra and Analytical Geometry

Code:

L.EGI002

Acronym:

ALGA

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Industrial Engineering and Management

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EGI	130	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

The curricular unit has two fundamental objectives: on the one hand, as it is a propaedeutic curricular unit, it has a didactic/scientific character, promoting the development of logical reasoning and methods of analysis; and, on the other hand, it aims to introduce and develop in theoretical terms a set of concepts that will be essential tools to support other curricular units of the course.
Fundamental concepts about Linear Algebra and Vector Algebra and Analytical Geometry are introduced, which are considered indispensable in the mathematical training of an Engineering student.

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) Use the matrix to represent a linear transformation and operate with linear transformations using matrix algebra;

n) Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations;
o) Define similar matrices and knowing properties;
p) 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

Howard Anton, Chris Rorres, Anton Kaul; Elementary Linear Algebra, Applications Version, 12th Edition, EMEA Edition, WILEY, 2019. ISBN: ISBN: 978-1-119-66614-1
José Augusto Trigo Barbosa; Noções sobre álgebra linear. ISBN: 978-972-752-142-5
José Augusto Trigo Barbosa; Noções sobre matrizes e sistemas de equações lineares. ISBN: 972-752-069-3 972-752-065-0
José Augusto Trigo Barbosa; Noções sobre Geometria Analítica e Análise Matemática, Efeitos Gráficos, 2018. ISBN: 978-989-99559-7-4

Complementary Bibliography

Carlos Silva Ribeiro, Luizete Reis, Sérgio da Silva Reis; Álgebra linear. ISBN: 972-8298-82-X
António Monteiro, Gonçalo Pinto ; colab. de Catarina Marques; Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
Gregório Luís, C. Silva Ribeiro ; prefácio de Bento J. F. Murteira; Álgebra linear. ISBN: 972-9241-05-8
Tom M. Apostol; Calculus. ISBN: 84-291-5001-3

Teaching methods and learning activities

Theoretical classes: detailed exposition of the program of the discipline illustrated by application examples.

Practical classes: application of the theoretical concepts in the resolution of several exercises that can be found in the proposed literature.

Exercises and other resources will be available in Moodle.

Software

Matlab

keywords

Physical sciences > Mathematics > Applied mathematics > Engineering mathematics
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	85,00
Frequência das aulas	77,00
Total:	162,00

Eligibility for exams

Admission to exam:
a) Attend to 75% of theoretical-practical classes, according to Article 6 of the Specific Rules for FEUP's Students Assessment;

Calculation formula of final grade

During the Distributed Assessment process, the student must take two written tests, each with the expected duration of 2 hours. Each test consists of a theoretical part, worth about 20% of its total mark, and a theoretical-practical part, worth the remaining mark. The final classification will be the average of the classifications obtained in the two tests carried out.

Students who, at the end of the distributed assessment process, obtained attendance but did not pass the course unit, may take a reassessment test, simply by appearing on the day and time set in the room(s) indicated in the meantime.
Students who, at the end of the distributed assessment process, have passed the curricular unit and intend to improve the classification obtained, must register with the Academic Services of FEUP (in appeal).
In both cases, the reassessment test is global and with an estimated duration of 2h +15min of tolerance.

During the assessment tests, it is not allowed to consult any type of form, cell phone, smartphone, smartwatch, calculator, microcomputer or similar.

Examinations or Special Assignments

Not applicable.

Special assessment (TE, DA, ...)

An exam at the special season, according to items 8.1, 8.3 c) and 8.4 c) in Article 8 of the Specific Rules for FEUP's Students Assessment.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to Article 11 of the Specific Rules for FEUP's Students Assessment.