Linear Algebra and Analytical Geometry

Code:

L.EM004

Acronym:

ALGA

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EM	285	Syllabus	1	-	6	65	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 linear combination of vectors, linear independence and subspace spanned by a set of vectors;
c) Define a basis and dimension of vector space; obtain the coordinates of a vector with respect to a given basis;
d) Define line and plane, properties and represent lines and planes;
e) Solve problems with lines and planes, such as distances, angles and relative positions;
f) Knowing basic matrix operations, properties and operations;
g) Define and calculate the rank of a matrix;
h) Define nonsingular matrix, properties of the inverse of a matrix and calculate the inverse of a matrix;
i) Define determinant of a matrix, properties and calculate it;
j) Analyse and solve linear systems of equations;
k) Define linear transformations, define and calculate kernel and algebraic operations;
l) Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations;
m) Define similar matrices and knowing properties;
n) 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
José Augusto Trigo Barbosa; Noções sobre Geometria Analítica e Análise Matemática, Efeitos Gráficos, 2018. ISBN: 978-989-99559-7-4
Cristina Faria M. Guedes; Apontamentos da disciplina (Aulas Teóricas), 2015
Anton, Howard; Elementary linear algebra. ISBN: 0-471-44902-4
Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3

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

Teaching methods and learning activities

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

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 (%)
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 75% of theoretical-practical classes, according to Article 6 of the Specific Rules for FEUP's Students Assessment;

Calculation formula of final grade

Any student can choosen to obtain approval by distributed evaluation or a global test revaluation which will cover all program of the course.
The distributed evaluation has three components that are tests written.
All assessment components are expressed on a scale of 0 to 20 (V-values). 

1. Distributed evaluation:

a)1st Test (P1) will cover the first part (half) of the program of the course.
b) 2nd Test (P2) will include the subject after the 1st test (the second part of the program of the course). In this test only are admited students that obtained a classification equal or greater than 6V in the 1st test (P1).
c) Global test with all program of the course, performed in simultaneous with P2.
d) Any student can choose to realize the global test.

The final classification (CF) will be calculated by :
CF=0,5*CP1 +0,5*CP2, if CP1≥6V AND CP2≥6V  (a minimum mark of 6V is required in P1 and P2)
or 

CF =CPG

in which
CP1: classification in the 1st test
CP2: classification in the 2nd test
CPG: classification in the global test
CF: classification final

To be approved, students have to achieve a final classification (CF) equal or greater than 10V.

2. Global test revaluation:

The global test revaluation includes all program of the course (whole program) and  can be realized by:

• Students who did not obtain approval in the distributed evaluation process.
• Students who have passed the end of the evaluation process distributed and intend to improve the classification obtained. In this case they should make their registration in Academic Services of FEUP, according to Article 11 of the Specific Rules for FEUP's Students Assessment.

Remarks:

i)Students with a final classification equal or greater than 18V will be subjected to an extra oral or written test.
ii) Classifications of tests from previous years are not considered.
iii) During the evaluation tests are not allowed to consult any type of mobile phones, calculators and microcomputer.

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.