Algebra

Code:

L.EC001

Acronym:

ALG

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 1S

Active?	Yes
Web Page:	http://https://moodle.up.pt/course/view.php?id=1441
Responsible unit:	Department of Civil and Georesources Engineering
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	201	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

Operate with matrices and calculate determinants. Solve linear systems.
Define vector spaces, bases of spaces (of finite dimension), linear transformations, values ​​and eigenvectors.
Define Euclidean space and master the main concepts of analytical geometry.
Determine these entities in concrete problems and solve problems that involve them and apply these concepts and the properties that involve their operability.
Discuss validity of solutions, distinguish problems with one or more solutions.
Formulate problems with algebraic components in mathematical terms.
Draw conclusions from the calculations carried out or
by applying known properties or theories.

Working method

Presencial

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

Program

1. Matrices

1.1 Matrix operations
   1.1.1 Addition; multiplication by a scalar; matrix multiplication.
   1.1.2 Transposed Matrix, conjugate and transconjugate of a matrix.
1.2 Special matrices
   1.2.1 Rectangular and square
   1.2.2 Identity matrix; diagonal; triangular; symmetrical; unitary and orthogonal.
1.3 Inverse matrix
   1.3.1 Definition 
   1.3.2 Properties 
1.4 Echelon and reduced echelon form of a matrix.
1.5 Resolution of matricial equations

2. Determinants

2.1 Definition and properties
2.2 Calculation of determinants
   2.2.1 Matrix condensation method
   2.2.2 Laplace's theorem
2.3 Minors, complementary minors and algebraic complements
2.4 Definition of an adjoint matrix
2.5 Calculation of the inverse matrix using determinants and condensation
2.6 Definition of rank matrix

3. Systems of Linear Equations.

3.1 Matricial form of a linear system
3.2 Classification of a linear system
3.3 Solving systems using the Gauss Method and Gauss-Jordan Method
3.4 Cramer's system and Cramer's rule
3.5 Systems discussion

4. Vector Spaces

4.1 Definition and properties
4.2 Subspaces of a vector space
4.3 Linear dependence and independence
4.4 Generator systems
4.5 Concepts of bases and dimension
4.6 Matrix of coordinate changes between bases

5. Linear Applications

5.1 Definition and Properties
5.2 Examples of linear applications of R^2 in R^2 and of R^3 in R^3, such as: Rotations; symmetries; contractions; dilations; etc. 
5.3 Core and image of a linear application
5.4 Matrix of a linear application
5.5 Surjective linear application
5.6 Injective linear application
5.7 Invertible linear application and composition of linear applications
5.8 Characteristic of a linear transformation
5.9 Some relevant theorems about linear applications

6. Vectors and Eigenvalues

6.1 Definition of eigenvector and eigenvalue
6.2 Definition of characteristic equation and characteristic polynomial
6.3 Proper subspace
6.4 Definition of diagonalizable matrix
6.5 Some relevant theorems about matrix diagonalization.

7. Euclidean spaces

7.1 Definition of real Euclidean space
7.2 Norm of a vector and Cauchy Schwarz inequality
7.3 Orthogonal and orthonormal set
7.4 Orthonormed basis in finite-dimensional Euclidean spaces
7.5 Orthogonal projection of a vector into a subspace of a finite-dimensional Euclidean space
7.6 Gram-Schmidt process
7.7 Definition and properties of inner and outer product in R^3

8. Analytical Geometry

8.1 Lines and planes in three-dimensional space
8.2 Non-metric problems: Incidence and parallelism
8.3 Metric problems. Distances and angles

Mandatory literature

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
Rorres, Chris e Anton, Howard; Álgebra Linear com Aplicações, Bookman. ISBN: 9788540701694

Complementary Bibliography

António Monteiro; Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
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

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

Software

matlab

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	103,50
Frequência das aulas	58,50
Total:	162,00

Eligibility for exams

Calculation formula of final grade

Evaluation formula:

The assessment will be distributed without a final exam and consists of 2 written tests (mandatory) and an assessment component (optional) to be carried out in theoretical classes on the Moodle platform.

CT1 = classification in the 1st test
CT2 = classification in the 2nd test
CQ = classification in theoretical class questions

Classification result (CF)  will be
CF = max{ 0.5*CT1+0.5*CT2 ; 0.4*CT1+0.4*CT2+0.2*CQ}.

A minimum score of 6 points is required in each test, CT1 and CT2.

Failure to obtain this grade forces the student to take the appeal exam.

If the student has missed or has not obtained the minimum score in one of the tests, in the appeal exam  can choose to take just that test or take a global test.

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

The 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

During any assessment, possession of any electronic device (e.g., cell phones, tablets, headphones, smartwatches, etc.) is strictly prohibited, with the exception of those expressly indicated by the teaching staff (e.g., calculators).