Matrices
Determinants
Vector spaces
Linear maps

The objective of this course is for students to master, illustrate, manipulate, and apply the basic concepts and techniques of linear algebra, namely:

• matrices—operating with matrices, reducing matrices, inverting (invertible) matrices

• systems of linear equations—discussing and solving systems of linear equations

• determinants—basic properties, invertibility criteria, determining the adjoint matrix, using determinant theory to discuss and solve systems of linear equations: Cramer’s rule

• finite-dimensional vector spaces—definitions and properties; concepts of vector subspace, linear dependence, linear independence, basis

• linear maps—definitions and properties; characterization of linear maps; concepts of kernel and image; matrix of a linear map; change-of-basis matrix

• eigenvalues and eigenvectors of a square matrix and an endomorphism—concept of an eigenspace, concepts of algebraic multiplicity and geometric multiplicity, diagonalizable matrices, characteristic equation, characteristic polynomial

By the end of this unit, students will be able to

• Recognize what matrices are and understand their basic properties. Perform operations on matrices, reduce matrices, and invert matrices (if invertible).

• Discuss and solve systems of linear equations, particularly using matrix methods.

• Calculate the determinant of a matrix, identify the basic properties of determinants, apply the invertibility criterion, determine the adjugate matrix, and use determinant theory to discuss and solve systems of linear equations.

• Recognize what vector spaces are and their properties. Acquire the concepts of vector subspaces, the (direct) sum and intersection of subspaces, linear dependence, linear independence, and bases. Be able to use matrix methods in the study of these concepts.

• Recognize what linear mappings are and their properties. Acquire the concepts of the kernel and image of a linear map. Be able to determine the matrix of a linear map and basis change matrices. Recognize what eigenvalues and eigenvectors of a square matrix and an endomorphism are. 

Acquire the concepts of eigenspace, algebraic multiplicity and geometric multiplicity, characteristic polynomial, and characteristic equation. Be able to determine whether a matrix is diagonalizable and, if so, obtain the similarity matrix.

Matrices and Linear Equations;
Determinants and Linear Equations;
Vector Spaces, Eigenvalues and Eigenvectors;
Linear Transformations and Matrices.

CABRAL, Isabel; PERDIGÃO, Cecília; SAIAGO, Carlos, Álgebra Linear, Escolar Editora, 2021,
ISBN 978-972-592-573-7
26 vídeos disponíveis em
https://www.math.tecnico.ulisboa.pt/~fcosta/videos_algebra_linear.html
 
 

In this course, the learning experience is designed to give students greater flexibility in managing their study time, without sacrificing support and interaction with instructors, tutors, and classmates. Activities take place primarily online and asynchronously through the Universidade Aberta learning platform, combining periods of independent study with opportunities to participate in forums, discussions, assignments, and other activities that apply knowledge in practice.

Students take an active role in their learning process and are encouraged to reflect, share experiences, solve problems, and develop skills relevant to their academic and professional paths. Throughout the course, they will have access to a variety of resources and guidance from the teaching staff, who provide regular support and feedback.

The teaching methods adopted emphasize autonomy, collaboration, inclusion, and flexibility, allowing students to balance their studies with their personal, family, and professional responsibilities.

To learn more about the Open University’s Pedagogical Model, visit: https://portal.uab.pt/modelo-de-ensino/

Assessment in this course is designed to support student learning throughout the semester, valuing not only the results achieved but also the process of knowledge construction. Assessment activities aim to promote critical reflection, the practical application of course content, active participation, and the development of skills relevant to students’ academic and professional paths.

Where applicable, assessment may incorporate different methods and tools, such as individual and collaborative activities, written assignments, projects, presentations, participation in forums, or exams, as defined in the Course Syllabus.

The assessment criteria, planned activities, and their respective weightings will be presented at the beginning of the course, ensuring transparency and allowing you to organize your learning journey in an informed manner.

To learn more about the principles and guidelines for assessment in the Open University’s Pedagogical Model, visit: https://portal.uab.pt/modelo-de-ensino/