Linear Algebra

Base Knowledge

Knowledge of the subject of Mathematics in secondary education.

Teaching Methodologies

The curricular unit has theoretical and theoretical-practical classes. Theoretical classes take place in an essentially expository way, approaching the themes foreseen in the program, prevailing a strong interaction between the concepts and their concrete application. The theoretical-practical classes will be aimed at solving problems and practical cases under the guidance of the teacher. The exercises will be performed individually or in small groups. The teaching of the curricular unit is complemented by student service periods.

Learning Results

The teaching of Mathematics in general should facilitate mathematical communication, reflective thinking, the application of mathematical techniques to problem solving, critical analysis of the results obtained, and finally, interdisciplinarity. One of the teaching objectives of the 1st year Linear Algebra discipline is to provide the basic foundations of mathematical methods, usually applied in the areas of Engineering, used by the various disciplines of the Degree in Eletromechanical Engineering.

It is intended that students develop abilities (skills) of algebraic manipulation and independent and analytical reasoning and the ability to apply mathematical concepts in solving practical problems.

Program

1. Complex Numbers (revision)

 2. Systems of Linear Equations and Matrices

  • Application of the study of linear equations systems to solving problems linked to Eletromechanical Engineering.
  • Matrix concept.
  • Special matrices (line matrix, column matrix, triangular matrix, diagonal matrix, scalar matrix, identity matrix, transposed matrix and conjugate matrix).
  • Operations with matrices and some properties.
  • Invertible matrices.
  • Matrix notation of  linear equations system.
  • Characteristic of a matrix. Characteristic calculation using the Gaussian elimination method.
  • Resolution of linear systems using the Gaussian elimination method.
  • Systems with parameters. Discussion of systems with parameters.
  • Inverse matrix. Calculation of the inverse matrix by the Gauss-Jordan method.

3. Vector spaces

  • Introduction of the concept of vector space from the set of solutions of a system of linear equations.
  • Vector spaces: definition and elementary properties.
  • The real vector space R^n.
  • Vector subspaces. Classification and characterization of subspaces.
  • Construction of subspaces: linear combination; linear expansion; subspace generated by a set of vectors; finite-dimensional space; intersection, sum and assemblage of subspace.
  • Dependency and linear independence.
  • Base of a vector space. Dimension of a vector space. Change of base.
  • Line spacing; column space and matrix null space.

4. Determinants

  • Introduction of determinant.
  • Second order determinant: definition and properties.
  • Third order determinant: definition using first-line development; development along one of the lines – “matrix of signals”. Sarrus Rule.
  • Determinant of order n; definition.
  • Minors, complementary minors, and algebraic complements. Laplace’s theorem and its generalization. Characteristics of a matrix and order of minors.
  • Calculation of the determinant of a matrix transforming it into a triangular matrix using the Gaussian elimination method.
  • Applications of determinants: adjunct matrix and inverse matrix; systems of linear equations. Cramer’s Rule.

5. Eigenvalues and eigenvectors

  • Introduction of the concept of eigenvalue and eigenvector of a linear application.
  • Definition of eigenvalue and eigenvector. Finding eigenvalues and eigenvectors.
  • Diagonal matrix representation of a linear application. Diagonalizing matrix and diagonalizing matrix concept.
  • Characteristic polynomial and characteristic equation of a matrix. Cayley-Hamilton theorem and some applications.

Curricular Unit Teachers

Internship(s)

NAO

Bibliography

Recommended:

Caridade, C.M.R., (2021). Apontamentos de Álgebra Linear e Geometria Analítica, DFM, ISEC.

Caridade, C.M.R. (2021, 27 setembro). e-MAIO (Módulos de Aprendizagem Interativa online). https://dfmoodle.isec.pt/

Caridade, C.M.R. (2021, 27 setembro). MOODLE ISEC – Álgebra Linear. https://dfmoodle.isec.pt/

Marcos, M.G., Oliveira, M.J.G.P, Barreiras, A.M.S., (2017). Álgebra linear e geometria analítica. Faro: Sílabas & Desafios. ISBN 9789898842152. Cota: 3-1-141 (ISEC).

Complementary:

Dias Agudo, F.R., (1996). Introdução à Álgebra Linear e Geometria Analítica, Escolar Editora, Lisboa. ISBN: 9789725920503. Cota: 3-1-130 (ISEC).

Ferreira, M.A, (2016). Álgebra Linear – Exercícios – Livro 1: Matrizes e determinantes, Edições Silabo. ISBN:9789726188506. Cota: 3-1-109 (ISEC).

Monteiro, A., (2010). Matrizes, Coleção Dashofer, Learning & Higher Education. ISBN 978-9896420833.

Monteiro, A., (2010). Álgebra Linear – Espaços vetoriais e transformações lineares, Coleção Dashofer, Learning & Higher Education. ISBN: 9789896420819

Strang, G., (2016). Introduction to Linear Algebra (fifth edition). Wellesley-Cambridge Press. ISBN: 97809802332776.

NEW:

López, C.P., (2014). MATLAB Linear Algebra. 1st. ed. Edition. Springer. Apress. ISBN: 9781484203224