# Linear Algebra

**Civil Engineering**

5 ECTS; 1º Ano, 1º Semestre, 30,0 T + 30,0 TP

**Lecturer**

**Prerequisites**

Not applicable.

**Objectives**

1. Acquisition of knowledge in Linear Algebra and Analytic Geometry mathematical areas.

2. Provide students with several algebraic tools that are necessary for modeling and solving problems related to engineering.

3. Development of logical, analytical and critical reasoning thinking skills.

**Program**

I. Matrices and systems of linear equations

1.1. Basic definitions. Some special types of matrices;

1.2. Matrix operations and properties;

1.3. Elementary operations (on rows). The rank of a matrix;

1.4. Systems of linear equations:

1.4.1. Matrix form of a system of linear equations;

1.4.2. Classification and discussion of a system of linear equations by use of the Rouché's theorem;

1.4.3. Solving systems of linear equations by use of the Gauss-Jordan's elimination method;

1.5. Inversion of matrices:

1.5.1. Singular and non-singular matrices;

1.5.2. Calculation of the inverse of a non-singular matrix by use of the Gauss-Jordan's method;

1.6. P^T LU decomposition:

1.6.1. Elementary matrices and permutation matrices;

1.6.2. P^T LU decomposition of a matrix;

1.6.3. Solving systems of linear equations by use of the P^T LU decomposition of the system's coefficient matrix.

II. Determinants

2.1. Definition. Calculation of second order determinants (crossed products rule);

2.2. Laplace's theorem;

2.2.1. Minor and cofactor of an entry of a square matrix;

2.2.2. Calculation of the determinant of a square matrix by use of the Laplace's theorem;

2.3. Some properties of determinants;

2.4. Calculation of the inverse of a non-singular matrix by use of its adjoint matrix.

2.5. Solving systems of linear equations by use of Cramer?s rule.

III. Vetor spaces

3.1. Introduction. Definition and examples of vector spaces;

3.2. Vector subspaces;

3.3. Linear combinations of vectors;

3.4. Subspace spanned by a set of vectors;

3.5. Linear dependence and independence of vectors;

3.6. Basis and dimension of a vector space;

3.7. Row space and column space of a matrix.

IV. Eigenvalues and eigenvectors

4.1. Eigenvalues and eigenvectors of square matrices: definitions, characteristic polynomial and algebraic multiplicity of an eigenvalue;

4.2. Eigenspace associated with an eigenvalue, and geometric multiplicity of an eigenvalue;

4.3. Calculation of eigenvalues and eigenvectors;

4.4. Eigenvalues properties;

4.5. Diagonalizable matrices and diagonalization of a matrix.

V. Analytic geometry

5.1. Inner product: definition and properties;

5.2. Vector and scalar triple products: definition, properties, applications to the calculation of the area of a parallelogram and of the volume of a parallelepiped;

5.3. Vector equation, parametric equations and cartesian equations of a straight line;

5.4. Vector equation, parametric equations and cartesian equation of a plane.

**Evaluation Methodology**

Continuous assessment: two written closed-book tests, each worthing 10 points, and a minimum score of 3 grade points in each test.

Exam assessment: one written closed-book test worthing 20 grade points including all taught topics.

**Bibliography**

- Dias Agudo, F. (1978). *Introdução à Álgebra Linear e Geometria Analítica*. Lisboa: Escolar Editora

- Smith, P. e Giraldes, E. (1995). *Curso de Álgebra Linear e Geometria Analítica*. Lisboa: McGraw-Hill

- T. Magalhães, L. (1989). *Álgebra Linear como Introdução à Matemática Aplicada*. Lisboa: Texto Editora

- Nicholson, W. (1995). *Linear Algebra with Applications*. Boston: PWS Publishing Company

- Leon, S. (2009). *Linear Algebra with Applications*. (pp. 1-552). USA: Pearson

- Ferreira, M. e Amaral, I. (2009). *Álgebra Linear: Espaços Vectoriais e Geometria Analítica*. (Vol. 2º). (pp. 1-160). Portugal: Edições Sílabo

- Ferreira, M. e Amaral, I. (2008). *Álgebra Linear: Matrizes e Determinantes*. (Vol. 1º). (pp. 1-240). Portugal: Edições Sílabo

**Method of interaction**

Theoretical and theoretical-practical lectures comprising content presentation and illustration.

**Software used in class**

Not applicable.