**Publication in the Diário da República**: Despacho nº 10764/2011 - 30/08/2011

5 ECTS; 1º Ano, 1º Semestre, 30,0 T + 30,0 TP , Cód. 91842.

**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 10/20.

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

The final grade (rounded to units) will be the sum of the assessments of the two written tests (unrounded marks). The student is exempted from sitting the examination if he obtains a final mark of 10 or more and if he obtains at least 3 marks in each of the two written examinations.

Assessment by examination: completion of a closed-book written test graded from 0 to 20 on all the subjects taught during the semester. The student passes if, in this exam, he/she reaches a grade higher than or equal to 10/20.

**Bibliography**

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

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

- Leon, S. (2010). *Linear Algebra with Applications*. (pp. 1-552). USA: Pearson

- Smith, P. e Giraldes, E. e Fernandes, V. (1997). *Curso de Álgebra Linear e Geometria Analítica*. (pp. 1-376). Lisboa: McGraw-Hill

**Teaching Method**

Lectures and theoretical-practical classes involving presentation and illustration of the subject matter.

**Software used in class**

Not applicable.

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