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# Algebra

### Presentation

The present curricular unit belongs to the curricular plan of this 1st cycle degree.

The main goal of this curricular unit is to provide students with fundamental knowledge in the context of algebra and mathematical and logical reasoning, that are essential in learning the contents of other subsequent curricular units. It is intended that, through various theoretical and practical strategies, students can apply and solidify the knowledge gained throughout the semester on vector spaces, matrices, determinants, systems of linear equations, complex numbers, eigenvectors and eigenvalues.

### Programme

Environmental Engineering

| Semestral | 6

### Year | Type of course unit | Language

1 |Mandatory |Português

ULP287-61

None.

Não

### Syllabus

REVISIONS: Mathematical symbols and sets. Resolution and classification of equation systems.

MATRIXES: Classification, properties and operations. Characteristics of a matrix; Condensation; Inverse of a square matrix; Resolution of matrix equations.

DETERMINANTS: Definitions and properties; Calculation of determinants by Sarrus's rule, Laplace's theorem and triangulation method; Obtain the inverse by the adjunct.

SYSTEMS OF EQUATIONS: Classification and resolution. Gaussian Method and Cramer's Rule.

VECTOR SPACES: Vectors and operations. Definition and properties; Linear combination; Linear dependence and independence; Vector subspace; Set of generators; Base and dimension of a vector space; Change of base.

EIGENVALUES AND EIGENVECTORS: Definition, properties and their determination. Diagonalization. Quadratic form. Applications.

COMPLEX NUMBERS. Algebraic, trigonometric and exponential forms. Conversion between different forms. Operations and properties.

### Objectives

At the end of this course, students should have acquired knowledge about:

- Operate with matrices to solve equations and calculate the matrix inverse of a matrix.

- Calculate the value of the determinant of a matrix.

- Solve a system of equations by applying the knowledge matrix.

- Analyze a system of equations using the knowledge matrix and on vector spaces, assessing their possible solution.

- Characterize real linear spaces, mastering the concept of linear dependence and independence of vectors, a basis to characterize and define the coordinates of a given vector basis.

- Operating with complex numbers in algebraic, trigonometric and exponential forms.

- Determine and work eigenvectors and eigenvalues.

### Teaching methodologies and assessment

METHODOLOGY: aim to encourage participation and to restore the initiative of the student in the educational process of their own training. The teaching structure is divided into theoretical sessions, using the lecture method, interactive, or, by audiovisual presentation of issues, and practical sessions, where students are confronted with problems, to be held individually or in groups, with the help of Professor and where they can apply the knowledge acquired. The ratio of student / teacher becomes much more favorable in these sessions since they met the appropriate conditions for greater participation in class, allowing and encouraging a closer relationship between faculty and students.

EVALUATION: continuous with two written tests, in accordance with Regulation Assessment of Knowledge ULP.

### References

- Giraldes, E., Fernandes, V., Smith, M. (2003), Curso de Álgebra Linear e Geometria Analítica, McGraw Hill, Portugal.

- Kreyszig, E. (2011), Advanced Engineering Mathematics (tenth edition), McGraw Hill, United States of America.

- Diversos textos de apoio a fornecer ao longo das sessões.

### Office Hours

 Nome do docente Horário de atendimento Sala Cândida Manuel Segunda feira a indicar