# Linear Algebra and Analytical Geometry

**Year**

1

**Academic year**

2022-2023

**Code**

01001636

**Subject Area**

Mathematics

**Language of Instruction**

Portuguese

**Other Languages of Instruction**

English

**Mode of Delivery**

Face-to-face

**Duration**

SEMESTRIAL

**ECTS Credits**

6.0

**Type**

Compulsory

**Level**

1st Cycle Studies

## Recommended Prerequisites

Mathematics A from the Portuguese High School Curriculum.

## Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasised. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

## Learning Outcomes

The student who successfully completes this course will be able to:

1. Solve and classify linear systems using Gauss elimination and matrix operations;

2. Compute 2 by 2 and 3 by 3 determinants and expand any determinant using the Laplace expansion;

3. Study the invertibility of a matrix using the rank or the determinant;

4. Compute the inverse of a matrix of order 2 or 3 using the Gauss-Jordan method;

5. Compute a basis and the dimension of a subspace in Rn and apply the Gram-Schmidt orthonormalisation process;

6. Use the method of least squares to determine approximate solutions of linear systems;

7. Compute eigenvalues and eigenvectors and determine whether a given matrix is diagonalisable;

8. Apply the acquired knowledge to solving problems in science and engineering.

## Work Placement(s)

No## Syllabus

1. Matrices and Linear Systems

2. Determinants

3. Vector Spaces and Linear Transformations

4. Vector Spaces with an Inner Product

5. Eigenvalues and Eigenvectors. Matrix Diagonalisation

6. Applications.

## Head Lecturer(s)

Ana Paula Jacinto Santana Ramires

## Assessment Methods

Continuous assessment

*2 or more midterm exams: 100.0%*

Final assessment

*Exam: 100.0%*

## Bibliography

[1] Ana Paula SANTANA, João QUEIRÓ (2010). Introdução à Álgebra Linear. Trajectos Ciência, 10. Gradiva.

[2] Luís T. MAGALHÃES (1989). Álgebra Linear como Introdução a Matemática Aplicada. Texto Editora.

[3] Chris RORRES, Howard ANTON (2014). Elementary linear algebra with supplemental applications, international student version, Hoboken, NJ: John Wiley & Sons, 11ª ed.

[4] David R. HILL e Bernard KOLMAN (2013). Álgebra Linear com Aplicações, Livros Téc. e Cient. Editora, 9ª ed.

[5] Gilbert STRANG (1988). Linear Algebra and its Applications, San Diego: Harcout Brace Jovanovich.