Subject: Linear Algebra

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - Our main goal is to promote learning of fundamental concepts of Vector Spaces, Linear Maps, Matrices, Determinants of Square Matrices, Eigenvalues and Eigenvectors as well as Solving Linear Systems.
2 - We also aim to develop the students' abstraction capacity by demonstrating theoretical results, as well as by providing examples and counterexamples of the structures analyzed.
3 - To promote analysis and discussion of problems.

# Syllabus:

1 - Vector space over a field.
2 - Linear combinations.
3 - Generators of a vector space.
4 - Equivalence of vector systems.
5 - Linear dependence. Bases. Dimension of a vector space.
6 - Vector subspaces. Intersection and sum of vector subspaces.
7 - Subspaces of a finite dimensional space.
8 - Direct sums of subspaces. The vector space quotient.
9 - Linear maps. Kernel and image of a linear map.
10 - Monomorphisms, epimorphisms, and isomorphisms.
11 - Operations with linear maps. Linear maps between finite dimensional vector spaces.
12 - Matrices. Matrix representation of a linear application. Matrix operations.
13 - Invertible matrices.
14 - Classification matrices.
15 - Determinant of square matrices. Calculation of determinants: the Rule of Sarrus. Laplace theorem.
16 - To calculate the inverse of an invertible matrix.
17 - Equivalent matrices. Characteristic of a matrix.
18 - Systems of linear equations. Resolution and discussion of systems of linear equations: the Gauss Elimination Method and Cramer's rule.
19 - Eigenvalues and eigenvectors. Characteristic polynomial. Geometric and algebraic multiplicities of an eigenvalue.

# Literature/Sources:

Serge Lang , Introduction to Linear Algebra , Springer Verlag
António J. Monteiro , 2001 , Álgebra Linear e Geometria Analítica , Mcgraw-Hill
F. Dias Agudo , 1992 , Introdução à Algebra Linear E Geometria Analítica , Escolar Editora

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Theoretical lectures follow tradition. Pratcila classes are based on solving problems, with the active participation of students. The evaluation is done through the realization of two tests (50%+50%). In a second season of exams, the assessment is on the entirety of the syllabus.