Description

Theory
2

Theory/Practice
2

Instructors

Amélia Caldeira

Contents

CP1-COMPLEX NUMBERS (CN)
-CN and their algebraic operations
-Representation of CN in the complex plane
-Representations of CN and operations
-Applications

CP2-MATRICES
-Def. and representation of a matrix
-Matrix op. and prop.
-Elementary row and column operations
-Matrix condensation
-Matrix rank; calculation using Gaussian Elimination Method (GEM)
-Inverse matrix
-Matrix eq.
-Applications

CP3-DETERMINANTS
-Def. and prop.
-Calculation of determinants
-Laplace's Theorem
-Applications

CP4-SYSTEMS OF LINEAR EQUATIONS (SLE)
-Def. and matrix form of an SLE
-Sarrus' Rule
-Classification and solving systs. depending on parameters
-Cramer's systems and homogeneous systs.
-Applications

CP5-REAL VECTOR SPACES
-Def. and prop.
-Vector subspaces
-Basis and dimension

CP6-LINEAR TRANSFORMATIONS
-Def. and prop.
-Matrix representation
-Kernel and image
-Eigenvalues and eigenvectors

CP7-VECTOR CALCULUS in R^3
-Algebraic operations with vectors
-Cross product and scalar triple product
-Lines and planesç

Learning Outcomes

GENERAL OBJECTIVES
The aim is for students to:
(P1) complement and consolidate their mathematical background acquired throughout their learning process;
(P2) develop their reasoning and abstraction skills;
(P3) cultivate mathematical thinking and acquire critical thinking skills;
(P4) become capable of applying mathematical techniques that are essential for understanding and interpreting topics taught in other undergraduate course units and fundamental to engineering.

SPECIFIC OBJECTIVES
Specifically, by the end of the semester, students should be able to:
(O1) work with and manipulate complex numbers, including their representation in the Argand plane;
(O2) perform basic matrix operations and understand their properties; define and determine the rank of a matrix and compute the inverse matrix; solve matrix equations;
(O3) compute determinants and manipulate them using their properties;
(O4) analyse and solve systems of linear equations;
(O5) identify and construct real vector spaces and work with vectors, particularly to verify whether they can be used as a basis of a vector space;
(O6) identify linear transformations, determine the associated matrices, and compute their eigenvalues and eigenvectors;
(O7) understand and apply the concepts of Euclidean spaces, including their definition and identification of examples, as well as compute norms, distances, and angles; apply the cross product and the scalar triple product in solving problems related to lines and planes.