Description

Theory
1

Theory/Practice
2

Instructors

João Emilio Matos

Contents

1. Matrix Calculus (3weeks,20%)
1.1. Definition and representation. Equality of matrices. Transpose of a matrix and symmetric matrix.
1.2. Matrix operations. Properties.
1.3. Elementary row operations. Equivalent matrices. Echelon form of matrices.
1.4. Rank of a matrix; Gauss Elimination?s Method (GEM).
1.5. Inverse of a matrix; computation by GEM.
 

2. Determinants(2weeks,20%)
2.1. Definition. Evaluating by Sarrus? Rule.
2.2. Laplace's Theorem.
2.3. Properties.
2.4. Computation of the inverse of a matrix using the adjoint matrix.
 

3. Systems of linear equations(2weeks,20%)
3.1. Matrix notation.
3.2. The Cramer's Rule.
3.3. Resolution by GEM.
 

4. Vector spaces(2weeks,20%)
4.1. Definition.
4.2. Subspaces.
4.3. Linear combinations.
4.4. Basis and dimension.
 

5. Linear transformations(3weeks,20%)
5.1. Definition and properties.
5.2. Matrix of a transformation.
5.3. Kernel and image.
5.4. Eigenvalues and eigenvectors.
 

Learning Outcomes

This course intends to provide the student with the language, the knowledge and the mathematical techniques of Linear Algebra; it also aims to promote the development of the reasoning and critical thinking, develop the capacity of abstraction. The student should also be able to understand problems and solve them, applying the learned techniques in specific problems of their area of Engineering.
The student must be able to:
CO1 - Perform the fundamental operations of matrix calculation, determine the inverse of a matrix (if it exists) and solve matrix equations; (Bloom level: 0 and 1)
CO2 - Evaluating determinants and using the properties to perform equalities between determinants (BL: 0 and 1).
CO3 - Use Gaussian elimination method and the determinants to solve and discuss systems of linear equations (BL: 1).
CO4 - Identify and generate real vector spaces; operate with vectors, in order to see, for example, if they can be used as a basis of a vector space (BL: 0 and 1).
CO5 - Identify linear transformations, determine the associated matrices and their eigenvalues and eigenvectors (BL: 0 and 1).
CO6 ? Present the resolution of exercises with rigor and detail, particularly using the structured presentation of reasoning underlying the resolution of the problem, the simplification of the result obtained and the presentation of an interpretative and critical comment on the solution obtained.