Linear Algebra

Recommended Prerequisites

Secondary School Mathematics.

Teaching Methods

Lectures and classes.  

Learning Outcomes

Overall objectives:
The aim of this course is to provide the students with the basic foundations of linear algebra which are usually necessary for studying and working in economics.
Specific objectives:
In this course we will present:
• basic concepts of matrices and matrix algebra
• methods of solving systems of linear equations
• the concept of and methods of computing determinants
• methods of computing and using eigenvalues and eigenvectors.
Generic competencies:
The students should acquire the ability to conceive a proof; to model a situation and to solve problems.
They should also be confident in reading and writing elementary results of the linear algebra course.
Specific competencies:
Students should be able to:
• solve systems of linear equations, manipulate matrix algebra and determinants and apply row operations;
evaluate eigenvalues and eigenvectors, evaluate algebraic multiplicity and geometric multiplicity and use them to diagonalize a matrix.
 

Work Placement(s)

No

Syllabus

I Matrices
1 Vectors.Definition and basic properties.Equality addition and multiplication by a scalar Linear combinations. Inner product and norm. Orthogonal and orthonormal sets.
2 Matrices.Definition and basic properties.Equality, addition and multiplication by a scalar Matrix multiplication.Algebraic properties
3 Partitioned matrices.Column space of a matrix
4 Matrix inverses

II Linear system of equations
1 Linear equations and linear systems.Existence and uniqueness of solutions.Gauss elimination.Rank of a matrix.Homogeneous systems.Null space of a matrix.Linear dependence and linear independence
2 Finding inverses by Gauss elimination

III Determinants
1 Introduction. Definition and elementary properties
2 Minors cofactors and adjoint matrix.Matrix inverses and Cramer’s Rule

IV Eigenvalues and eigenvectors
1 Definition and basic properties.Eigenspace algebraic multiplicity and geometric multiplicity of an eigenvalue
2 Diagonalization of matrices. Symmetric matrices 

Head Lecturer(s)

Pedro André Ribeiro Madeira Cerqueira

Assessment Methods

Assessment
Periodic or by final exam as given in the course information: 100.0%

Bibliography

BRETSCHER Otto - Linear algebra with applications. 2nd ed.. Upper Saddle River : Prentice-Hall 2001. [BP 512 BRE]
LAY, David C. - Linear algebra and its applications. 2nd ed., update. Reading, Mass. : Addison-Wesley Publishing Company, 2000. [BP 512 LAY]
LIMA, Teresa Pedroso de - Lições de álgebra linear. 2ª ed.. Coimbra : Imprensa da Universidade de Coimbra, 2014. [BP 512 LIM]
LIMA, Teresa Pedroso de ; VÍTÓRIA, José - Álgebra linear. Lisboa : Universidade Aberta 1998. [BP 512 LIM]
STRANG Gilbert - Introduction to linear.