Mathematics II

Recommended Prerequisites

Secondary Education Mathematics. Knowledge of the limits of derivative and integral functions studied in Mathematics II.

Teaching Methods

In theoretical lessons, the subjects will be presented using a chalkboard. Examples will be provided and problems will be solved. The professor will be the center of the task developing process. On theoretical and practical lessons, we will solve exercises that were previously proposed to students. The students will take initiative when choosing and solving problems that will be studied in each of the lessons.

Learning Outcomes

Develop the ability to calculate limits of successions. Determinate the nature and the sum of series. Determinate the series development of a function. Use elementary matrix development. Solve systems and calculate determinants. Understand the notion of linear dependency and interdependency in Rⁿ and the notion of basis and dimension of a subspace in Rⁿ.

Use the notion of inner product and the method of least squares.

Determinate the eigenvalue and the eigenvector of a matrix.

Work Placement(s)

No

Syllabus

Sequences: notion of limit of a sequence; limited sequence; subsequence; rules of limits; frame sequences; monotonic sequence; indeterminations.

Series: definition and general properties; series of positive terms; convergence criteria; approximate calculation of the sum of a series; term series of any sign.

Series of Functions: definition of sequence and of series of functions; Weirestrass criterion; potency series and Taylor series.

Matrices and Linear equations systems: matrices (operations with matrices); linear equations systems; Gauss elimination method; LU decomposition and systems resolution; matrices inversion; Gauss.Jordan algorithm; determinants.

Vector Space Rn: Vector subspaces in Rn; linear dependence and independence; base and dimension; characteristic of a matrix; linear transformations.

Inner product in Rn: orthogonal projection; Gram-Schmidt orthogonalization methods; least square method.

Eigenvalues and eigenvectors: Diagonalization of matrices.

Head Lecturer(s)

Carlos Manuel Franco Leal

Bibliography

GOODAIRE, Edgar (2003). Liner Algebra A Pure and Applied First Course. Prentice Hall, Pearson Education Inc.,

LEON, Steven J. (2002). Linear Algebra with Applications. New Jersey: Prentice Hall.

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

SILVA, Jaime Carvalho e (1999). Princípios de análise Matemática Aplicada. Mcgraw – Hill.

SILVA, Jaime Carvalho e;  LEAL, Carlos (1999). Análise Matemática Aplicada. Mcgraw – Hill.

STEWART, James (2001). Cálculo. Volume II. Thomson.

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