Transformações lineares no plano e aplicações

• Autor(a) principal: Nogueira, Leonardo Bernardes
• Data de Publicação: 2013
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/0013000003sgv
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/3123
• topic: Álgebra linear; Teorema espectral; Secções cônicas; Linear algebra; Spectral Theorem; Conic Section; MATEMATICA::MATEMATICA APLICADA

This paper begins with a brief history about the development of vector spaces and linear transformations, then presents fundamental concepts for the study of Linear Algebra, with greater focus on linear operators in the R2 space. Through examples it explores a wide range of operators in R2 in order to show other applications of matrices in high school and prepares the ground for the presentation a version of Spectral Theorem for selfadjoint operators in R2, which says that for every operator self-adjoint T : E!E in finite dimensional vector space with inner product, exists an orthonormal basis fu1; : : : ;ung E formed by eigenvectors of T, and culminates with their applications on the study of conic sections, quadratic forms and equations of second degree in x and y; on the study of operators associated to quadratic forms, a version of Spectral Theorem could be called as The Main Axis Theorem albeit this nomenclature is not used in this paper. Thereby summarizing a study made by Lagrange in "Recherche d’arithmétique ", between 1773 and 1775, which he studied the property of numbers that are the sum of two squares. Thus he was led to study the effects of linear transformation with integer coefficients in a quadratic form in two variables.