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This thesis is a study of linear spaces and linear transformations in normed linear spaces. The notion of a field, in particular the complex number field, is assumed in this paper.

It is well documented that undergraduate students struggle with the more formal and abstract concepts of vector space theory in a first course on linear algebra. Some of these students continue on to classes in abstract algebra, where they learn about algebraic structures such as groups. It is clear to the seasoned mathematician that vector spaces are in fact groups, and so linear transformations are group homomorphisms with extra restrictions. This study explores the question of whether or not students see this connection as well. In addition, I probe the ways in which students’ stated understandings are the same or different across contexts, and how these differences may help or hinder connection making across domains. Students’ understandings are also briefly compared to those of mathematics professors in order to highlight similarities and discrepancies between reality and idealistic expectations. The data for this study primarily comes from clinical interviews with ten undergraduates and three professors. The clinical interviews contained multiple card sorts in which students expressed the connections they saw within and across the domains of linear algebra and abstract algebra, with an emphasis specifically on linear transformations and group homomorphisms. Qualitative data was analyzed using abductive reasoning through multiple rounds of coding and generating themes. Overall, I found that students ranged from having very few connections, to beginning to form connections once placed in the interview setting, to already having a well-integrated morphism schema across domains. A considerable portion of this paper explores the many and varied ways in which students succeeded and failed in making mathematically correct connections, using the language of research on analogical reasoning to frame the discussion. Of particular interest were the ways in which isomorphisms did or did not play a role in understanding both morphisms, how students did not regularly connect the concepts of matrices and linear transformations, and how vector spaces were not fully aligned with groups as algebraic structures.

In this thesis a machine vision based indoor navigation system is presented. This is achieved by using rotationally independent optimized color reference labels and a geometrical camera calibration model which determines a set of camera parameters. All reference labels carry one byte of information (0 to 255), which can be designed for different values. An algorithm in Matlab has been developed so that a machine vision system for N number of symbols can recognize the symbols at different orientations. A camera calibration model describes the mapping between the 3-D world coordinates and the 2-D image coordinates. The reconstruction system uses the direct linear transform (DLT) method with a set of control reference labels in relation to the camera calibration. The least-squares adjustment method has been developed to calculate the parameters of the machine vision system. In these experiments it has been demonstrated that the pose of the camera can be calculated, with a relatively high precision, by using the least-squares estimation.

Orientador: Marcelo de Carvalho Borba / Banca: Maria Aparecida Viggiani Bicudo / Banca: Nilson José Machado / Banca: Siobhan Victoria Healy / Banca: Marcus Vinicius Maltempi / Resumo: Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and "funcionetes" (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of "funcionetes planos" is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below) / Doutor

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-11-03Bitstream added on 2014-06-13T20:02:50Z : No. of bitstreams: 1 figueiredo_oa_dr_rcla.pdf: 599343 bytes, checksum: 2107a2036f27ecde0dcfdbff1f4a2bf3 (MD5) / Os processos perceptivos que fundamentam a experiência humana podem nos parecer absolutamente naturais. Devido a isso, costumamos não tematizá-los. Este trabalho é um esforço de evidenciação da percepção na educação matemática, mais especificamente na geometria dinâmica e no ensino de funções com auxílio de representações dinâmicas. Percepção é entendida em uma concepção fenomenológica. Sustenta-se que: (a) é da natureza humana certa capacidade de perceber comportamentos de dependências entre eventos do mundo físico, isto é, existe um sentido de percepção de dependência; (b) as representações dinâmicas de funções, como os Dynagraphs (conhecidos na literatura) e os funcionetes (propostos no trabalho), são depreendidas pelo sentido de percepção de dependência; (c) o emprego de representações dinâmicas no auxílio ao ensino de funções abre novos sentidos para funções matemáticas, conceitos, propriedades e teoremas correlatos, justificando o interesse em sua aplicação; além disso, os sentidos abertos são perceptivos e, por isso, diretos, imediatos e evidentes (conforme a fundamentação fenomenológica); (d) existe um sentido de percepção de restrições ou impedimentos; (e) na resolução interativa (geometria dinâmica) de sistemas de restrições geométricas, o sentido de percepção de restrições apresenta, ao trazer perceptivamente as restrições para primeiro plano, as construções geométricas como uma combinação de restrições. No desenvolvimento dessas ideias: apresentam-se os funcionetes planos e sua aplicação na construção de uma abordagem pedagógica para o conceito (da álgebra linear) transformação linear, que é um tipo de função; abordam-se os tópicos: autovetores de um operador linear, propriedade de linearidade e núcleo de uma transformação linear, inclusive o teorema do núcleo e da imagem... / The perceptive processes that provide the basis for human experience can seem absolutely natural. Therefore we do not have the habit of focusing on them as the object of study. The present theoretical study aimed to make perception evident in the context of mathematics education, specifically dynamic geometry and teaching of functions using dynamic representations. Perception is understood as a phenomenological conception. It is maintained that: (a) it is of human nature to be able to perceive dependent behaviors among events in the physical world, i.e. a sense of perception of dependence; (b) dynamic representations of functions, such as Dynagraphs (known in the literature) and “funcionetes” (proposed here), are ascertained through the sense of perception of dependence; (c) the use of dynamic representations to aid in the teaching of functions opens up new senses for mathematical functions, concepts, properties and correlated theorems, justifying interest in its application; in addition, these newly-opened senses are perceptive in nature, and therefore direct, immediate and evident (according to foundations of phenomenology); (d) there exists a sense of perception of constraints or impediments; (e) in the interactive resolution (dynamic geometry) of constraint systems for geometric domain, the sense of perception of constraints presents geometric constructions as a combination of constraints as it perceptively brings the constraints to the foreground. The concept of “funcionetes planos” is presented and their use proposed as part of an approach for teaching the concept (from linear algebra) of linear transformation, which is a type of function. Topics addressed include: eigenvectors of a linear operator, the property of linearity and nucleus of a linear transformation, including the theorem of nucleus and of image, presented in a perceptive sense... (Complete abstract click electronic access below)

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-19T16:07:51Z No. of bitstreams: 1 josimarmoreirarocha.pdf: 2068448 bytes, checksum: 720b45220091d2a8fe3fb2740d4b4cf9 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-26T13:50:01Z (GMT) No. of bitstreams: 1 josimarmoreirarocha.pdf: 2068448 bytes, checksum: 720b45220091d2a8fe3fb2740d4b4cf9 (MD5) / Made available in DSpace on 2016-02-26T13:50:01Z (GMT). No. of bitstreams: 1 josimarmoreirarocha.pdf: 2068448 bytes, checksum: 720b45220091d2a8fe3fb2740d4b4cf9 (MD5) Previous issue date: 2014-08-16 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O propósito deste trabalho é dissertar sobre as possibilidades do uso da planilha eletrônica como recurso didático no ensino de Matemática. Para exemplificar esse uso, é apresentada uma sequência didática sobre multiplicação de matrizes, abordada sob a perspectiva de transformações no plano. Este trabalho tem um duplo caráter: mostrar que a planilha eletrônica é uma eficiente ferramenta a ser utilizada na intermediação do processo de ensino e aprendizagem de diversos conteúdos e sugerir uma abordagem diferenciada do tópico multiplicação de matrizes no Ensino Médio. É um trabalho de natureza teórica, fundamentado na pesquisa da bibliografia disponível sobre o assunto. Espera-se que sua leitura estimule o uso da planilha eletrônica em sala de aula e, em particular, que ela forneça elementos para o tratamento significativo da multiplicação de matrizes. De uma maneira mais geral, a importância deste trabalho está em promover a inserção das planilhas eletrônicas no contexto escolar, reconhecendo seu potencial pedagógico no ensino de Matemática e no ensino de outras áreas do conhecimento. / The purpose of this work is to discuss the possibilities of using the electronic spreadsheet as a teaching resource in the teaching of Mathematics. To illustrate this use, a didactic sequence on matrix multiplication is presented, approached from the perspective of changes in the plan. This work has a dual character: to show that the electronic spreadsheet is an efficient tool to be used in the intermediation of the teaching and learning process of several contents and suggest a differentiated approach to the matrix multiplication topic in high school. It is a work of theoretical nature, based on the research of the literature available on the subject. It is expected that its reading will stimulate the use of the electronic spreadsheet in the classroom and, in particular, that it will provide evidence for the significant treatment of matrix multiplication. In a more general way, the importance of this work is to promote the integration of electronic spreadsheets in the school context, recognizing its pedagogical potential in the teaching of mathematics and in the teaching of other areas of knowledge.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2016-09-14T18:15:51Z No. of bitstreams: 1 Maria Eliana Santana Da Cruz Silva.pdf: 2630796 bytes, checksum: 59dcb18293cd1765db482b9ff7efb016 (MD5) / Made available in DSpace on 2016-09-14T18:15:51Z (GMT). No. of bitstreams: 1 Maria Eliana Santana Da Cruz Silva.pdf: 2630796 bytes, checksum: 59dcb18293cd1765db482b9ff7efb016 (MD5) Previous issue date: 2016-03-21 / This doctoral dissertation reports an investigation of the conceptions of linear transformation held by teaching certification students in mathematics following a course of linear algebra devised to involve students in the search for elements to build the concept of this mathematical object. The analysis of these conceptions was carried out according to the APOS theory by Dubinsky and Lewin (1986) and Asiala et al. (1996), in articulation with premises of didactic engineering (MACHADO, 2010). The investigation was a qualitative, case-study type, which, according to Bogdan and Biklen (1994) and Lüdke and André (1986), enables the context studied to be investigated and interpreted, given that all qualitative studies seek to portray reality in a more dense fashion and represent the different perspectives present in a given social situation. Centered on this understanding of qualitative research, classes were organized with the aim of actively involving the study participants to facilitate data collection, which was carried out three semesters after the linear algebra course. Premises of didactic engineering were employed to prepare and conduct the activities, which were discussed and solved by paired students. Their interactions were audio recorded and also noted by an observer. The analysis revealed that the subjects paired for the activities experienced different stages in terms of conception (action, process, and object) in most of the activities. We concluded that the students investigated did not always show a conception of object during the activities, but were all able to demonstrate a conception of process for the notion of linear transformation / Esta tese apresenta uma pesquisa que busca investigar as concepções de transformação linear de alunos de uma licenciatura em matemática ao vivenciarem um curso de álgebra linear elaborado com a intenção de envolvê-los na busca de elementos para a construção desse objeto matemático. A análise das concepções se embasou principalmente em partes da teoria APOS, de Dubinsky e Lewin (1986) e Asiala et al. (1996), articulada com alguns pressupostos da engenharia didática (MACHADO, 2010). A pesquisa se caracteriza como qualitativa, abordagem que, segundo Bogdan e Biklen (1994) e Lüdke e André (1986), possibilita investigar e interpretar o contexto investigado, uma vez que todo estudo qualitativo visa retratar a realidade de forma densa, procurando representar as diferentes perspectivas presentes em dada situação social. A partir dessa compreensão de pesquisa qualitativa, foi possível organizar aulas com o objetivo de envolver ativamente os participantes da pesquisa, de forma a facilitar a coleta de dados, que foi realizada após três semestres do curso em questão. Utilizamos pressupostos da engenharia didática na preparação e execução das atividades, discutidas e resolvidas em duplas, atividades essas audiogravadas e descritas por um observador. As análises revelaram que os sujeitos que compuseram as duplas durante a realização das atividades vivenciaram diferentes etapas em termos de concepção (ação, processo e objeto) na maioria das atividades propostas. Concluímos que os sujeitos da pesquisa nem sempre demonstraram uma concepção de objeto no desenvolvimento das atividades, mas todos foram capazes de demonstrar uma concepção de processo da noção de transformação linear

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho, consideramos combinaÃÃes afins de vetores de um espaÃo vetorial com especiais aplicaÃÃes no ensino mÃdio atravÃs da mÃdia aritmÃtica ponderada e da desigualdade de Jensen. Verificamos caracterÃsticas de transformaÃÃes lineares de conjuntos especÃficos nos espaÃos vetoriais como conjuntos convexos e variedades afins, atravÃs do nÃcleo e da imagem das transformaÃÃes. Estabelecemos relaÃÃes entre transformaÃÃes afins, combinaÃÃes afins e transformaÃÃes lineares. Discutimos a dimensÃo do hiperplano relacionando-o como variedade afim. Vemos que todo subespaÃo vetorial de Rn com dimensÃo n - 1 Ã um hiperplano, assim como o nÃcleo de um funcional linear. / In this paper, we consider combinations of related vectors of a vector space with special applications in high school through the weighted arithmetic mean and the Jensen inequality. We observed characteristics of specific sets of linear transformations in the vector spaces as convex sets and related varieties through the core and image transformations. Established relations between affine transformations, combinations thereof and linear transformations. We discuss the size of the hyperplane relating it as affine variety. We see that all of Rn vector subspace with dimension n - 1 is a hyperplane, as the core of a linear functional.

Assessment of finger range of motion (ROM) is often required for monitoring the effectiveness of rehabilitative treatments and for evaluating patients' functional impairment. There are several devices which are used to measure this motion, such as wire tracing, tracing onto paper and mechanical and electronic goniometry. These devices are quite cheap, excluding electronic goniometry; however the drawbacks of these devices are their lack of accuracy and the time- consuming nature of the measurement process. The work described in this thesis considers the design, implementation and validation of a new medical measurement system utilized in the evaluation of the range of motion of the human finger joints instead of the current measurement tools. The proposed system is a non-contact measurement device based on computer vision technology and has many advantages over the existing measurement devices. In terms of accuracy, better results are achieved by this system, it can be operated by semi-skilled person, and is time saving for the evaluator. The computer vision system in this study consists of CCD cameras to capture the images, a frame-grabber to change the analogue signal from the cameras to digital signals which can be manipulated by a computer, Ultra Violet light (UV) to illuminate the measurement space, software to process the images and perform the required computation, a darkened enclosure to accommodate the cameras and UV light and to shield the working area from any undesirable ambient light. Two calibration techniques were used to calibrate the cameras, Direct Linear Transformation and Tsai. A calibration piece that suits this application was designed and manufactured. A steel hand model was used to measure the fingers joint angles. The average error from measuring the finger angles using this system was around 1 degree compared with 5 degrees for the existing used techniques.