CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho abordaremos alguns tÃpicos matriciais e determinantes e sua aplicaÃÃo no Ensino MÃdio. Em especial a Matriz de Gram em uma transformaÃÃo linear que pode ser apli-cada, por exemplo, para calcular a Ãrea de um triÃngulo em funÃÃo dos seus lados e tambÃm o Gramiano (determinante da Matriz de Gram) que permite calcular o volume de um paralele-pÃpedo. Ambos podem ser aplicados no ensino mÃdio. Nesse trabalho tembÃm fazemos uma generalizaÃÃo do produto vetorial e algumas de suas propriedades envolvendo determinantes. Por fim mostramos a Identidade de Lagrange. / In this paper we discuss some topics and determinants matrix and its application in high school. In particular, the Gram matrix in a linear transformation which can be applied, for example, to calculate the area of a triangle in terms of their sides and also Gramiano (Gram matrix determinant) for calculating the volume of a parallelepiped. Both can be applied in high school. In this work we tembÃm a generalization of the vector product and some of its properties involving determinants. Finally we show the identity of Lagrange.

Furthering our understanding of the physics of flapping flight has the potential to benefit the field of micro air vehicles. Advancements in micro air vehicles can benefit applications such as surveillance, reconnaissance, and search and rescue. In this research, flapping kinematics of a ladybug was explored using a direct linear transformation. A flapping mechanism design is presented that was capable of executing ladybug or other species-specific kinematics. The mechanism was based on a differential gear design, had two wings, and could flap in harsh environments. This mechanism served as a test bed for force analysis and optimization studies. The first study was based on a Box-Behnken screening design to explore wing kinematic parameter design space and manually search in the direction of flapping kinematics that optimized the objective of maximum combined lift and thrust. The second study used a Box-Behnken screening design to build a response surface. Using gradient-based techniques, this surface was optimized for maximum combined lift and thrust. Box-Behnken design coupled with response surface methodology was an efficient method for exploring the mechanism force response. Both methods for optimization were capable of successfully improving lift and thrust force outputs. The incorporation of the results of these studies will aid in the design of more efficient micro air vehicles and with the ultimate goal of leading to a better understanding of flapping wing aerodynamics and the development of aerodynamic models.

Submitted by admin tede (tede@pucgoias.edu.br) on 2018-11-05T16:55:20Z No. of bitstreams: 1 ALINE MOTA DE MESQUITA ASSIS.pdf: 6688952 bytes, checksum: 94c9e4c183a133d7dbafd52f7e741501 (MD5) / Made available in DSpace on 2018-11-05T16:55:20Z (GMT). No. of bitstreams: 1 ALINE MOTA DE MESQUITA ASSIS.pdf: 6688952 bytes, checksum: 94c9e4c183a133d7dbafd52f7e741501 (MD5) Previous issue date: 2018-08-30 / This work falls into the category of research into Theories of Education and Pedagogical Processes, and has as its main investigative focus, the teaching-learning process according to the algebraic concept of linear transformation, based on V.V. Davydov´s theory of developmental teaching. The question it seeks to clarify is : what are the repercussions for teaching the concept of linear transformation, based on the historical-cultural theory, in specific, Davydov's developmental theory, in the process of concept formation by students? Specifically, it aims to analyze the history of the logical development of the concept of linear transformation in order to grasp the relations present in it and the forms of mental movement displayed, towards identifying the mental actions to be contemplated in the planning and conduct of the activity of study; to carry out the study activity through the development of a didactic formation experiment to understand, in the course of the teaching-learning process of the concept of linear transformation, elements that indicate qualitative and quantitative changes in the development of student thinking. To this end research was carried out that consisted of a teaching experiment in a class of Linear Algebra at the Federal Institute of Education, Science and Technology of Goiás - Câmpus Goiânia, based on the assumptions of Davydov. This was completed with fourteen students of the Bachelor in Electrical Engineering graduate course and done so according to the structure of the study activity proposed by Davydov. The procedures for collecting the data were as follows: a written record of semistructured interviews with the teacher, socio-cultural questionnaires completed by the students, a diagnostic instrument for evaluation, an experimental teaching plan and the notes from non-participant direct observers. Data analysis focuses on the process of concept formation and the elements involved in this process from the following categories: transformation of task data into the identification of the general principle of the concept of linear transformation; from modeling to transformation of a model to the concept of linear transformation and the use of the concept of linear transformation as a mental tool. The results showed: the motivation of students during the experimental teaching; an understanding of algebraic concepts after logical-historical analysis by the majority of the research subjects; indicators of the zone of proximal development of the students in relation to the concepts of matrix, function and vector space - considered here as the prerequisites for the formation of the concept of linear transformation, developing the ability to think Mathematically according to the logic of this science; evidence of qualitative changes in the development of theoretical thinking of the research subjects, again, regarding the concept of linear transformation. The main contribution of this research was to show an alternative way of organizing the teaching of the concept of linear transformation, and consequently Linear Algebra. It is believed that even with the contradictions present in the curricular structure of the courses in the areas of the exact and world sciences and in engineering, as well as in the students' school formation, it is possible to carry out teaching based on the theory of developmental teaching and contribute to the theoretical thought formation in the majority of students. / Este trabalho, inscrito na linha de pesquisa Teorias da Educação e Processos Pedagógicos, tem como principal foco investigativo o processo de ensino-aprendizagem do conceito algébrico de transformação linear, fundamentando-se na teoria do ensino desenvolvimental de V. V. Davydov. A questão que se buscou esclarecer foi: que repercussões teriam, no processo de formação de conceitos pelos alunos, o ensino do conceito de transformação linear fundamentado na teoria histórico-cultural, em específico, na teoria do ensino desenvolvimental de Davydov? Especificamente, objetiva-se: analisar a história do desenvolvimento lógico do conceito de transformação linear a fim de apreender as relações nele presentes e o tipo de movimento mental que ele contém para identificar as ações mentais a serem contempladas no planejamento e na condução da atividade de estudo; proceder à realização da atividade de estudo mediante o desenvolvimento de um experimento didático formativo; apreender, no decorrer processo de ensino-aprendizagem do conceito de transformação linear, elementos que indicam mudanças qualitativas e quantitativas no desenvolvimento do pensamento do aluno. Para tanto, realizou-se uma pesquisa que consistiu em um experimento de ensino, baseado nos pressupostos de Davydov, em uma turma de Álgebra Linear do Instituto Federal de Educação, Ciência e Tecnologia de Goiás – Câmpus Goiânia, desenvolvido com quatorze alunos do curso de Bacharelado em Engenharia Elétrica e seguindo a estrutura da atividade de estudo proposta por Davydov. Os procedimentos para a coleta dos dados foram: roteiro de entrevista semiestruturada com o professor, questionário sociocultural dos alunos, instrumento de avaliação diagnóstica, plano de ensino experimental e roteiro de observação direta não participante. A análise dos dados enfoca o processo de formação de conceitos e os elementos intervenientes nesse processo a partir das seguintes categorias: transformação dos dados da tarefa na condução da identificação do princípio geral do conceito de transformação linear; da modelação à transformação de um modelo para o conceito de transformação linear e o uso do conceito de transformação linear como ferramenta mental. Os resultados obtidos revelaram: motivação dos alunos durante o ensino experimental; compreensão dos conceitos algébricos, após a análise lógico-histórica, pela maioria dos sujeitos da pesquisa; indícios de progresso da zona de desenvolvimento proximal dos alunos no que tange aos conceitos de matriz, função e espaço vetorial, considerados aqui como os pré-requisitos para a formação do conceito de transformação linear, desenvolvendo a capacidade de pensar a Matemática de acordo com a forma de pensar desta ciência; indícios de mudanças qualitativas no desenvolvimento do pensamento teórico dos sujeitos da pesquisa quanto ao conceito de transformação linear. A principal contribuição desta pesquisa consistiu em mostrar um caminho alternativo de organização do ensino do conceito de transformação linear, consequentemente, da Álgebra Linear. Acredita-se que, mesmo com as contradições presentes na estrutura curricular dos cursos das áreas de Ciências Exatas e da Terra e Engenharias, bem como na formação escolar dos alunos, é possível realizar um ensino embasado na teoria do ensino desenvolvimental e contribuir para a formação do pensamento teórico da maioria dos alunos

Tato disertační práce se zabývá odhadem pózy kamery z korespondencí 3D a 2D přímek, tedy tzv. perspektivním problémem n  přímek (angl. Perspective- n -Line, PnL). Pozornost je soustředěna na případy s velkým počtem čar, které mohou být efektivně řešeny metodami využívajícími lineární formulaci PnL. Dosud byly známy pouze metody pracující s korespondencemi 3D bodů a 2D přímek. Na základě tohoto pozorování byly navrženy dvě nové metody založené na algoritmu přímé lineární transformace (angl. Direct Linear Transformation, DLT): Metoda DLT-Plücker-Lines pracující s korespondencemi 3D a 2D přímek a metoda DLT-Combined-Lines pracující jak s korespondencemi 3D bodů a 2D přímek, tak s korespondencemi 3D přímek a 2D přímek. Ve druhém případě je redundantní 3D informace využita k redukci minimálního počtu požadovaných korespondencí přímek na 5 a ke zlepšení přesnosti metody. Navržené metody byly důkladně testovány za různých podmínek včetně simulovaných a reálných dat a porovnány s nejlepšími existujícími PnL metodami. Metoda DLT-Combined-Lines dosahuje výsledků lepších nebo srovnatelných s nejlepšími existujícími metodami a zároveň je značně rychlá. Tato disertační práce také zavádí jednotný rámec pro popis metod pro odhad pózy kamery založených na algoritmu DLT. Obě navržené metody jsou definovány v tomto rámci.

Autoencoders, a type of artificial neural network, have gained recognition by researchers in various fields, especially machine learning due to their vast applications in data representations from inputs. Recently researchers have explored the possibility to extend the application of autoencoders to solve nonlinear differential equations. Algorithms and methods employed in an autoencoder framework include sparse identification of nonlinear dynamics (SINDy), dynamic mode decomposition (DMD), Koopman operator theory and singular value decomposition (SVD). These approaches use matrix multiplication to represent linear transformation. However, machine learning algorithms often use convolution to represent linear transformations. In our work, we modify these approaches to system identification and forecasting of solutions of nonlinear differential equations by replacing matrix multiplication with convolution transformation. In particular, we develop convolution-based approach to dynamic mode decomposition and discuss its application to problems not solvable otherwise.

L’introduction aux concepts unificateurs dans l’enseignement des mathématiques privilégie typiquement l’approche axiomatique. Il n’est pas surprenant de constater qu’une telle approche tend à une algorithmisation des tâches pour augmenter l’efficacité de leur résolution et favoriser la transparence du nouveau concept enseigné (Chevallard, 1991). Cette réponse classique fait néanmoins oublier le rôle unificateur du concept et n’encourage pas à l’utilisation de sa puissance. Afin d’améliorer l’apprentissage d’un concept unificateur, ce travail de thèse étudie la pertinence d’une séquence didactique dans la formation d’ingénieurs centrée sur un concept unificateur de l’algèbre linéaire: la transformation linéaire (TL). La notion d’unification et la question du sens de la linéarité sont abordées à travers l’acquisition de compétences en résolution de problèmes. La séquence des problèmes à résoudre a pour objet le processus de construction d’un concept abstrait (la TL) sur un domaine déjà mathématisé, avec l’intention de dégager l’aspect unificateur de la notion formelle (Astolfi y Drouin, 1992). À partir de résultats de travaux en didactique des sciences et des mathématiques (Dupin 1995; Sfard 1991), nous élaborons des situations didactiques sur la base d’éléments de modélisation, en cherchant à articuler deux façons de concevoir l’objet (« procédurale » et « structurale ») de façon à trouver une stratégie de résolution plus sûre, plus économique et réutilisable. En particulier, nous avons cherché à situer la notion dans différents domaines mathématiques où elle est applicable : arithmétique, géométrique, algébrique et analytique. La séquence vise à développer des liens entre différents cadres mathématiques, et entre différentes représentations de la TL dans les différents registres mathématiques, en s’inspirant notamment dans cette démarche du développement historique de la notion. De plus, la séquence didactique vise à maintenir un équilibre entre le côté applicable des tâches à la pratique professionnelle visée, et le côté théorique propice à la structuration des concepts. L’étude a été conduite avec des étudiants chiliens en formation au génie, dans le premier cours d’algèbre linéaire. Nous avons mené une analyse a priori détaillée afin de renforcer la robustesse de la séquence et de préparer à l’analyse des données. Par l’analyse des réponses au questionnaire d’entrée, des productions des équipes et des commentaires reçus en entrevus, nous avons pu identifier les compétences mathématiques et les niveaux d’explicitation (Caron, 2004) mis à contribution dans l’utilisation de la TL. Les résultats obtenus montrent l’émergence du rôle unificateur de la TL, même chez ceux dont les habitudes en résolution de problèmes mathématiques sont marquées par une orientation procédurale, tant dans l’apprentissage que dans l’enseignement. La séquence didactique a montré son efficacité pour la construction progressive chez les étudiants de la notion de transformation linéaire (TL), avec le sens et les propriétés qui lui sont propres : la TL apparaît ainsi comme un moyen économique de résoudre des problèmes extérieurs à l’algèbre linéaire, ce qui permet aux étudiants d’en abstraire les propriétés sous-jacentes. Par ailleurs, nous avons pu observer que certains concepts enseignés auparavant peuvent agir comme obstacles à l’unification visée. Cela peut ramener les étudiants à leur point de départ, et le rôle de la TL se résume dans ces conditions à révéler des connaissances partielles, plutôt qu’à guider la résolution. / Introduction to unifying concepts in the teaching of mathematics typically adopts the axiomatic approach. It is not surprising that under these conditions, tasks tend to become more algorithmic in order to help students’ performance and favor apparent transparency of the new concept (Chevallard, 1991). This classical response makes forget the unifying role of the concept and does not encourage its powerful use. In order to improve the learning of a unifying concept, this thesis aimed at studying the relevance of a didactical sequence in the formal training of future engineers, centered on a unifying concept of linear algebra: the linear transformation (LT). The idea of unification and the question of meaning are addressed through the development of problem solving competencies. The sequence of problems to solve is aimed at constructing an abstract concept (the LT) on a domain which is already mathematized, with the intent of abstracting the unifying aspect of the formal notion (Astolfi y Drouin, 1992). Building on the work of Dupin (1995) and Sfard (1991), in mathematics and science education, we have designed didactical situations with elements of modelling, by articulating two ways of conceiving the notion (« procedural » and « structural ») in order to find a safest, more economical and reusable solving strategy. In particular, we have situated the notion in various mathematical domains where it is applicable: arithmetics, geometry, algebra and analysis. The sequence aims at developing connections between different mathematical frameworks, and between various representations of the LT in the different mathematical registers, with the historical development of the notion as a source of inspiration. Moreover, the didactical sequence aims at achieving a balance between the practical aspect of the tasks in the foreseen professional practice and the theoretical aspect required to structure the concepts. The study was conducted in Chile, with engineering students in the first linear algebra course of the program. We had completed a detailed a priori analysis of the sequence in order to reinforce its robustness and prepare for data analysis. With the analysis of answers to the entry questionnaire, team productions to the tasks, and comments received in students interview, we were able to identify the mathematical competencies and the levels of communication (Caron, 2004) put at work in their use of the LT. Results show emergence of the unifying role of the LT, even with students whose problem solving habits in mathematics have been marked by a procedural influence in the teaching and the learning. The didactical sequence showed its effectiveness in the progressive construction by students of the linear transformation concept (LT), with its specific meaning and properties: the TL has appeared as an economical means of solving problems outside of linear algebra, which helped students in abstracting its underlying properties. In contrast, we have also observed that some previously taught concepts could act as obstacles to the desired unification. In these cases, students could revert to their old habits, and their use of the LT would rather reveal their partial understanding than help guide the resolution. / La introducción de conceptos unificadores en la enseñanza de las matemáticas privilegia comúnmente el enfoque axiomático. No es sorprendente que la utilización de este concepto así definido, opera a menudo sobre la algoritmización de tareas para aumentar la eficacia de las resoluciones y promover la transparencia del nuevo objeto enseñado (Chevallard, 1991). Esta respuesta clásica hace sin embargo olvidar el rol unificador y no favorece la utilización de su poder. A fin de mejorar el aprendizaje de un concepto unificador, este trabajo de tesis estudia la pertinencia de una secuencia didáctica en la formación de ingenieros sobre un concepto unificador del álgebra lineal: la transformación lineal (TL). La noción de unificación y la pregunta del sentido lineal son tratadas bajo el ángulo de la adquisición de competencias en situación de resolución de problemas. La secuencia de los problemas a resolver está centrada en el proceso de construir un concepto abstracto (la TL) sobre un dominio ya matematizado, con el fin de retener el aspecto unificador de la noción formal (Astolfi y Drouin, 1992). A partir de resultados de trabajos de didácticas de las ciencias y de las matemáticas (Dupin 1995; Sfard 1991), elaboramos situaciones didácticas en base a elementos de modelización articulando las dos formas de concebir el objeto: “procedural” y “estructural” que permitan encontrar un medio de resolución más seguro, más económico y reutilizable. En particular, tratamos de hacer interactuar las aplicaciones de la noción situándonos en diversos dominios matemáticos; aritmético, geométrico, algebraico y analítico. La secuencia pone atención al desarrollo de conexiones entre diferentes marcos, y entre las representaciones de la TL en los distintos registros, inspirándose en particular del desarrollo histórico de la noción. Por sí misma, la secuencia didáctica, se encarga de mantener un equilibrio entre el lado aplicable de las tareas a su dominio práctico profesional y el lado teórico para ayudar a la estructuración de los conceptos. El estudio concierne a estudiantes chilenos en un primer curso de álgebra lineal. Valoramos un análisis a priori bien detallado para reforzar la secuencia y al mismo tiempo preparar el análisis de los datos. Con el análisis de las respuestas al cuestionario de entrada, de las producciones de los equipos y de los comentarios recibidos en entrevista, pudimos identificar las competencias matemáticas y los niveles de explicitación (Caron, 2004) en la utilización de la TL. Los resultados obtenidos muestran la emergencia del papel unificador de la TL, incluso en aquellos cuyas costumbres en resolución de problemas matemáticos están marcadas por los enfoques procedurales de aprendizaje y de enseñanza. La secuencia didáctica ha mostrado ser eficaz para el desarrollo progresivo de la herramienta lineal (TL) con sentido y propiedades propias: la TL aparece como un medio económico para resolver problemas fuera del álgebra lineal, lo que parece otorgar a los estudiantes una abstracción de las propiedades subyacentes. Por otra parte, observamos que los procesos atados por la enseñanza a ciertos conceptos pueden actuar como obstáculos a la unificación. Eso puede hacer volver a los estudiantes al punto de entrada, y el papel de la TL resulta más bien “revelar” un conocimiento parcial que conducir el proceso de la solución.

Traditionally, the task of determining aircraft position and attitude for automatic control has been handled by the combination of an inertial measurement unit (IMU) with a Global Positioning System (GPS) receiver. In this configuration, accelerations and angular rates from the IMU can be integrated forward in time, and position updates from the GPS can be used to bound the errors that result from this integration. However, reliance on the reception of GPS signals places artificial constraints on aircraft such as small unmanned aerial vehicles (UAVs) that are otherwise physically capable of operation in indoor, cluttered, or adversarial environments. Therefore, this work investigates methods for incorporating a monocular vision sensor into a standard avionics suite. Vision sensors possess the potential to extract information about the surrounding environment and determine the locations of features or points of interest. Having mapped out landmarks in an unknown environment, subsequent observations by the vision sensor can in turn be used to resolve aircraft position and orientation while continuing to map out new features. An extended Kalman filter framework for performing the tasks of vision-based mapping and navigation is presented. Feature points are detected in each image using a Harris corner detector, and these feature measurements are corresponded from frame to frame using a statistical Z-test. When GPS is available, sequential observations of a single landmark point allow the point's location in inertial space to be estimated. When GPS is not available, landmarks that have been sufficiently triangulated can be used for estimating vehicle position and attitude. Simulation and real-time flight test results for vision-based mapping and navigation are presented to demonstrate feasibility in real-time applications. These methods are then integrated into a practical framework for flight in GPS-denied environments and verified through the autonomous flight of a UAV during a loss-of-GPS scenario. The methodology is also extended to the application of vehicles equipped with stereo vision systems. This framework enables aircraft capable of hovering in place to maintain a bounded pose estimate indefinitely without drift during a GPS outage.

L’introduction aux concepts unificateurs dans l’enseignement des mathématiques privilégie typiquement l’approche axiomatique. Il n’est pas surprenant de constater qu’une telle approche tend à une algorithmisation des tâches pour augmenter l’efficacité de leur résolution et favoriser la transparence du nouveau concept enseigné (Chevallard, 1991). Cette réponse classique fait néanmoins oublier le rôle unificateur du concept et n’encourage pas à l’utilisation de sa puissance. Afin d’améliorer l’apprentissage d’un concept unificateur, ce travail de thèse étudie la pertinence d’une séquence didactique dans la formation d’ingénieurs centrée sur un concept unificateur de l’algèbre linéaire: la transformation linéaire (TL). La notion d’unification et la question du sens de la linéarité sont abordées à travers l’acquisition de compétences en résolution de problèmes. La séquence des problèmes à résoudre a pour objet le processus de construction d’un concept abstrait (la TL) sur un domaine déjà mathématisé, avec l’intention de dégager l’aspect unificateur de la notion formelle (Astolfi y Drouin, 1992). À partir de résultats de travaux en didactique des sciences et des mathématiques (Dupin 1995; Sfard 1991), nous élaborons des situations didactiques sur la base d’éléments de modélisation, en cherchant à articuler deux façons de concevoir l’objet (« procédurale » et « structurale ») de façon à trouver une stratégie de résolution plus sûre, plus économique et réutilisable. En particulier, nous avons cherché à situer la notion dans différents domaines mathématiques où elle est applicable : arithmétique, géométrique, algébrique et analytique. La séquence vise à développer des liens entre différents cadres mathématiques, et entre différentes représentations de la TL dans les différents registres mathématiques, en s’inspirant notamment dans cette démarche du développement historique de la notion. De plus, la séquence didactique vise à maintenir un équilibre entre le côté applicable des tâches à la pratique professionnelle visée, et le côté théorique propice à la structuration des concepts. L’étude a été conduite avec des étudiants chiliens en formation au génie, dans le premier cours d’algèbre linéaire. Nous avons mené une analyse a priori détaillée afin de renforcer la robustesse de la séquence et de préparer à l’analyse des données. Par l’analyse des réponses au questionnaire d’entrée, des productions des équipes et des commentaires reçus en entrevus, nous avons pu identifier les compétences mathématiques et les niveaux d’explicitation (Caron, 2004) mis à contribution dans l’utilisation de la TL. Les résultats obtenus montrent l’émergence du rôle unificateur de la TL, même chez ceux dont les habitudes en résolution de problèmes mathématiques sont marquées par une orientation procédurale, tant dans l’apprentissage que dans l’enseignement. La séquence didactique a montré son efficacité pour la construction progressive chez les étudiants de la notion de transformation linéaire (TL), avec le sens et les propriétés qui lui sont propres : la TL apparaît ainsi comme un moyen économique de résoudre des problèmes extérieurs à l’algèbre linéaire, ce qui permet aux étudiants d’en abstraire les propriétés sous-jacentes. Par ailleurs, nous avons pu observer que certains concepts enseignés auparavant peuvent agir comme obstacles à l’unification visée. Cela peut ramener les étudiants à leur point de départ, et le rôle de la TL se résume dans ces conditions à révéler des connaissances partielles, plutôt qu’à guider la résolution. / Introduction to unifying concepts in the teaching of mathematics typically adopts the axiomatic approach. It is not surprising that under these conditions, tasks tend to become more algorithmic in order to help students’ performance and favor apparent transparency of the new concept (Chevallard, 1991). This classical response makes forget the unifying role of the concept and does not encourage its powerful use. In order to improve the learning of a unifying concept, this thesis aimed at studying the relevance of a didactical sequence in the formal training of future engineers, centered on a unifying concept of linear algebra: the linear transformation (LT). The idea of unification and the question of meaning are addressed through the development of problem solving competencies. The sequence of problems to solve is aimed at constructing an abstract concept (the LT) on a domain which is already mathematized, with the intent of abstracting the unifying aspect of the formal notion (Astolfi y Drouin, 1992). Building on the work of Dupin (1995) and Sfard (1991), in mathematics and science education, we have designed didactical situations with elements of modelling, by articulating two ways of conceiving the notion (« procedural » and « structural ») in order to find a safest, more economical and reusable solving strategy. In particular, we have situated the notion in various mathematical domains where it is applicable: arithmetics, geometry, algebra and analysis. The sequence aims at developing connections between different mathematical frameworks, and between various representations of the LT in the different mathematical registers, with the historical development of the notion as a source of inspiration. Moreover, the didactical sequence aims at achieving a balance between the practical aspect of the tasks in the foreseen professional practice and the theoretical aspect required to structure the concepts. The study was conducted in Chile, with engineering students in the first linear algebra course of the program. We had completed a detailed a priori analysis of the sequence in order to reinforce its robustness and prepare for data analysis. With the analysis of answers to the entry questionnaire, team productions to the tasks, and comments received in students interview, we were able to identify the mathematical competencies and the levels of communication (Caron, 2004) put at work in their use of the LT. Results show emergence of the unifying role of the LT, even with students whose problem solving habits in mathematics have been marked by a procedural influence in the teaching and the learning. The didactical sequence showed its effectiveness in the progressive construction by students of the linear transformation concept (LT), with its specific meaning and properties: the TL has appeared as an economical means of solving problems outside of linear algebra, which helped students in abstracting its underlying properties. In contrast, we have also observed that some previously taught concepts could act as obstacles to the desired unification. In these cases, students could revert to their old habits, and their use of the LT would rather reveal their partial understanding than help guide the resolution. / La introducción de conceptos unificadores en la enseñanza de las matemáticas privilegia comúnmente el enfoque axiomático. No es sorprendente que la utilización de este concepto así definido, opera a menudo sobre la algoritmización de tareas para aumentar la eficacia de las resoluciones y promover la transparencia del nuevo objeto enseñado (Chevallard, 1991). Esta respuesta clásica hace sin embargo olvidar el rol unificador y no favorece la utilización de su poder. A fin de mejorar el aprendizaje de un concepto unificador, este trabajo de tesis estudia la pertinencia de una secuencia didáctica en la formación de ingenieros sobre un concepto unificador del álgebra lineal: la transformación lineal (TL). La noción de unificación y la pregunta del sentido lineal son tratadas bajo el ángulo de la adquisición de competencias en situación de resolución de problemas. La secuencia de los problemas a resolver está centrada en el proceso de construir un concepto abstracto (la TL) sobre un dominio ya matematizado, con el fin de retener el aspecto unificador de la noción formal (Astolfi y Drouin, 1992). A partir de resultados de trabajos de didácticas de las ciencias y de las matemáticas (Dupin 1995; Sfard 1991), elaboramos situaciones didácticas en base a elementos de modelización articulando las dos formas de concebir el objeto: “procedural” y “estructural” que permitan encontrar un medio de resolución más seguro, más económico y reutilizable. En particular, tratamos de hacer interactuar las aplicaciones de la noción situándonos en diversos dominios matemáticos; aritmético, geométrico, algebraico y analítico. La secuencia pone atención al desarrollo de conexiones entre diferentes marcos, y entre las representaciones de la TL en los distintos registros, inspirándose en particular del desarrollo histórico de la noción. Por sí misma, la secuencia didáctica, se encarga de mantener un equilibrio entre el lado aplicable de las tareas a su dominio práctico profesional y el lado teórico para ayudar a la estructuración de los conceptos. El estudio concierne a estudiantes chilenos en un primer curso de álgebra lineal. Valoramos un análisis a priori bien detallado para reforzar la secuencia y al mismo tiempo preparar el análisis de los datos. Con el análisis de las respuestas al cuestionario de entrada, de las producciones de los equipos y de los comentarios recibidos en entrevista, pudimos identificar las competencias matemáticas y los niveles de explicitación (Caron, 2004) en la utilización de la TL. Los resultados obtenidos muestran la emergencia del papel unificador de la TL, incluso en aquellos cuyas costumbres en resolución de problemas matemáticos están marcadas por los enfoques procedurales de aprendizaje y de enseñanza. La secuencia didáctica ha mostrado ser eficaz para el desarrollo progresivo de la herramienta lineal (TL) con sentido y propiedades propias: la TL aparece como un medio económico para resolver problemas fuera del álgebra lineal, lo que parece otorgar a los estudiantes una abstracción de las propiedades subyacentes. Por otra parte, observamos que los procesos atados por la enseñanza a ciertos conceptos pueden actuar como obstáculos a la unificación. Eso puede hacer volver a los estudiantes al punto de entrada, y el papel de la TL resulta más bien “revelar” un conocimiento parcial que conducir el proceso de la solución.