Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.

Sequential development of algebra knowledge : a cognitive analysis

Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students' readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students' understanding of relevant knowledge is discussed.

Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

'The war of Canudos: a Euclidean interpretation.' [Guerra de Canudos: uma leitura euclidiana]

The publication of the book The interior, in 1902, would change the course of thinking about the War of Canudos, who for many years, had been known simply as' the history of Euclid. President Getulio Vargas became interested in the backwoods bloodbath after reading the book avenger-Euclidean. Liked the work he visited the place of occurrence of war promising enjoy the river poured-Barris with the construction of the weir Cocorobo. Euclides da Cunha lived and produced his work in a time of great change in thought, politics and technology. Despite having worked in the press throughout his life, was best known as an engineer, for having exercised the office during the reconstruction of the bridge, in Sao Jose do Rio Pardo. This article aims to illuminate the event of war in light of the Euclidean work. We will examine the trajectory of Euclides da Cunha in journalism. Your learning process to execute the office newsreader and war correspondent, the newspaper O Estado de S. Paul, as well as their reports and work-monument the hinterlands. Resumo: A publicação da obra Os sertões, em 1902, mudaria os rumos do pensamento sobre a Guerra de Canudos, que, por muitos anos, ficara conhecida, simplesmente, como ‘história de Euclides’. O presidente Getúlio Vargas interessou-se pela hecatombe sertaneja após ter lido o livro-vingador euclidiano. Gostou tanto da obra que visitou o lugar de acontecimento da guerra prometendo aproveitar as águas do rio Vaza-Barris com a construção do açude de Cocorobó. Euclides da Cunha viveu e produziu a sua obra em um momento de grandes transformações no pensamento, na política e na tecnologia. Apesar de ter atuado na imprensa ao longo de toda a sua vida, ficou mais conhecido como engenheiro, por ter exercido o ofício, durante a reconstrução da ponte, em São José do Rio Pardo. O presente artigo visa iluminar o acontecimento da guerra à luz da obra euclidiana. Examinaremos a trajetória de Euclides da Cunha no jornalismo. O seu processo de aprendizagem para exercer o ofício de noticiarista e correspondente de guerra, pelo jornal O Estado de S. Paulo, bem como, as suas reportagens e obra-monumento Os sertões.

RcppArmadillo : accelerating R with high-performance C++ linear algebra

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-to-use template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

Object tracking via non-Euclidean geometry : a Grassmann approach

A robust visual tracking system requires an object appearance model that is able to handle occlusion, pose, and illumination variations in the video stream. This can be difficult to accomplish when the model is trained using only a single image. In this paper, we first propose a tracking approach based on affine subspaces (constructed from several images) which are able to accommodate the abovementioned variations. We use affine subspaces not only to represent the object, but also the candidate areas that the object may occupy. We furthermore propose a novel approach to measure affine subspace-to-subspace distance via the use of non-Euclidean geometry of Grassmann manifolds. The tracking problem is then considered as an inference task in a Markov Chain Monte Carlo framework via particle filtering. Quantitative evaluation on challenging video sequences indicates that the proposed approach obtains considerably better performance than several recent state-of-the-art methods such as Tracking-Learning-Detection and MILtrack.