Linear Algebra Concepts Research Articles

Some Results on Shannon Wavelets and Wavelets Frames

In this article, Shannon wavelets are studied together with their properties and we focused on Shannon wavelets and scalar functions through linear algebraic concept such as linearly independent, nested properties and direct product etc. Also, we generated Laguerre wavelets frames and discussed some results on it.

On Modified Self Organizing Feature Maps

This paper presents optimization of self-organizing feature maps by adjusting tunable parameters and in the iterative process by utilizing linear algebra concepts. A gradient rule is applied on the weights matrix singular value decomposition values during convergence phases to optimize the algorithm while achieving sufficient statistical accuracy. Tunable parameters such as the learning rate, and neighborhood radius are adjusted to support the learning algorithm. The algorithm presented herein is tested on the self-organization of the standard Iris dataset, in the literature.

As representações algébrica e geométrica na aprendizagem de mudança de base

Na aprendizagem de conceitos de Álgebra Linear os alunos do ensino superior tendem a manifestar dificuldades devido à sua natureza abstrata, o que podem ser minimizadas caso se explore a conexão entre as diferentes representações de um mesmo conceito. Recorrendo a um ensino que explorou as representações algébrica e geométrica de conceitos de Álgebra Linear, procuramos averiguar como os alunos compreendem as transformações entre as diferentes representações, algébrica e geométrica, dos conceitos de coordenadas e mudança de base na resolução de uma tarefa após esse ensino. Adotamos uma abordagem qualitativa e interpretativa na procura de compreender como os alunos operacionalizam os conceitos de coordenadas e de mudança de base nessas representações. A análise das respostas dos alunos à tarefa proposta aponta para um melhor desempenho da transição da representação algébrica para a geométrica do que nas transformações realizadas dentro da própria representação algébrica. As maiores dificuldades relacionam-se com a interpretação equivocada dos conceitos envolvidos, a linguagem simbólica e com o uso inadequado de conceitos e procedimentos advindos da Geometria Analítica ou já abordados na própria Álgebra Linear.

Varieties in the use of geometry in the teaching of linear algebra

Geometry is used in different ways in the teaching of linear algebra. In this paper, I offer a typology of these ways, which I call varieties, and address three central questions. The first question is, What varieties of use of geometry in the teaching of linear algebra exist? This question is addressed through an analysis of six linear algebra textbooks, republished in multiple editions in the last decade or so. The analysis resulted in seven varieties, which can be used by researchers to investigate systematically the use of geometry in the teaching of linear algebra, in and outside the classroom. The second question is, What are the different impacts of these varieties on students’ ability to accomplish the following: (a) abstract geometrically-based linear-algebraic concepts into general representations? (b) Extend geometrically-based linear-algebraic concepts to their counterparts in other models, such as space of polynomials, functions, and matrices? This question is addressed through analyses of a sample of various results reported in the literature. The third question is, What might account for these impacts? A conceptual basis underlying the results discussed in this paper is theorized.

PROBLEMA DAS LUZES APAGADAS

Linear Algebra presents a very important role in the areas of accuracy, and through it we can show its utility in modeling a problem that involves a simple game of erasing and lighting lights, after modeling the problem, we will solve it by implementing an algorithm of elimination of Gauss in Scilab (Scilab is free and open source software, focused on numerical computation similar to Matlab). The game, Lights Out, is a famous game of the 90's, which consists of 25 keys illuminated and arranged in the form of a 5x5 matrix, where it has an initial state and we must delete all keys by pressing a correct sequence of keys, this sequence will be provided by the program implemented in Scilab, ie we will have an immediate solution to erase all lights in the game. The modeling uses Linear Algebra concepts: matrices, determinants and linear systems, together with the concept of the set of the rest of the division by 2(ℤ2).

Norm method to define and evaluate robustness of self-compacting concrete due to component quantity variations

Presently, self-compacting concrete (SCC) has been widely used and has many incomparable advantages compared to the traditional vibrated concrete. While it also has many disadvantages among which the robustness of SCC seems to be a serious problem at the production stage. In this study, both the fresh and hardened properties of a low binder content SCC were studied in the cases of the quantity variations of components. Meanwhile, a new robustness assessment method based on the concepts of linear algebra (using norm analysis of vectors) was proposed in order to define and evaluate the robustness of SCC. The results show that the robustness of the hardened properties is higher than that of the fresh-related properties. Additionally, by adopting the proposed method, the robustness of the SCC mixtures can be described and evaluated in a more intuitive way and both the variations of moisture content and aggregate quantities, such as the volume fraction and the gradation, strongly affect the fresh properties of the low-binder SCC. Finally, a good consistency was found between the proposed method and a previous robustness evaluation method thus proving that the proposed methods can be regarded as a concise and effective tool to define and evaluate the robustness of the SCC mixtures.

Discrete-Time, Linear Periodic Time-Varying System Norm Estimation Using Finite Time Horizon Transfer Operators

The norm is one of the fundamental concepts of linear algebra and functional analysis. The notion of the norm is often employed in engineering, e.g. in control engineering, where main application is calculating the norm of the transfer function. Unfortunately existing methods are applicable for systems that can be described using Laplace transform, i.e. linear time-invariant (LTI) systems. An operational equivalent of the transfer function for linear time-varying systems is transfer operator. The transfer operator defined for finite time horizon can be described by finite dimensional matrix. Although for infinite time horizon the operator is infinite dimensional. In the paper a method for norm estimation of transfer operator defined on infinite time horizon is proposed. The method is applicable for linear time-varying, discrete-time systems given in general state-space form. The method take advantage of the properties of the transfer operator norm on a finite time horizon. Theoretical considerations are complemented by numerical examples.

Phase Retrieval by Linear Algebra

The null vector method, based on a simple linear algebraic concept, is proposed as an initialization method for nonconvex approaches to the phase retrieval problem. For the stylized measurement with random complex Gaussian matrices, a nonasymptotic error bound is derived, stronger than that of the spectral vector method. Numerical experiments show that the null vector method also has a superior performance for the realistic measurement of coded diffraction patterns in coherent diffractive imaging.

A characterization of a unified notion of mathematical function: the case of high school function and linear transformation

As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. An overarching goal of this study was to examine how linear algebra students see linear transformation as related to high school function. Analysis of these data led to a characterization of student responses into three categories of mathematical structures used to discuss function: properties, computations, and a series of five interrelated clusters of metaphorical expressions. In this paper, we use this analytic framing for exploring the question: to what extent does each of the students in this study have a unified concept image of function across two mathematical contexts, high school algebra and their study of linear algebra? We found that students who expressed a unified notion of function used metaphorical language to bridge any gaps between the notion of function from high school and the notion of linear transformation from linear algebra. We conjecture that the framing of computations, properties, and metaphorical clusters could be extended to discussions of functions in contexts with other mathematical domains. Future research that further examines the extent to which undergraduate students develop a unified concept image of function could then lead to various design research efforts aimed at explicitly fostering such a unified understanding.

Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

ABSTRACTIn this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.

Developing conceptual understanding and definitional clarity in linear algebra through the three worlds of mathematical thinking

Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall’s three worlds (embodied, symbolic and formal) of mathematical thinking, and use a framework combining these with Dubinsky’s Action, Process, Object and Schema (APOS) theory to analyse students’ resulting levels of understanding. Through interviews and analysis of test and examination scripts, we investigate students’ understanding of the basic concepts of linear algebra, their ability to use and explain these concepts and their relationship to definitional clarity. The results show that, while students tend not to learn definitions by rote and can be imprecise when expressing them in words, they seem to understand the concepts, can talk sensibly about them and are able to use their essential features in solving problems.

Mathematical modelling in engineering: A proposal to introduce linear algebra concepts

The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts: span and spanning set. This was applied to first year engineering students. Results suggest that this type of instructional design contributes to the construction of these mathematical concepts and can also favour first year engineering students understanding of key linear algebra concepts and potentiate the development of higher order skills.

Exploiting data representation for fault tolerance

Incorrect computer hardware behavior may corrupt intermediate computations in numerical algorithms, possibly resulting in incorrect answers. Prior work models misbehaving hardware by randomly flipping bits in memory. We start by accepting this premise, and present an analytic model for the error introduced by a bit flip in an IEEE 754 floating-point number. We then relate this finding to the linear algebra concepts of normalization and matrix equilibration. In particular, we present a case study illustrating that normalizing both vector inputs of a dot product minimizes the probability of a single bit flip causing a large error in the dot product's result. Furthermore, the absolute error is either less than one or very large, which allows detection of large errors. Then, we apply this to the GMRES iterative solver. We count all possible errors that can be introduced through faults in arithmetic in the computationally intensive orthogonalization phase of GMRES, and show that when the matrix is equilibrated, the absolute error is bounded above by one.

Vector Spaces of New Special Magic Squares: Reflective Magic Squares, Corner Magic Squares, and Skew-Regular Magic Squares

The definition of a regular magic square motivates us to introduce the new special magic squares, which are reflective magic squares, corner magic squares, and skew-regular magic squares. Combining the concepts of magic squares and linear algebra, we consider a magic square as a matrix and find the dimensions of the vector spaces of these magic squares under the standard addition and scalar multiplication of matrices by using the rank-nullity theorem.

MODELING IN MAPLE AS THE RESEARCHING MEANS OF FUNDAMENTAL CONCEPTS AND PROCEDURES IN LINEAR ALGEBRA

The article is devoted to binary technology and "fundamental training technology." Binary training refers to the simultaneous teaching of mathematics and computer science, for example differential equations and Maple, linear algebra and Maple. Moreover the system of traditional course of Maple is not performed. The use of the opportunities of Maple-technology in teaching mathematics is based on the following fundamental concepts of computer science as an algorithm,program, a linear program, cycle, branching, relative perators, etc. That’s why only a certain system of command operators in Maple is considered. They are necessary for fundamental concepts of linear algebra and differential equations studying in Maple-environment. Relative name - "the technology of fundamental training" reflects the study of fundamental mathematical concepts and procedures that express the properties of these concepts in Maple-environment. This article deals with the study of complex fundamental concepts of linear algebra (determinant of the matrix and algorithm of its calculation, the characteristic polynomial of the matrix and the eigenvalues of matrix, canonical form of characteristic matrix, eigenvectors of matrix, elementary divisors of the characteristic matrix, etc.), which are discussed in the appropriate courses briefly enough, and sometimes are not considered at all, but they are important in linear systems of differential equations, asymptotic methods for solving differential equations, systems of linear equations. Herewith complex and voluminous procedures of finding of these linear algebra concepts embedded in Maple can be performed as a result of a simple command-operator. Especially important issue is building matrix to canonical form. In fact matrix functions are effectively reduced to the functions of the diagonal matrix or matrix in Jordan canonical form. These matrices are used to rise a square matrix to a power, to extract the roots of the n-th degree of a square matrix, to calculate matrix exponent, etc. The author creates four basic forms of canonical models of matrices and shows how to design matrices of similar transformations to these four forms. We introduce the programs-procedures for square matrices construction based on the selected models of canonical matrices. Then you can create a certain amount of various square matrices based on canonical matrix models, it allows to use individual learning technologies. The use of Maple-technology allows to automate the cumbersome and complex procedures for finding the transformation matrices of canonical form of a matrix, values of matrices functions, etc., which not only saves time but also attracts attention and efforts on understanding the above mentioned fundamental concepts of linear algebra and procedures for investigation of their properties. All these create favorable conditions for the use of fundamental concepts of linear algebra in scientific and research work of students and undergraduates using Maple-technology.

A Framework to Gauge Mathematical Understanding: A Case Study on Linear Algebra Concepts

AbstractA framework is arrived at to help gauge the level of mathematical understanding that could be suggested by an examination of a student’s written responses to mathematical problems. This framework has been to make conclusions on two students’ level of mathematical understanding and use of mathematical symbolism to communicate their understanding of operations on matrices in an external form to others. It was found that for those two students mathematical symbolism served largely as an instrumental role to keep track of their thinking. At times the incorrect use of a mathematical symbol, for example the equal to sign, led the student to make illogical conclusions. Further it was found that drawing students’ attention to their incorrect use of mathematical symbolism and the lack of explicit communication of their thinking to others could help to clarify and even modify existing schema that they use to find possible solutions to given tasks.

Organizational climate research: a proposed approach focused on banking institutions

Purpose– The use of the general satisfaction index (GSI) concept is a great way to measure the organizational climate. However, the traditional concept has an inaccurate calculation approach. In order to review this concept, the purpose of this paper is to propose a methodology to obtain this index through a more appropriate mathematical approach, supported by some linear algebra concepts. For this purpose, an organizational climate survey was carried out in a bank agency to exemplify application of this method. Each step of the application was detailed, as well as the intermediate results achieved. Among its principal benefits are: improved accuracy to obtain the GSI; the possibility of a comparison with a minimum required rate for the decision-making process; and the possibility of identifying the order of priority of the organizational climate dimensions that mostly need improvements.Design/methodology/approach– The working method was divided into three parts. The first part consists of the collection of data on employee satisfaction through a questionnaire based on Luz (2003) considerations and presented in a research performed by Maciel (2011), which uses 27 pertinent questions directed to the context of a banking system. These questions were answered on a scale from 0 to 4, where 0 means a very negative satisfaction condition and 4 a very positive one. The second part dealt with the application of the methodology for calculating GSI. The third and final part consisted of proposing improvements to increase the GSI of the studied bank agency.Findings– In general, the proposed method has proved to be a great ally to climate research and also to identify the need for improvements in banking context. Finally, the use of the 5W2H tool has proved important to improve weak points observed in the climate analysis, providing an initial guide to raise the agency’s GSI and achieve the goals pursued by the company. It is noteworthy that, due to the fact that improvement actions have been inferred based on direct observation, there is still a need for further analysis and/or debates about the impact and viability of these solutions.Originality/value– The proposed method turns GSI into a more accurate and appropriate tool, making it even more valuable in decision-making process faced by human resources departments focussed on improving or maintaining the organizational climate. To illustrate the application of the new approach regarding this tool, an organizational climate study in a bank branch is conducted.

Interference Networks With no CSIT: Impact of Topology

We consider partially connected $K$ -user interference networks, where the transmitters have no knowledge about the channel gain values, but they are aware of network topology. We introduce several linear algebraic and graph theoretic concepts to derive new topology-based outer bounds and inner bounds on the symmetric degrees-of-freedom of these networks. We evaluate our bounds for two classes of networks to demonstrate their tightness for most networks in these classes, quantify the gain of our inner bounds over benchmark interference management strategies, and illustrate the effect of network topology on these gains.

Estimation of integral curves from high angular resolution diffusion imaging (HARDI) data

We develop statistical methodology for a popular brain imaging technique HARDI based on the high order tensor model by Özarslan and Mareci [10]. We investigate how uncertainty in the imaging procedure propagates through all levels of the model: signals, tensor fields, vector fields, and fibers. We construct asymptotically normal estimators of the integral curves or fibers which allow us to trace the fibers together with confidence ellipsoids. The procedure is computationally intense as it blends linear algebra concepts from high order tensors with asymptotical statistical analysis. The theoretical results are illustrated on simulated and real datasets.This work generalizes the statistical methodology proposed for low angular resolution diffusion tensor imaging by Carmichael and Sakhanenko [3], to several fibers per voxel. It is also a pioneering statistical work on tractography from HARDI data. It avoids all the typical limitations of the deterministic tractography methods and it delivers the same information as probabilistic tractography methods. Our method is computationally cheap and it provides well-founded mathematical and statistical framework where diverse functionals on fibers, directions and tensors can be studied in a systematic and rigorous way.

Some issues about the introduction of first concepts in linear algebra during tutorial sessions at the beginning of university

Certain mathematical concepts were not introduced to solve a specific open problem but rather to solve different problems with the same tools in an economic formal way or to unify several approaches: such concepts, as some of those of linear algebra, are presumably difficult to introduce to students as they are potentially interwoven with many types of difficulties as formal ones and far away from the actual knowledge of the students. The purpose of this paper is to propose a methodology for studying the introduction of such concepts in linear algebra during tutorial sessions at the beginning of university, the wording of the concepts being yet presented during lectures. For this purpose, we amend a general methodology of Pariès, Robert and Rogalski inside the general framework of Activity Theory. This methodology lets us take into account several specificities of these concepts and studies the mathematical activity the teacher organises for students and the way he manages the relationship between students’ actual activities and mathematical tasks. We also present an implementation of this methodology based on a French university course to illustrate our approach and discuss its possibilities.