Linear Algebra Concepts Research Articles

Mat-Rix-Toe: Improving Writing Through a Game-based Project in Linear Algebra

The Mat-Rix-Toe project utilizes a matrix-based game to deepen students’ understanding of linear algebra concepts and strengthen students’ ability to express themselves mathematically. The project was administered in three classes using slightly different approaches, each of which included some editing component to encourage the improvement of the students’ mathematical thinking and writing. Differences in the implementation of the project illustrate the benefits and drawbacks of various methods of editing in the mathematics classroom and highlight recommendations for improvements in future implementations of the project.

Effective gene prediction by high resolution frequency estimator based on least-norm solution technique.

Linear algebraic concept of subspace plays a significant role in the recent techniques of spectrum estimation. In this article, the authors have utilized the noise subspace concept for finding hidden periodicities in DNA sequence. With the vast growth of genomic sequences, the demand to identify accurately the protein-coding regions in DNA is increasingly rising. Several techniques of DNA feature extraction which involves various cross fields have come up in the recent past, among which application of digital signal processing tools is of prime importance. It is known that coding segments have a 3-base periodicity, while non-coding regions do not have this unique feature. One of the most important spectrum analysis techniques based on the concept of subspace is the least-norm method. The least-norm estimator developed in this paper shows sharp period-3 peaks in coding regions completely eliminating background noise. Comparison of proposed method with existing sliding discrete Fourier transform (SDFT) method popularly known as modified periodogram method has been drawn on several genes from various organisms and the results show that the proposed method has better as well as an effective approach towards gene prediction. Resolution, quality factor, sensitivity, specificity, miss rate, and wrong rate are used to establish superiority of least-norm gene prediction method over existing method.

Sage 행렬계산기와 선형대수학 Sage 콘텐츠

For over 20 years, the issue of using an adequate CAS tool in teaching and learning of linear algebra has been raised constantly. And a variety of CAS tools were introduced in many linear algebra textbooks. In Korea, however, because of some realistic problems, they have not been introduced in the class and the theoretical aspect of linear algebra has been focused on in teaching and learning of it. In this paper, we suggest Sage as an alternative for CAS tools overcoming the problems mentioned above. And, we introduce full contents for linear algebra and matrix calculator that Sage was used to develop. Taking advantage of them, almost all the concepts of linear algebra can be easily covered and the size of matrices can be expanded without difficulty.

ReFresh: Retaining First Year Engineering Students and Retraining for Success

Student retention and support are key priorities at the University of Toronto Faculty of Applied Science and Engineering, as is evidenced by a first year to second year average retention rate of 91% over the past 4 years. At U of T Engineering, academic standing and registration are determined on a term-by-term basis. As a result, student performance in the fall term can result in obligatory withdrawal from their studies in January, including first year students. While approximately only 4% of the first year class have to withdraw, the impact can be very distressing for the individual student, as his/her plans for the year are disrupted and their confidence shaken. Withdrawal from studies can occur for many reasons such as insufficient academic background from their high school, personal crisis, low resilience and lack of study skills and these struggles can be especially acute for international students. The ReFresh Program gives these students an opportunity to continue their education while learning from their mistakes. A small classroom environment allows students to integrate into a collaborative community, create study groups with classmates, communicate easily with their instructors and TA’s and benefit from constant support from First Year Office staff. This format helps students deal with the personal and academic challenges that prevented them from performing at their best in the fall term, relearn the foundations and key concepts of Calculus, Physics, Computer Programming, Chemistry and Linear Algebra and develop a plan to be successful for the next fall to repeat their first semester. This paper will discuss the structure of the ReFresh program as well as the impact on student success, student development and retention.

Sage와 GeoGebra를 이용한 선형대수학 개념의 Visual-Dynamic 자료 개발과 활용

수학적 개념의 시각화는 단순히 학생들의 개념에 대한 이해를 돕는 것에서 그치는 것이 아니라 학생으로 하여금 시각화 과정을 통하여 스스로 발견하며 깨우치는 교육이 가능하도록 하는 것을 추구한다. 따라서 시각화 자료는 세심한 교육적 고려를 바탕으로 준비되어야 한다. 이를 위하여 본 연구에서는 동적 시각화 및 대수 계산에 적합한 Sage와 GeoGebra를 선택적으로 활용하여, 선형대수학을 수강하는 학생들의 수학적 개념의 이해를 돕기 위해 교재의 순서를 따라가면서, 이론에 추가되는 양방향의 시각적 도구를 개발하였다. 본 논문에서는 이 과정에서 개발된 선형대수학 수업에 필요한 시각적 이해를 돕는 다양한 도구들을 소개한다. Sage와 GeoGebra를 이용한 선형대수학 개념의 시각화에서 얻어진 경험은 다른 대학 수학 강좌뿐만 아니라 중 고등 수학에도 적용될 수 있다. This work started with recent students' conception on Linear Algebra. We were trying to help their understanding of Linear Algebra concepts by adding visualization tools. To accomplish this, we have developed most of needed tools for teaching of Linear Algebra class. Visualizing concepts of Linear Algebra is not only an aid for understanding but also arouses students' interest on the subject for a better comprehension, which further helps the students to play with them for self-discovery. Therefore, visualizing data should be prepared thoroughly rather than just merely understanding on static pictures as a special circumstance when we would study visual object. By doing this, we carefully selected GeoGebra which is suitable for dynamic visualizing and Sage for algebraic computations. We discovered that this combination is proper for visualizing to be embodied and gave a variety of visualizing data for undergraduate mathematics classes. We utilized GeoGebra and Sage for dynamic visualizing and tools used for algebraic calculation as creating a new kind of visual object for university math classes. We visualized important concepts of Linear Algebra as much as we can according to the order of the textbook. We offered static visual data for understanding and studied visual object and further prepared a circumstance that could create new knowledge. We found that our experience on visualizations in Linear Algebra using Sage and GeoGebra to our class can be effectively adopted to other university math classes. It is expected that this contribution has a positive effect for school math education as well as the other lectures in university.

Asset Pricing, Financial Markets, and Linear Algebra

SummaryConcepts from asset pricing and financial markets theory are used to illustrate concepts of linear algebra and linear programming.

Spherical projections with OSXStereonet

OSXStereonet is a Macintosh application to plot spherical data (lines, planes and small circles) in 2D, equal angle or equal area stereonets of any view orientation, or in 3D. The program can also operate on the data calculating angles and intersections between lines or planes, lines (poles) from planes and vice versa, or performing rotations and statistical analyses (cylindrical or conical best fit, mean vectors and contouring) of the data. The stereonet and most of the program's methods are based on linear algebra. We illustrate the functionality of the program as well as its close relation to linear algebra concepts such as transformations through several examples including down plunge projections, map calculations, stress orientations and paleomagnetic reversal tests.

On the equivalence of direct mechanisms and structurally minimal pathways

A reaction-pathway identification procedure has two distinct phases. The first phase enumerates exhaustively the feasible candidate pathways, and the second phase identifies the ultimate feasible pathway or pathways among them. Probably the most efficient way to execute the first phase is to algorithmically generate the networks of feasible candidate pathways from a predefined set of plausible elementary reactions. The available algorithmic methods for this purpose can be roughly grouped into two major classes, one based on graph theory and the other on linear algebra. Both classes of methods consider any chemical reaction system as a network of elementary reactions, thereby implying that the two classes are interrelated. This paper studies the linear algebraic concept termed direct mechanism introduced in the mid-eighties and the graph-theoretical concept termed structurally minimal pathway introduced two decades later. Herein, it has been formally proven that the two concepts are equivalent.

Graphic Representation and Cognition Diagnosis on Concepts of Linear Algebra

The purpose of this study is to integrate pathfinder and item response theory so as to manage concept structures. Concept structure is one important issue of knowledge management as to human knowledge storage. Pathfinder and item response theory are based on graph theory and psychometrics respectively and this integrated method should be feasible to represent concept structures. Besides, fuzzy clustering technique is adopted to provide features of concept structures based on homogeneity of sample. In this study, the empirical data is the assessment of linear algebra for university students. The important concepts of linear algebra consist of subspace, spanning, linear independent, R2 and R3 and many literatures indicate concept structures of linear algebra will influence advanced mathematics. However, little is known about the concept structure and cognition diagnosis on linear algebra. In this study, it shows that lack of concrete examples in general dimensional space will prevent the development of the general theory. There are some limitations for students to use some materials to clarify complicated mathematic concepts perfectly. Most students could not be able to use the geometric insight and apply the Pythagorean of R2 or R3. It shows that methodology of the pathfinder and item response theory will reveal important information of concept structures for students. In addition, fuzzy clustering could distinguish characteristics of concept structures on linear algebra. Finally, some limitations and suggestions as to this study are discussed.

Matrices: un modelo para las fotografías digitales

<p>En este trabajo se resalta la relación que existe entre dos disciplinas, aparentemente muy distintas: la fotografía digital y las matemáticas. En efecto, las imágenes digitales -que todos manejamos hoy en día- se modelizan matemáticamente como matrices. Más aún, todas las manipulaciones sobre fotografías digitales se expresan mediante operaciones matriciales. En concreto, aquí se plantea un problema de compresión de imágenes digitales, basado en la DVS (descomposición en valores singulares) que los alumnos aprenden en clase de Álgebra Lineal. De esta forma, los alumnos descubren por sí mismos algunas realidades importantes que los docentes queremos transmitirles: en primer lugar, que los conocimientos teóricos adquiridos en Álgebra Lineal tienen una aplicación directa a la tecnología que les rodea diariamente en este mundo digital; y en segundo lugar, cómo la modelización matemática es una poderosa herramienta para resolver problemas prácticos reales</p>

Using an economics model for teaching linear algebra

We present an approach for teaching linear algebra using models. In particular, we are interested in analyzing the modeling process under an APOS perspective. We will present a short illustration of the analysis of an economics problem related to production in a set of industries. This problem elicits the use of the concepts of linear combination, linear independence, among other linear algebra concepts related to vector space. We describe cycles of students’ work on the problem, present an analysis of the learning trajectory with emphasis on the constructions they develop, and discuss the advantages of this approach in terms of students’ learning.

Construct Concept Structure for Linear Algebra Based on Cognition Diagnosis and Clustering with Mahalanobis Distances

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. The purpose of this study is improved Fuzzy C-Means algorithm based on Mahalanobis distance to identify concept structure for Linear Algebra. In addition, Concept structure analysis (CSA) could provide individualized knowledge structure. CSA algorithm is the major methodology and it is based on fuzzy logic model of perception (FLMP) and interpretive structural modeling (ISM). CSA could display individualized knowledge structure and clearly represent hierarchies and linkage among concepts for each examinee. Each cluster of data can easily describe features of knowledge structures. The results show that there are five clusters and each cluster has its own cognitive characteristics. In this study, the author provide the empirical data for concepts of linear algebra from university students. To sum up, the methodology can improve knowledge management in classroom more feasible. Finally, the result shows that Algorithm based on Mahalanobis distance has better performance than Fuzzy C-Means algorithm.

Visualization approach in teaching process of linear algebra

The aim of this study is to present a way of visualization to teaching and learning linear algebra at a level suitable for freshman and sophomore students. In this study, it has been shown how to use visualization approach through vector geometry in teaching and learning linear algebra concepts. This study concentrates basic linear algebra concepts such as determinant, vector subspace, linear dependence and linear independence, basis and dimension, linear transformation, Eigen value, dot product and projection.

Identifying the Mastery Concepts in Linear Algebra by Using FCM-CM Algorithm

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters by employing Mahalanobis distance in objective function, however, both of them need to add some constrains for Mahalanobis distance. In this paper, the authors’ improved Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is used to identify the mastery concepts in linear algebra, for comparing the performances with other four partition algorithms; FCM-M, GG, GK, and FCM. The result shows that FCM-CM has better performance than others.

Student learning of basis, span and linear independence in linear algebra

One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have major difficulties with concepts of span and linear independence which form the requirements for a set of vectors to form a basis. In this research we applied a theoretical framework based on Tall's three worlds of mathematics learning and action-process-object-schema (APOS) theory to the learning of the concept of basis by a group of second year university students. The results suggest that an emphasis on matrix processes may not help students understand the concept, and embodied, visual ideas that could be valuable were usually lacking.

Time modes and nonlinear systems

A vector-space implementation of the equation that has been known as Hamilton's Law of Varying Action is used with a non-virtual perspective to generate direct solutions to nonlinear initial value response problems in the time domain. This novel vector-space perspective unleashes the power of linear algebra to provide new capabilities with Hamilton's Law of Varying Action and new insights into nonlinear dynamics. In particular, this perspective permits the use of unconstrained temporal-basis-functions without the need to augment Hamilton's Law of Varying Action with Lagrange multipliers, and provides a linearly independent set of spatiotemporal functions spanning the solution space of the response. These spatiotemporal functions are descriptively denoted as fundamental-time-modes. By providing this basis of fundamental-time-modes, the concepts of linear algebra can be taken advantage of to accomplish model reduction and superposition for nonlinear systems. In particular, classical model reduction techniques are used to eliminate non-dominant fundamental-time-modes from the solution process, resulting in model reduction in the time domain. In addition, using the novel concept of eigen-direction iteration, linearly independent fundamental-time-mode response trajectories along eigen-directions are superposed to yield new response trajectories for nonlinear systems. These capabilities of the vector-space approach are demonstrated for nonlinear systems exhibiting nonlinear normal mode response. The article concludes with an Appendix A entitled “The Law of Evolutionary Energy, Hamilton's Law of Varying Action and the Principle of Virtual Work”, providing essential insight to the non-virtual working equation of the authors’ previous articles and the current article, referred to as Hamilton's Law of Varying Action, to distinguish it from virtual concepts and virtual variational principles of classical dynamics. It is shown that this particular equation referred to as Hamilton's Law of Varying Action is a special case of the Law of Evolutionary Energy which inherently encompasses the First Law of Thermodynamics involving quantifiable real changes (alterations) along the real dynamic path, the application of which to mechanical systems yields both the current non-virtual working form of Hamilton's Law of Varying Action and the original form of the only enunciated (non-virtual) “law of varying action” by Hamilton which are not the same expressions.

TopoToolbox: A set of Matlab functions for topographic analysis

TopoToolbox contains a set of Matlab functions that provide utilities for relief analysis in a non-Geographical Information System (GIS) environment. The tools have been developed to support the work flow in combined spatial and non-spatial numerical analysis. They offer flexible and user-friendly software for hydrological and geomorphological research that involves digital elevation model analysis and focuses on material fluxes and spatial variability of water, sediment, chemicals and nutrients. The objective of this paper is to give an introduction to the linear algebraic concept behind the software that employs sparse matrix computations for digital elevation model analysis. Moreover, we outline the functionality of the toolbox. The source codes are freely available in Matlab language on the authors' webpage ( physiogeo.unibas.ch/topotoolbox).

Student connections of linear algebra concepts: an analysis of concept maps

This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students’ concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.

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 takes advantage of the properties of the transfer operator norm on a finite time horizon. Theoretical considerations are complemented by numerical examples.

A framework for mathematical thinking: the case of linear algebra

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the learning and teaching of basic linear algebra concepts. This article highlights some of the overall findings of this research with regard to student understanding of these concepts, and suggests applications of the framework for learning and teaching in undergraduate mathematics classrooms.