Linear Algebra Concepts Research Articles

An In-Depth Evaluation of Educational Burst Games in Relation to Learner Proficiency

Game-based learning assessments rely on educational data mining approaches such as stealth assessments and quasi mixed methods that help gather data on student learning proficiency. Rarely do we see approaches where student proficiency in learning is woven into the game’s design. Educational burst games (EBGs) represent a new approach to improving learning proficiency by designing fast-paced, short, repetitive, and skill-based games. They have the potential to be effective learning interventions both during instruction in the classroom and during after-school activities such as assignments and homework. Over five years, we have developed two EBGs aimed at improving linear algebra concepts among undergraduate students. In this study, we provide the results of an in-depth evaluation of the two EBGs developed with 45 participants that represent our target population. We discuss the role of EBGs and their design constructs, such as pace and repetition, the effect of the format (2D vs. 3D), the complexity of the levels, and the influence of prior knowledge on the learning outcomes.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

Creating and sharing linear algebra metaphors as an assessment for engaging students beyond the cognitive domain

In this research, we investigated how a summative assessment of creating and sharing metaphors for linear algebra concepts supported undergraduate students’ affective, behavioral, and cognitive engagement. All seven research participants were enrolled in an introductory linear algebra course designed to develop students’ geometric understanding of linear algebra concepts in R2 and R3. Using embodied cognition and an engagement framework, we analyzed students’ written responses and video-taped their focus group discussion. Our findings suggest that this summative assessment (a) privileged mathematical aesthetics and the affective domain of learning, (b) engaged students in binding formal aspects of linear algebra concepts with metaphors that they enacted via embodiment, and (c) was an opportunity to demonstrate learning and higher-order cognition. Thus, illustrating that assessments can focus on aesthetic and affective domains of mathematics while simultaneously integrating serious mathematical cognition. We conclude by offering adaptations of this assessment for other mathematics courses.

An Application Approach to Teaching Linear Algebra

This paper shares insights and experiences in developing and teaching a first-year applied linear algebra course. In addition to incorporating innovative pedagogical techniques like active and inquiry-based teaching and growth grading systems, this course implements modules and other activities which connect linear algebra concepts to applications such as computer graphics and sports analytics. These activities enhance students' experiences by providing them with clear applications of linear algebra concepts and the opportunity to learn accessible modeling techniques used by researchers. These application-centered activities have been incorporated as 50-minute lab modules or as short in-class activities and have been used across multiple modalities including asynchronous and synchronous online classes. This paper provides accessible examples and resources which can help instructors incorporate interesting applications and activities into a linear algebra course. We also provide suggestions and tips for incorporating these activities and pedagogical techniques.

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

Understanding of Linear Algebra Concepts through the TAPPS (Thinking Aloud Pair Problem Solving) Learning Model at the Faculty of Computer Science, Dharma AUB University of Surakarta

Education aims to improve the quality of human life which is technically through lectures. A good lecture produces a positive influence on students' ability. The interaction between lecturers and students in the class is an important role in achieving the goals. Linear Algebra is an important subject in the Mathematic. Many students think that Linear Algebra are difficult. Linear Algebra at the Faculty of Computer Science, Dharma AUB University of Surakarta have not been able to improve students' understanding of Linear Algebra concepts. Because of class that are centered on lecturers. Based on the description above, the author applies the TAPPS (Thinking Aloud Pair Problem Solving) cooperative learning model to increase concept understanding of Linear Algebra. Formulation of the problem in this research: "Is there an increase concept understanding of Linear Algebra after using the TAPPS learning model?". This research is a qualitative descriptive group with the type of Classroom Action Research. The data used are field notes and documentation. Based on the research results, it can be concluded that there is an increase in understanding of the concept of Linear Algebra through the TAPPS (Thinking Aloud Pair Problem Solving) model.

LINEAR ALGEBRA AS THE MATHEMATICAL FOUNDATION OF ARTIFICIAL INTELLIGENCE: CONCEPTS, APPLICATIONS, AND FUTURE PROSPECTS

Linear algebra, a fundamental tool in the field of mathematics, plays an essential part in the mathematical foundation of the groundbreaking arena of Artificial Intelligence (AI). This mathematics branch offers immense possibilities for capturing complex relationships within datasets through matrices and vectors. These mathematical tools facilitate the concise representation of large volumes of data, contributing to efficient data analysis between them, bringing immense value to AI algorithms and models. This research paper investigates the pivotal role of linear algebra as the fundamental mathematical framework underpinning artificial intelligence. This paper also explores the foundational concepts of linear algebra and its direct applications in various AI algorithms, demonstrating how matrices, vectors, and transformations provide the essential mathematical scaffolding for the development and optimization of intelligent systems.

Learning Eigenvalues and Eigenvectors with Online YouTube Resources: A Journey in the Embodied, Proceptual-symbolic, and Formal Worlds of Mathematics

Linear algebra is one of the service mathematics courses that students in many programs take as part of their undergraduate study. Previous studies indicate that linear algebra courses are often challenging for undergraduate students, leading some to seek help from other resources, including online resources such as YouTube. Through a multiple case study, I focused on identifying the opportunities that YouTube resources could offer students to learn about eigenvalues and eigenvectors, which are among the key concepts in introductory linear algebra. I utilized a combination of APOS (Action-Process-Object-Schema) and Tall’s three worlds of mathematics to analyze two highly viewed YouTube videos on this topic. The two analyzed files had different emphases (one more on the embodied world, and the other on the proceptual-symbolic world), catering to students with different levels of understanding of linear algebra. The findings suggest that, overall, these resources could provide students with opportunities to learn about eigen theory in the embodied and proceptual-symbolic worlds, with more focus on action. Lecturers and teacher assistants could benefit from familiarizing themselves with these available resources and considering the possible integration of these resources into their teaching and/or support they provide for students to meet their academic needs and preferences.

Role of Mathematics in Data Science - Machine learning

The mathematical concepts help with identifying patterns and algorithmic development. Mathematics is an important part of many modern scientific areas. It is also a field of major significance in data science. Mainly because of the concepts they propose and the tools they allow, statistics and arithmetic have such an essential role in data science. A specific type of mathematical reasoning is needed for analysing data, in addition to the basic concepts of calculus, discrete mathematics, and linear algebra. To understand how to apply such algorithms in data science, one must have a solid understanding of many different statistical and probability theory ideas. Modern data science methods have a strong mathematical basis for almost all of their techniques, including machine learning.

A linear algebra quiz game as a supplementary learning tool in tertiary-level

This article presents a study investigating the use of games in undergraduate mathematics instruction, with a specific focus on students’ perspectives regarding post-instructional quiz games. The study explores students’ opinions on the implementation of a linear algebra quiz game as a supplementary learning tool. A survey was administered to 78 students to examine their perceptions of the game’s usefulness in preparing for their linear algebra examinations. Students reported high levels of attention and interest while engaging with the game and described it as fun, dynamic, exciting, and beneficial for reviewing linear algebra concepts. The findings suggest that quiz games can be a valuable addition to undergraduate mathematics education, fostering student engagement and providing an enjoyable and effective means for reinforcing course material.

Linear algebra concepts with SageMath for systems engineering students

This work presents an innovative proposal for the teaching and learning of systems of linear equations, graphical interpretation in the plane, in space and orthogonal projection, in the subject of Linear Algebra, mediated by the SageMath software as a technology of learning and knowledge. The proposal was carried out in some Linear Algebra courses in the Systems Engineering career at the University of Cauca, helping to increase its quality, reduce the retention rate and contribute to educational innovation in the university student population. The methodology for the development of this work consisted of the design and application of activities that comprise a set of problems which require the creation of functions in the Python programming language under the SageMath, environment to obtain possible solutions and correlate algebraic and geometric representation registers of the mathematical objects necessary for its solution. We conclude at the end of this educational research the relevance of the integration of technological tools within the classroom in conjunction with active methodological strategies to stimulate the understanding of some concepts of Linear Algebra; since the student can visualize, manipulate and observe these abstract mathematical objects, eliminating unnecessary manual actions and focusing on the analysis of logical deduction for the solution of the proposed activities.

Another Geometric Interpretation of Cramer’s Rule

Summary We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.

Stability analysis for scheduled air traffic flow

PurposeModeling and system performance analysis play an important role in the planning of air traffic system. In particular, this paper aims to concentrate on the modeling and the performance evaluation of air transportation system.Design/methodology/approachIt has shown that system matrices inherent in the airspace network can be acquired based on the service times of flights. This paper deals with a logical modeling which can avoid temporal synchronization conflict of resources. As a result, a class of queuing system is investigated to obtain a representation of the airspace network dynamics using dioid model.FindingsThe analysis of the air traffic system is conducted by solving the system state model. This indicates that it is feasible to use eigenvalues of system model for acquiring characteristics of the considered air transport systems.Practical implicationsThe departure scheduling prototype presented in this paper can offer air traffic controller a decision support tool to build optimal departure sequences for aircraft.Originality/valueThe properties of max-plus algebra described allow us to apply linear algebra concepts like eigenvalue and eigenvectors to obtain a solution to the air traffic system being modeled. By solving and analyzing the dioid model, this paper evaluated some performance measures of airspace network.

Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

For the use of embodied notions in teaching linear algebra, some studies indicate that it is helpful, whereas other studies indicate that it could be problematic or become an obstacle. Hence, additional research is needed. This study is focused on linear (in)dependence and basis, and investigates the relation between their understandings in the embodied and symbolic worlds. We also examine whether students' conceptions in the embodied world can be improved by the instruction emphasizing geometric images, as our previous studies identified some limitations of students' understanding in the embodied world. To address these issues, we designed four tasks aiming to assess students' conceptions of linear (in)dependence, basis, and dimension, and also designed linear algebra lessons emphasizing geometric images of these concepts. These tasks were conducted during the lessons and the data of 38 engineering students was collected. The analysis for the data showed that conceptions in the embodied world was positively associated with conceptions in the symbolic world; however, students' conceptions in the embodied world were not sufficiently improved by the geometric instruction implemented in this study.

Applications of Mathematical in Computer Science

Abstract: Mathematics (The QUEEN’s mother of all Sciences), is the foundation of Computer Science. Mathematics can be perceived in our garden or park from symmetry of leaves, flowers, fruits etc. and by so many examples of Geometry and symmetry can be seen in nature. Scientists and researchers cannot ideally accomplish their work without the inclusion of mathematics. Mathematics is sociable for analytical skills needed in Computer Disciplines like; Concepts of binary number system, Boolean algebra, Calculus, Discrete mathematics, linear algebra, number theory, and graph theory are the most applicable to the subject of computer science with the accessional emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality make the future of mathematics grow endless. Mathematics has been an important intellectual preoccupation of man for a long time. Computer Science as a formal discipline is about seven decades young. Is the almost spontaneous use of computing? In this article, this paper convey to the frontage the many close connections and parallels between the Mother and daughter sciences. The paper underscores the strong interplay and interactions by looking at some exciting contemporary results from number theory and combinatorial mathematics and algorithms of computer science.

Investigating difficulties and enhancing understanding in linear algebra: Leveraging SageMath and ChatGPT for (orthogonal) diagonalization and singular value decomposition.

We explored some common challenges faced by undergraduate students when studying linear algebra, particularly when dealing with algorithmic thinking skills required for topics such as matrix factorization, focusing on (orthogonal) diagonalization and singular value decomposition (SVD). To address these challenges, we introduced SageMath, a Python-based open-source computer algebra system, as a supportive tool for students performing computational tasks despite its static output nature. We further examined the potential of dynamic ChatGPT, an AI-based chatbot, by requesting examples or problem-solving assistance related to (orthogonal) diagonalization or the SVD of a specific matrix. By reinforcing essential concepts in linear algebra and enhancing computational skills through effective practice, mastering these topics can become more accessible while minimizing mistakes. Although static in nature, SageMath proved valuable for confirming calculations and handling tedious computations because of its easy-to-understand syntax and accurate solutions. However, although dynamic ChatGPT may not be fully reliable for solving linear algebra problems, the errors it produces can serve as a valuable resource for improving critical thinking skills.

Violence Detection Approach based on Cloud Data and Neutrosophic Cognitive Maps

Violence has remained a momentous problem since time immemorial. Various scientific studies are conducted in the recent past to identify the stimuli causing violent behavior among the masses and to achieve the target of cloud data protection. Given the inherent ambiguity or indeterminacy in human behaviour, this study in the area of violence detection appears to be effective, as it finds a variety of stimuli and character qualities that contribute to violent conduct among masses. This uncertainty of traits causing violence can easily be seen in surveillance data present over the cloud and also from the data collected using academic research. Therefore, for the purpose of identifying violent behavior we have considered the factors (data) from existing research and from data over clouds. The factors that lead to violent behavior and are identified by algorithms running over clouds are termed as determinate or certain factors. The factors that were not considered and least identified by the cloud algorithms and given less importance are termed indeterminate factors or uncertain factors. The indeterminate factors are also considered based on the expert’s opinion where the experts are not in a condition to provide a clear stance or when they are neutral in their opinion. Tests are performed using Neutrosophic Cognitive Maps (NCMs) to model the violent behavior taking into consideration both determinate and indeterminate factors. Earlier these tests were performed using Fuzzy Cognitive Maps (FCMs) where indeterminate or uncertain factors were not considered. Therefore, we provide a brief comparison between NCMs and FCMs and show how effective NCMs are when we need to consider the uncertainty of concepts while carrying out tests for identifying violent behavior. Later results are obtained by forming a Neutrosophic adjacency matrix which is evaluated using the concepts of linear algebra. The obtained results in the form of 1 ∗ n vector (1 I I I I 1 I 1 I I I I I I I I I I I ) clearly shows the presence of indeterminate factor ‘I’ in the vector which was absent in earlier models when designed using FCMs. This shows how these indeterminate or uncertain factors play a significant role in cultivating violent behavior which was not shown in the previous study. The study is significant since it takes into account factors from cloud data, experts’ opinions, and also from literature, and shows how these factors are taken into consideration at the data level itself so that they will not impact the modeling stage, and machine learning algorithms will perform well because uncertain and indeterminate information is taken care of at training phase itself. Hence uncertainty could be reduced in machine learning algorithms and in the overall recognition of violent behavior.

IN SERVICE TEACHERS MENTAL CONSTRUCTION OF LINEAR ALGEBRA CONCEPTS. AN UNDERGRADUATE CASE STUDY

This study investigated how undergraduate in-service mathematics teachers understand linear algebra concepts. The significance of the study is to reveal the difficulties and type of mental constructions they make when they are solving linear independence concepts by using inspection. The action, process, object, schema (APOS) theory was used to analyse the data collected. A descriptive qualitative case study design was used. The participants were 73 in-service teachers studying a Bachelor of Science Education Honours Degree in mathematics. Data was collected through a structured activity sheet and semi-structured interviews. The results showed that the teachers had an inadequate understanding of linear independence concepts. Many of the students developed the action conceptions of linear algebra as they were engrossed in step-by-step procedures trying to show that given vectors are linearly independent. The findings indicated that the schemata for determinants and the various theorems for determining linear independence promote learning. The modified genetic decomposition proposed has implications for teaching the concept and students should be given different kinds of sets to manipulate so that they can develop their mental constructions at the process level.

Development and Delivery of an Electric Circuits Course Featuring Competency Based Assessment for First Year Engineering

Most common-core first year engineering programs in Canada include an introduction to electric circuits and electromagnetic physics. The launch of the RE-ENGINEERED first year program at the University of Saskatchewan has provided an opportunity to try something different in this arena. The RE-ENGINEERED program includes a “spine” of electric circuit analysis and the related physics that runs through both semesters of the first year.
 The modular and highly integrated structure of the RE-ENGINEERED program has allowed for accelerated courses that take advantage of timely learning in other courses. In the fall term, students are introduced to direct-current, resistive circuit analysis in a six-week, fifteen-contact-hour module. In the winter term, they experience an accelerated physics course which covers the electricity, magnetism, capacitance, and inductance concepts often taught in tandem with basic circuit analysis. The students then finish the winter term with an intensive course on alternating-current circuit analysis.
 The fall term course fully adopts the competency based assessment system of the RE-ENGINEERED program, and uses in-house-developed quizzes and tutorials on the most basic concepts and calculations to scaffold students to solving more complex circuit analysis problems. The course forgoes a hands-on lab component and focuses on circuit simulation using an open-source simulation package. Concurrent math and MATLAB courses introduce required linear algebra concepts just in time for use in the circuit analysis problems.
 This paper describes the development and delivery of the fall term course, including how the learning outcomes were synthesized and then used as the basis for the development of all other aspects of the course to ensure constructive alignment. Instructor and student impressions of the first delivery of the course are presented along how lessons learned will be applied to modify the course for future offerings.

Bidimensional Jordan form and linear discrete dynamical systems with application in red blood cell production

Jordan canonical form (JCF) is one of the most important, and useful, concepts in linear algebra. Mathematics, physics, biology, science and engineering undergraduates often find the first application of real JCF in the discipline of differential equations (continuous models) to solving systems of differential equations. In this work, we apply real JCF to understand, through analytical and numerical methods, the two-dimensional linear discrete dynamical systems. After that, we present a schematic model of red blood cell production. An advantage of discrete models over continuous models is that they readily yield numerical exploration, whether by calculator or computer (in our case we will use Mathematica software). This fact provides us with a valuable pedagogical resource because we can find a simple model to show the links between mathematics and other areas of knowledge without the need for sophisticated mathematical concepts beyond linear algebra. Following Paul Halmos's suggestion, ‘The only way to learn mathematics is to do mathematics’, some classroom exercises are given to better understand the subject, and a project for further student investigations is suggested.