Linear Algebra Concepts Research Articles

Computational Simulation: Multidisciplinary Teaching of Dynamic Models from the Linear Algebra Perspective

The technological development has stablish a new learning perspective in the Linear Algebra Concepts. In progress is being made in multidisciplinary teaching in engineering careers, since the proposal is that students learn Math taking into account the benefits provided from the new technologies that analyse different aspects of an engineering system. In this paper are presented a classroom experience in the Computer and Multidisciplinary Laboratory of Basic Sciences where a fluid flow system is modelled when we want to deal with the Eigenvalues and Eigenvectors topic, content that belongs to Algebra and Analytic Geometrical program. The possibilities that the computational systems offers have unchained a reformulation of methodological focus of educative programs, since they propose to intensify learning through the acquisition of skills by students. The developed experience proposes a learning methodology that guarantees an academic knowledge according to the new developments, and assert that Engineering students approach new mathematical subjects trough adequate applications.

Examining Learning Outcomes of Inquiry-Oriented Instruction in Introductory Linear Algebra Classes

This study evaluates the effectiveness of inquiry-oriented instruction in introductory linear algebra classes by comparing the performance of students who learned introductory linear algebra concepts in inquiry-oriented settings (TIMES students) with the students who attended other linear algebra classes (Non-TIMES students). We used the assessment data from 461 students (271 TIMES and 190 Non-TIMES students) which were collected from 19 linear algebra classes at 15 institutes across the country. The linear algebra assessment was given as a post-test to all students, and TIMES students performed significantly better than Non-TIMES students. Overall, the difference in the performance of both groups was statistically significant in the entire assessment, procedural subscale, and conceptual scale of the assessment. In a pair-wise comparison of TIMES and Non-TIMES classes, only those TIMES classes performed significantly better than the Non-TIMES classes on the linear algebra assessment where instructors have more experience with inquiry-oriented teaching.

Polynomials Whose Zeros Are Powers of a Given Polynomial’s Zeros

We describe a straightforward method to obtain a polynomial whose zeros are integer powers of the zeros of a given polynomial. The method uses linear algebra concepts, namely, companion matrices of both scalar and matrix polynomials, and is illustrated by applications.

Williamson theorem in classical, quantum, and statistical physics

In this work, we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show that applying this theorem reveals the normal-mode coordinates and frequencies of the system in the Hamiltonian scenario. A modest introduction of the symplectic formalism in quantum mechanics is presented, using the theorem to study quantum normal modes and canonical distributions of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.

Application of Mathematics in Computer Science

No one escape the learning of mathematics in one way or other, ranging from our kitchen to our journey from earth to Moon or Mars. Mathematics persists everywhere around us. It 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. God used mathematics in creation of the universe in one form or the other. Likewise, Mathematics is the queen of all sciences. Scientists and researchers can not perfectly accomplish their work without including mathematics. Mathematics is the foundation of Computer Science. If one is eager to learn any arena of Computer Science, first he/she has to imbibe a love of Mathematics that will be supportive for progressive learning of the said subject. Mathematics is friendly for analytical skills needed in Computer Science. 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 emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality.

Повышение эффективности преподавания линейной алгебры посредством методики укрупнения дидактических единиц для студентов с ограниченными возможностями здоровья

Presented is enlarged didactic units developed by P.M. Erdniev for teaching mathematics, that improves the ability of students to understand educational material. This effect is achieved as a result of organizing the educational process according to the principles of complementarity of teaching methods and the spatial and temporal combination of interrelated elements of knowledge. Among the basic mathematical disciplines taught at Bauman Moscow State Technical University for students with disabilities the most difficult discipline is Linear Algebra, which is due to its content and various problems with students’ health. Linear Algebra is largely based on the theory from Analytical Geometry that is quite well perceived by students with disabilities. A comparative analysis of these disciplines shows that the enlarged didactic units linking these disciplines together can significantly simplify the study of Linear Algebra. However, due to difficulties with scheduling the time gap between these disciplines can be up to two semesters, which makes it impossible to directly use the enlarged didactic units. To solve this problem, we introduce the supporting discipline "Cognitive technologies for supporting the discipline Linear Algebra" which runs in parallel with Linear Algebra and which is based on the material of Analytical Geometry. Forming its content in close connection with Linear Algebra in accordance with the main principles of the enlarged didactic units, we smoothly lead students to the understanding of abstract concepts of Linear Algebra, reducing the complexity of learning.

Interactionist perspective on negotiation processes of students’ different understandings during small group work on linear algebra

Student group work represents a central learning setting within mathematics programs at the university level. In this study, a theoretical perspective on collaboration is adopted in which the differences between students’ interpretations of a mathematical concept are seen as an opportunity for individual restructuring processes. This so-called interactionist perspective is applied to student group work on linear algebra. The concepts of linear algebra at the university level are characterized by a versatility of different modes of expression and interpretation. For students of linear algebra, the flexible transitions between the different interpretations of linear algebra concepts usually pose a challenge. This study focuses on how students negotiate their different interpretations during group work on linear algebra and how transitions between interpretations might be stimulated or hindered. Video recordings of eight student groups working on a task that required flexible transition between interpretations of homomorphisms were sampled. The recordings were analyzed from an interactionist perspective, focusing on interaction situations in which the participating students expressed and negotiated different interpretations of homomorphisms. The analyses of students’ interactions highlight a phenomenon whereby differences in students’ interpretations remain implicit in group discussions, which constitutes an obstacle to the negotiation process.

Undergraduate Students’ Learning of Linear Algebra Through Mathematical Modelling Routes

Mathematical modelling has acquired relevance at all educational levels in the last decades since integrating this activity in instruction provides significant contexts for improving students’ learning, including in linear algebra courses that have a notable presence in many undergraduate courses from different fields, including engineering and sciences. This paper reports a study aiming to characterise the distinct modelling routes performed by Costa Rican undergraduate students when solving a mathematical modelling task involving the concept of system of linear equations (SLE). In analysing those modelling routes, it was possible to identify their learning of linear algebra concepts and their modelling competencies as well as the associated difficulties that students faced. Data collection included participant observation, with audio recording of the students’ discussions, their written work on the task, and digital files of their work with technology. The results show that non-linear routes are associated with a greater mobilisation of students’ knowledge on SLE concepts and with their development of modelling competencies. The results also highlight the need to improve the students’ competency of validating results, an important step that they did not take, and suggest the need to make technology relevant to the students’ work on modelling tasks.

The Geometric Meaning of Several Concepts in Linear Algebra

The Geometric Meaning of Several Concepts in Linear Algebra

Role of Mathematics in Machine Learning

Context: Machine learning is forming an algorithm that proposes a model that can learn from data and do predictions. It is used to know why something works and why one model is better than another. Machine learning is a built-in mathematical prerequisite. Machine learning is powered by a combination of statistics, calculus, linear algebra, and probability. Objective: This paper aims to examine the mathematics behind machine learning. Method: Different digital repositories are searched for research articles. Studies that are most relevant to the objective of the paper are selected for analysis. Result: Selected studies show that machine learning models are designed to perform calculation operations like matrix manipulation of the data for which linear algebra is used. The concept of probability is the dominant framework for dealing with uncertainty. Learning algorithms use the concept of probability to create an analysis on a given training dataset for creative prediction. Machine learning is intrinsically data-driven. The initial data acquired from different sources is in the form of junks, to search out the right data, statistical analysis is performed. Calculus helps to find the direction of change. In which direction should the unknown variable be changed such that the prediction is more optimal with minimum error? Conclusion: Findings of this paper are that the mathematical rules of designing models of machine learning are based on concepts of linear algebra, calculus, statistics, and probability. Statistics is widely used for the estimation of the value of a population parameter. Calculus tells us how to learn and optimize models. Linear algebra ensures the feasibility of algorithms to run on massive data sets. Probability predicts the likelihood of the occurrence of an event.

A Review On Some Neutrosophic Algebraic Linear Structures

This paper is dedicated to reviewing some of the basic concepts in neutrosophic linear algebra and its generalizations, especially neutrosophic vector spaces, refined neutrosophic, and n-refined neutrosophic vector spaces. Also, this work gives the interested reader a strong background in the study of neutrosophic matrix theory and n-refined neutrosophic matrix theory. We study elementary properties of these concepts such as Kernel, AH-Quotient, and dimension.

On Euler-Lagrange's Equations: A New Approach

A new formalism is proposed to study the dynamics of mechanical systems composed of N connected rigid bodies, by introducing the concept of $6N$-dimensional composed vectors. The approach is based on previous works by the authors where a complete formalism was developed by means of differential geometry, linear algebra, and dynamical systems usual concepts. This new formalism is a method for the description of mechanical systems as a whole and not as each separate part. Euler-Lagrange's Equations are easily obtained by means of this formalism.

On Euler-Lagrange’s Equations: A New Approach

A new formalism is proposed to study the dynamics of mechanical systems composed of N connected rigid bodies, by introducing the concept of $6N$-dimensional composed vectors. The approach is based on previous works by the authors where a complete formalism was developed by means of differential geometry, linear algebra, and dynamical systems usual concepts. This new formalism is a method for the description of mechanical systems as a whole and not as each separate part. Euler-Lagrange's Equations are easily obtained by means of this formalism.

Non-Hermitian physics

A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Generalizing Multi-agent Graph Exploration Techniques

The system of multiple agents working in coordination for a given task has several advantages on faster completion, fault-tolerance, etc. To develop a multi-agent graph exploration technique, the following are defined: 1) the way agents communicate, 2) initial placement of agents and 3) the role of each agent in a systematic search of (unexplored) vertices/edges in the graph. However, the general concepts and requirements are not developed. Hence, we attempt to generalize the multi-agent system for graph exploration. The graph may or may not be known a priori. The proposed representation for the multi-agent system is aimed to provide general conditions for declaring completion and finite time completion applicable to any multi-agent system for graph exploration. Results are proved using linear algebraic concepts on graphs. We also propose modifications in the incidence matrix as a data structure for organized information exchange between the agents. Further, a generic algorithm for the system of multiple agents exploring a graph is developed. The algorithm is based on the proposed generalized framework and modified incidence matrix. Simulation videos show the applicability of proposed generalization in exploring a graph using multiple agents for various combinations of initial placements of agents, search strategy, and interagent communication method. Further, We use the generalization to show the application of the proposed method in multi-robot exploration and map building of an unknown environment represented by a graph.

Developing Mathematical Thinking among Polytechnic Students in Linear Algebra through Peer Tutoring Strategy

Thinking is unique to human beings in a way that it makes them the super power creatures of all time. One of the highlighted aspects in producing human capital of high quality is through the concept of thinking skills as it is use in describing thinking that is purposeful, reasoned, and goal directed. The students’ ability to think mathematically and use it in solving linear algebra problems is among the ultimate goal of teaching mathematics at Polytechnic level. Previous research works indicated that lack of pedagogical approach is among the causes of students' misunderstanding of the basic concepts of linear algebra, which gives them the opportunity to generate their own ideas and become effective problem solvers. This research is therefore targeted at developing students’ mathematical thinking processes in linear algebra through peer tutoring. In this study, a quasi-experimental design was adopted and the samples were purposively selected involving 73 NDII students. A self-developed Linear Algebra Test 1&2were used as the research instruments (pretest and posttest). Peer tutoring approach was found to enhance students’ mathematical thinking processes using mean, SD, and independent t-test statistics. The study suggested that students should form group discussions and practice together at the end of each subject so as to: enhance their thinking abilities and overcome challenges and issues in linear algebra.

An APOS analysis of solving systems of equations using the inverse matrix method

The concept of determinant plays a central role in many linear algebra concepts and is also applied to other branches of mathematics and science. In this study, we focus on the application of determinant and inverse matrix concepts, in solving systems of equations by a group of 116 in-service mathematics teachers who were studying the topic at a Zimbabwean university as part of a first course in linear algebra. The purpose of the study was to explore the role of prerequisite concepts of determinant and matrix inverse in solving systems of equations using the inverse matrix method. The Action-Process-Object-Schema (APOS) theoretical framework was used to analyse the 116 participants’ written responses to a questionnaire together with interview responses of 13 of the participants. Across the group of participants, it was found that only a small number had constructed prerequisite concepts of determinant and that of matrix inverse as Objects. The difficulties they had with prerequisite concepts prevented them from working more efficiently with the higher level concept of solving systems of equations using the inverse matrix method. These results suggest that instructors should help students consolidate their understanding of prerequisite concepts so that they are better placed to work with applications of these concepts.

Teaching Principle and Instrutional Design Analyze in Linear Algebra Course Based on Problem-based Learning

Linear algebra is an important compulsory course for undergraduate students majoring in science and engineering, as well as some economic and management students. However,the teaching effect is not satisfiedin the current teaching and learning.In order to improve the teaching effect, this paper researches the basic teaching principles of problem-based learning in linear algebra course based on the analysis of the current teaching and learning. It finds out that because the highly abstract of this course, the traditional teaching pattern is not suitable for linear algebra curse, while problem-based learning can achieve better result. What’s more, it also provides several examples of instructional design using problem-based learning on the main concepts of linear algebra, such as determinant, matrix and vector space. These examples discuss how to design the problems to help the students for understanding the abstract concepts and construct a stable and precise concept map.The analysis and the examples are meaningful for both teaching theory and practice.

Interpretación de transformaciones lineales en el plano utilizando GeoGebra

En el estudio de conceptos de Algebra Lineal, algunas de las dificultades cognitivas que se observan en los alumnos son derivadas de la falta de articulación de diferentes representaciones de un objeto matemático, especialmente entre el registro simbólico y su correspondiente correlación geométrica en los espacios vectoriales (R2.+.R,*) o (R3.+.R.*), prevaleciendo, en consecuencia, procedimientos de carácter algorítmicos en desmedro de lo conceptual. Fundados en la teoría de la Visualización Matemática, se propone a tales efectos el uso de software GeoGebra como herramienta didáctica en el proceso de aprendizaje de contenidos de algebra Lineal, en caso particular de transformaciones lineales. El carácter dinámico de este aplicativo permite la interpretación geométrica de la transformación, y de conceptos asociados como su núcleo e imagen, a partir de su respectiva matriz asociada, se pretende además que descubran las relaciones entre la singularidad de la matriz, con las dimensiones del núcleo e imagen de la transformación lineal.

Mental constructions in linear algebra

The aim of this paper is on the one hand to discuss from an APOS (Action–Process–Object–Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research instruments and implementation of didactic interviews are discussed, supported by empirical data. Detailed analysis of transcripts from an interview with a student focuses on strategies used in interviewing as well as the mental stages involved in the construction of some linear algebra concepts. Due to the strategies employed, during the interview it is possible to witness the transition between different conceptions. A discussion of the relationships and interactions between different mental structures and mechanisms that play a role in the development of knowledge is provided, including theoretical considerations on the matter. Recommendations about pedagogical strategies are included.