Topics In Linear Algebra Research Articles

Student Co-Creation of Resources in a Second Year Linear Algebra Course

In this project, we investigated how designing their own tutorial questions could help level 2 mathematics students negotiate the often-challenging transition from level 1 in the Scottish curriculum. One student from the level 2 linear algebra course at the University of Aberdeen volunteered to create practice questions on topics of linear algebra they selected. Through analysis of the participant’s interview, we showed, in accordance with the literature, that the activity benefited them in terms of study skills (motivation, focus, independence), in terms of mathematics learning (deep learning and mathematical knowledge construction). We also found that the activity had contributed to improving the volunteer’s resilience to learning mathematics, and their sense of legitimate participation in the mathematics community. This has not been discussed in the literature and is a significant finding considering the known difficulty for mathematics students, in particular women, to feel they ‘belong’. This research further suggests that if the activity took place in groups, it could also help develop a peer-support community which students could rely on to negotiate the transition to level 2.

Enhancing Linear Algebra Learning through Computational Thinking: A Project-Based Approach

This article discusses the integration of computational thinking concepts, including algorithm, coding, abstraction, decomposition, debugging, and pattern recognition, into a Linear Algebra course in a community college in the fall of 2022. Through the implementation of project-based learning (PBL), we aimed to enhance students' understanding of linear algebra topics while familiarizing them with essential computational thinking concepts. A pre-and post-survey assessed the students' familiarity with the concepts. The results indicated a significant improvement in the students' understanding of computational thinking concepts and their application in linear algebra.

Generative Adversarial Networks: A Brief History and Overview

Over the past decade, research in the field of Deep Learning has brought about novel improvements in image generation and feature learning; one such example being a Generative Adversarial Network. However, these improvements have been coupled with an increasing demand on mathematical literacy and previous knowledge in the field. Therefore, in this literature review, I seek to introduce Generative Adversarial Networks (GANs) to a broader audience by explaining their background and intuition at a more foundational level. I begin by discussing the mathematical background of this architecture, specifically topics in linear algebra and probability theory. I then proceed to introduce GANs in a more theoretical framework, along with some of the literature on GANs, including their architectural improvements and image-generation capabilities. Finally, I cover state-of-the-art image generation through style-based methods, as well as their implications on society.

Linear functionals on hypervector spaces

The study of linear functionals, as an important special case of linear transformations, is one of the key topics in linear algebra and plays a significant role in analysis. In this paper we generalize the crucial results from the classical theory and study main properties of linear functionals on hypervector spaces. In this way, we obtain the dual basis of a given basis for a finite-dimensional hypervector space. Moreover, we investigate the relation between linear functionals and subhyperspaces and conclude the dimension of the vector space of all linear functionals over a hypervector space, the dimension of sum of two subhyperspaces and the dimension of the annihilator of a subhyperspace, under special conditions. Also, we show that every superhyperspace is the kernel of a linear functional. Finally, we check out whether every basis for the vector space of all linear functionals over a hypervector space V is the dual of some basis for V.

Making connections among representations of eigenvector: what sort of a beast is it?

Many studies provide insights into students’ conceptions of various linear algebra topics and difficulties they face with multiple modes of thinking needed for conceptualization. While it is important to understand students’ initial conceptions, students’ transfer of learning of these conceptions to subsequent courses can provide additional information to structure meaningful curricular materials. This study explores physics students’ transfer of learning of eigenvalues and eigenvectors from prerequisite experiences to quantum mechanics. Data analysis focused on three task-based interviews with undergraduate students, observations of physics courses, and students’ course artifacts. Existing studies on students’ conceptions of linear algebra topics indicate the necessity of developing flexible shifts between different modes of thinking in order to grasp linear algebra. This study’s participants, who had initial learning experiences of linear algebra, were also observed to struggle with such shifts prior to quantum courses. It seems that various contexts in quantum courses, and explicit instructional methods, provided opportunities for students to enhance this initial learning of eigenvalues and eigenvectors. In particular, the explicit reasoning of one of the quantum courses’ instructors concerning the choice of certain representations during problem solving in class, seemed to facilitate students’ construction of similarities, thus providing evidence for actor-oriented transfer. Results of this study align with goals for recently developed instructional materials and interventions that emphasize opportunities for students to inquire and connect multiple modes of thinking.

The use of the wxMaxima linear algebra module on Gauss elimination lesson for mathematics education students

Students’ difficulties in solving problems in Linear Algebra course could be assisted by the use of wxMaxima Linear Algebra module. This paper focuses on the learning on Gauss elimination topic using the wxMaxima Linear Algebra module. The wxMaxima Linear Algebra module has been developed using the ADDIE development model. The participants were 18 mathematics education students of UIN Ar-Raniry Banda Aceh. This study used the lecturer’s self-reflection sheet to see the relevance of the lesson plan and the learning implementation. The learning process implemented direct instruction model and the time allocated for the class session was 150 minutes. The learning activities began with the explanation about elementary row operations (ERO) and Gauss elimination (GE) by the lecturer and followed by solving problems manually solving using the paper-pencil method. Furthermore, the solutions obtained manually were checked using the wxMaxima software. Learning activities were then continued to the concept of Gauss Jordan elimination method. Although the students spent much time to solve the Gauss elimination problems manually, overall the learning process went well as the time allocated in the lesson plan. The findings suggest that the wxMaxima Linear Algebra module could be used for other linear algebra topics.

University Students’ Solution Processes in Systems of Linear Equation

The focus of the study is system of linear equations which is one of the important topics of linear algebra. The aim is to investigate how university students’ skills in the process of solving systems of linear equations by focusing on how they perform operations with matrices. This study is a case study based on a non-positivist paradigm with interpretivist approach. Findings indicate that candidates face differing levels difficulties in conceptualizing and doing operations in their solutions.

Basic mathematics for economics, business, and finance

This book can help overcome the widely observed math-phobia and math-aversion among undergraduate students in these subjects. The book can also help them understand why they have to learn different mathematical techniques, how they can be applied, and how they will equip the students in their further studies. The book provides a thorough but lucid exposition of most of the mathematical techniques applied in the fields of economics, business and finance. The book deals with topics right from high school mathematics to relatively advanced areas of integral calculus covering in the middle the topics of linear algebra; differential calculus; classical optimization; linear and nonlinear programming; and game theory. Though the book directly caters to the needs of undergraduate students in economics, business and finance, graduate students in these subjects will also definitely find the book an invaluable tool as a supplementary reading. The website of the book – ww.emeacollege.ac.in/bmebf – provides supplementary materials and further readings on chapters on difference equation, differential equations, elements of Mathematica®, and graphics in Mathematica®, . It also provides materials on the applications of Mathematica®, as well as teacher and student manuals.

Principles of linear algebra with Maple

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

Teaching and Learning using Mathematics Software “The New Challenge”

Teaching and learning in mathematics curriculums in Universities by using mathematical software can be a difficult and demanding task, especially for novice learners. This paper presents efficient mathematical tool for teaching and learning of Linear Algebra courses. MAPLE software tool was used for teaching and learning of parts of the Linear Algebra course. Using MAPLE in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. In this paper we provide mathematical examples by using mathematical software and necessary steps as evidences that didactically it increases mathematical skills. By using interactive Maple worksheets and animated graphics, students can find the opportunity of numerous experiments that provide well understanding. Further the use of Maple provide conceptual and meaningful understanding for the student, several Maple can be designed to see, geometrical application of Linear Algebra topics. Indeed, utilizing ICT and particular the use of interactive facilities of Maple in teaching and learning which will provide a new challenge to both mathematics educators as well as students.

Interactive linear algebra: a laboratory course using Mathcad

This is a completely interactive linear algebra course. Developed by the authors and class-tested at Penn, Temple and Duke Universities, Interactive Linear Algebra runs in MathCad (Windows environment). The subject is taught in a laboratory setting, with or without additional lectures, and students are taught that through this technology-centered approach, mathematics becomes an experimental science. This electronic text guides students through the standard topics in linear algebra, with a carefully planned series of computer-based discussions, examples, questions and projects. MathCad provides the digital tools needed for developing, visualizing, connecting and applying important concepts.

Using PC‐Matlab in the teaching of linear algebra

Traditionally, computers were used mainly for arithmetic operations. Nowadays, they are a useful tool in the creation of various teaching‐learning environments. Computers have been used successfully as teaching aids in many spheres of education. In this paper we describe how we used a microcomputer package, PC‐Matlab, in the teaching of some topics in linear algebra.

Visualization in linear algebra

The use of a software package to enhance the visualization of topics in linear algebra is discussed. This material is suitable for use in an investigative laboratory situation, and ideas are outlined for worksheets in the areas of transformation matrices and the eigenvalue problem.

Nonnegative Matrices and Applicable Topics in Linear Algebra.

Nonnegative Matrices and Applicable Topics in Linear Algebra.

Non-Negative Matrices and Applicable Topics in Linear Algebra.

Non-Negative Matrices and Applicable Topics in Linear Algebra.

Applications of Graph Theory in Linear Algebra

Graph theory has existed for many years not only as an area mathematical study but also as an intuitive and illustrative tool. The use graphs in wiring diagrams is a straightforward representation the physical elements an electrical circuit; a street map is also a graph with the streets as edges, intersections streets as vertices, and street names as labels the edges. The graphs resemble the physical object that they represent in these cases, and so the application (and sometimes the genesis) the graph-theoretic ideas is immediate. A flow diagram a computer program and a road map with one way streets are examples graphs which contain the concept direction or flow to the edges; these are called directed graphs. There are applications graphs and directed graphs in almost all areas the physical sciences and mathematics, many them known for fifty years or more, but very few these ideas have percolated down to the undergraduate student. The purpose this article is to apply graph-theoretic ideas to some the fundamental topics in linear algebra. While there are many such applications, we shall focus on only two the most elementary ones, i.e., matrix multiplication and the theory determinants. The presentation is usable as a supplement to the usual classroom lectures; in fact, this paper grew out notes used in a linear algebra course for sophomores. The main tool to be used is the directed graph. Intuitively this can be thought as a set points (or vertices) with arrows (or arcs) joining some the points. A label may be put on an arc. More formally, a digraph consists a set vertices V and a subset ordered pairs vertices called the arcs. A labelling the digraph is a function from the arcs to the real numbers. A labelled digraph is usually visualized by considering the vertices as points with arcs as arrows going from vertex i to vertexj whenever (i, j) belongs to the sets arcs. The ith vertex the arc (i, j) is called its initial vertex, while the jth vertex is called its terminal vertex. The arc is then given a label which is the image that arc under the labelling function. When the initial and terminal vertices are identical, the arc is called a loop. We shall sometimes say that an arc goes out of its initial vertex and that it goes its terminal vertex. The number arcs that go out a vertex is called its outdegree, and the number arcs that go into that vertex is called its