Study Of Linear Algebra Research Articles

Conditions for The Saturation of a Matrix Inequality

Matrix inequalities play a vital role in the study of linear algebra and information theory. In this paper, the author investigates the condition when a proven matrix inequality is saturated, and Pi’s are linearly independent matrices of the same size, and Si’s are also linearly independent matrices of the same size. The author constructs the conditions when Pi is a column vector or a 2 2 matrix, and Si is an arbitrary square matrix. The author finds a very potential result when investigating the column situation. The author also gets massive result in the 2 2 situation. In particular, the inequality is not saturated when Pi is a 2 2 matrix and K=2 or 3 and it could be saturated under the situation when K=4 if only if two certain matrices can enable the product of and form two established matrices. At the end of the essay, the author tries some more complex cases such as 2 3 matrices Pi when K=2.

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.

ECONOMIC APPLIED TASKS AS A WAY TO ENLARGE MOTIVATION WHEN LEARNING ADVANCED MATHEMATICS IN TECHNICAL UNIVERSITY

The problems of students' motivation in studying advanced mathematics by students of economic specialties are considered. The ways of its increase are shown by the use of applied tasks related to the problems of the region.The authors analyzed the research motives, which encourage students of the first year to study, based on the problem of increasing motivation. One of the ways of its solution suggested by the authors is the application of applied tasks at studying of advanced mathematics. Among several classes of applied tasks, the application of applied tasks that are directly related to the future professional activity of students are considered and based on statistical data taken from specific enterprises of the region. The authors identified the main stages of solving such tasks and designated the main difficulties that may arise. For example, mathematical solution of an economic task can be given in the form of the following steps: the construction of a mathematical model which would reflect those properties of the object that really affect the investigated indicator and are consistent with known standard mathematical schemes; the choice of a solution method or a research of a model that directly depends on the quality of the model; directly the process of solution, which is purely mathematical; economic interpretation of the obtained result. As one of the key stages the authors consider the construction of a mathematical model of this process. In the article the role of mathematical modeling for increasing students' motivation in the study of advanced mathematics is considered. Basing on many years of experience, the authors showed the application of applied tasks in the study of linear algebra by students of economic specialties in the Poltava region. Also there is highlighted a number of specific tasks that the teacher can solve, using the proposed approach in the classes. Namely: the consolidation of theoretical knowledge, the formation of skills for the application of theoretical knowledge in practice, the development of skills for analysis, synthesis, induction, deduction and model construction, increasing motivation to study. Thus, in order to increase the efficiency of the perception of theoretical material and to increase the motivation to study advanced mathematics, it is useful for students to use not only applied tasks related to the specialty on which students study, but also to fill them with specific statistical data taken from enterprises of a certain region.

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.

Norms, Isometries, and Isometry Groups

The study of linear algebra has become more and more popular in the last few decades. People are attracted to this subject because of its beauty and its connections with many other pure and applied areas. In theoretical development of the subject as well as in many applications, one often needs to measure the “length” of vectors. For this purpose, norm functions are considered on a vector space. In this expository article, we explain why one would want to study different kinds of norms on a real vector space. We then focus on the problem of how to identify different norms using linear isomorphisms with particular attention to a group theory method used by several authors recently. One may view the first half of this article as a gentle introduction to the theory of norms and the second half as an illustration of how group theory can be applied to questions in linear algebra.

Inductive problem‐sequences for independent study of linear algebra

Three chapters of the syllabus of a basic mathematics course for engineering students at the Technion† were adapted for the student's independent development of mathematical knowledge by means of solving a chain of inductive problem‐sequences: (i) vectors and scalars in the plane; (ii) vectors in the three‐dimensional space; (iii) planes and lines in the 3‐dimensional space. The main principles which guided the adaptation of course materials for this purpose are presented. In particular an inductive problem‐sequenceis defined as a sequence consisting of problems which lead the learner to investigate particular cases, discover a common regularity, make a conjecture about a unifying rule, and finally prove or refute the conjecture. The major part of the paper is an annotated analysis of an excerpt of the materials. † Technion is the oldest and largest school of engineering in Israel.

The microcomputer in linear algebra

Menu-driven software is used to augment the teaching and study of linear algebra. The software allows the exploration of possibilities that laborious calculations would otherwise exclude. Examples are provided to show how the software helps to understand and apply the theory.

An algorithmic approach to linear systems

This article treats the solution of linear algebraic equations as a typical subject for a mathematical laboratory. Following a brief description of such a laboratory, equipped with a set of microcomputers, an algorithm for solving linear systems is presented. No knowledge of matrices, vectors and the underlying theory is assumed. Pedagogical considerations guided the choice of material, style and level of presentation, while emphasizing the learning process in a mathematical laboratory environment. Special attention is given to possible loss of accuracy, sensitivity to minor changes in the data, pivoting, pre‐scaling and computational efficiency. Since the laboratory participants are introduced to these concepts without using matrix theory, the laboratory sessions can be carried out even before the study of linear algebra.