《工程数学——线性代数》简介

<正> 我所编写的《工程数学——线性代数》经高等学校工科数学课程教学指导委员会第十三次全体会议评审,被选定为高等工业学校《线性代数》教材,即将由高等教育出版社出版。     

线性代数教材内容与体系结构改革的思考与实践--《大学数学基础教程--线性代数与空间解析几何》简介

2002年初,我们承担了普通高等教育“十五”国家级规划教材《大学数学基础教程》的编写工作.它包括:一元函数微积分,多元函数微积分,线性代数与空间解析几何,概率论与数理统计,数学实验教程等五分册.我们编写这套书的主要指导思想为:适应当前高等教育从精英教育到大众化教育的需要,主要针对一般高等院校的教学实际,选择适当的教学定位,将数学建模的思想与方法渗透到教材内容中去,注重学生应用数学知识解决实际问题的意识与能力的培养;注重吸收国内外优秀教材的长处,将传统的教材内容与体系结构作较大幅度的调整,使教材更便于施教.线性代数与…     

线性代数与空间解析几何新形态教材建设的实践与思考

从线性代数与空间解析几何课程发展历史引入,对比分析传统教材内容与体系的差异,在此基础上探讨信息技术与教育教学深度融合背景下新形态教材出版的必要性,进而阐述线性代数与空间解析几何新形态教材出版创新的探索与实践,最后浅谈新形态教材建设的几点思考.     

《线性代数（非数学专业）》整体教学的实践与认识

首先明确了《线性代数（非数学专业）》整体教学的目的和实践的过程,其次从学生构建《线性代数》知识、技能和思想方法的角度总结了《线性代数》整体教学实践的一些体会,最后指出《线性代数》整体教学应把握数学观念,更好地将启发式教学与问题解决结合起来．     

线性代数课程思政建设与教学实践

探讨了线性代数课程思政建设与教学实践.通过优化课程思政教学目标,从线性代数相关历史人物、课程知识点及教学拓展案例三个方面,积极挖掘课程思政元素切入点.加强课程思政教学团队建设,将思想政治教育融入到线性代数课堂混合式教学,增加对学生思政学习的过程性评价.教学实践表明,线性代数课程思政教育有利于培养学生的综合素质,受到学生...     

教材优点明显 问题值得商榷——简评同济大学应用数学系编《微积分》教材

同济大学应用数学系编《微积分》(上、下册 )是高等教育出版社精心组织和推出的一本面向 2 1世纪的课程教材 .这本教材自 1 999年出版以来 ,在全国工科数学界产生了积极的影响 ,得到了高等理工科院校从事高等数学教学同仁们的关注和重视 .作为探索高等数学教材改革的一种模式 ,正如该书编者在前言中所指出 ,该教材在突出微积分的基本思想和基本方法 ,渗透现代数学思想 ,促进微积分与线性代数及其他数学课程的结合 ,为学生进一步学习现代数学知识提供一些“接口”,把数学软件的学习、使用融合进教材 ,尝试将微积分的教学与计算机应用的有机结…     

让线性代数课程易教易学

就在线性代数课程教学中遇到的困难和问题进行了分析,并提出了应对方法,即以问题为主线,把握宏观定义和微观定义,融入数学建模思想,从而使线性代数课程易教易学.     

让线性代数课程易教易学

就在线性代数课程教学中遇到的困难和问题进行了分析,并提出了应对方法,即以问题为主线,把握宏观定义和微观定义,融入数学建模思想,从而使线性代数课程易教易学.     

线性代数教学改革点滴

线性代数是一门较抽象的数学课程 .传统的线性代数教材通常是基于入门学科立论的准确和证明的严格来编写的 ,其抽象性跃然纸上 .但是 ,线性代数除其抽象性之外 ,还具有另外一个同样重要的特点 :实用性 ,它同微积分一样极富实用价值 .实际上 ,线性代数更适于将数学的两大因素——抽象和应用结合起来 .由于电子计算机的飞速发展和广泛应用 ,线性代数已成为越来越多的科技工作者必不可少的数学工具 ,为培养跨世纪高级工程技术人才 ,教好学好线性代数是十二分重要的 .基于这样的理由 ,我在线性代数教学中采取了一些不同于传统的教法 ,一方面力求…     

对比国外优秀教材 探索我国线性代数课程改革的新思路

大学数学课程的教学改革是当前的研究热点.本文通过中外线性代数教材的内容结构、建模应用等方面进行对比、分析,针对目前我国线性代数教材的不足,提出了我国线性代数课程改革的新思路.     

关于数学分析在线性代数中某些应用的数学札记

线性代数中有不少难度较高的问题,若从代数学内部系统出发处理起来,相当繁冗.但是若从另一角度出发,运用数学分析的方法去处理这些代数问题,可能有着事半功倍之效.为此,本文运用数学分析的有关知识和方法论证线性代数中的某些问题.供大家在教学时参考,不妥之处,敬请斧正.     

面向独立学院学生的线性代数课程“可视化”教学研究

针对独立学院培养目标和学生的特点,研究如何通过案例教学,具体将线性代数的概念、定理、应用、习题"可视化",完成将枯燥数学知识由抽象到直观的转化,使得线性代数理论与数学建模紧密结合,培养应用型人才.     

“金课”背景下强化线性代数应用教学的实践与探索

线性代数课程具有高度的抽象性,再加上课时等多方面限制,使得学生学起来感觉非常枯燥.针对此问题,在线性代数课程中,适当增加应用的讲解,以增加学生对于相关知识应用的认知,提高学生综合利用线性代数知识解决问题的能力.通过调查问卷,分析了在线性代数课程中增加应用知识讲解的经验、存在的问题及应对措施.     

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.     

Max-Plus Algebra and Mathematical Fear in Dynamic Optimization

Max-plus algebra, cost measures, and mathematical fear have proved useful tools in dynamic optimization. Indeed, the first two have even become a central tool in some fields of investigation such as discrete event systems. We first recall the fundamentals of max-plus algebra with simple examples of max-plus linear models, and simple consequences of that remark. We then introduce cost measures, the natural equivalent of probability measures in the max-plus algebra, and their fundamental properties, including the definition of the mathematical fear (the equivalent of the mathematical expectation), induced measures and conditioning. Finally, we concentrate on those aspects that are put in use in dynamical optimization and state a separation theorem which was first derived using these tools.     

有关解线性方程组的一些思考

解线性方程组是线性代数课程的最重要内容之一,目前工科线性代数的大纲和教材一般不包括不相容方程组,其实这部分内容具有广泛的应用.本文用微积分方法给出不相容方程组的最小二乘解以及极小范数最小二乘解,可供线性代数课程的教学改革作参考.建议待条件成熟时,将不相容方程组的最小二乘解纳入工科线性代数的教学大纲和教材.     

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.     

Using Prediction to Promote Mathematical Understanding and Reasoning

Research has shown that prediction has the potential to promote the teaching and learning of mathematics because it can be used to enhance students' thinking and reasoning at all grade levels in various topics. This article addresses the effectiveness of using prediction on students' understanding and reasoning of mathematical concepts in a middle school algebra context. In the treatment classroom, prediction questions were utilized at the launch of each algebra lesson, and in the control classroom such questions were not used. Both classrooms were taught by the same teacher and used the same curriculum. After completing each of the linear and exponential units, the two classrooms were compared in terms of their mathematical understanding and reasoning through unit assessments. Overall, the treatment classroom outperformed the control classroom on the unit assessments. This result supports that prediction is a valid construct with respect to enhanced conceptual understanding and mathematical reasoning.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

On the Riemann-Hilbert transformations for a Galilean invariant system

Summary As an application of the Riemann-Hilbert (RH) problem to mathematical physics, the RH transformations are considered for a Galilean invariant nonlinear system. Algebraic RH transformation gives rise to new solutions from the old via a calculation in linear algebra. It is proved that the infinitesimal RH transformations form an infinite-dimensional Lie algebra without using a hierarchy of potentials.