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从一道线性代数习题出发,举例说明常见教材中关于由矩阵A的特征值确定ψ(A)的特征值的结论不够完备,进而分析问题关键,运用求解特征多项式的方法推导出矩阵多项式的特征值. 相似文献
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从问题出发引入线性代数概念 总被引:4,自引:0,他引:4
线性代数是大学最重要的基础课程之一.它的内容比起另一门最重要的基础课一微积分要少得多,但是还是有很多学生感到线性代数难学,特别是难以入门,其主要原因在于线性代数一开始就从天而降许多抽象的概念,将初学者先打了“一百杀威棒”.在我们通过建设国家精品课程《线性代数》形成的十五规划教材《线性代数》(李尚志著,高教出版社,2006)中,为了帮助初学者克服学习抽象概念时所遇到的困难,我们不是从定义出发而是从问题出发来组织课程内容,首先提出一些重要而又能引起学生兴趣的问题,引导学生一步步建立数学模型来描述和解决这些问题,在解决… 相似文献
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从灵活解答2010年三道线性代数考研试题,谈理解数学基本概念及培养灵活运用数学知识处理问题能力的重要性. 相似文献
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In this paper we characterize students’ conceptions of span and linear (in)dependence and their mathematical activity to provide insight into their understanding. The data under consideration are portions of individual interviews with linear algebra students. Grounded analysis revealed a wide range of student conceptions of span and linear (in)dependence. The authors organized these conceptions into four categories: travel, geometric, vector algebraic, and matrix algebraic. To further illuminate participants’ conceptions of span and linear (in)dependence, the authors developed a categorization to classify the participants’ engagement into five types of mathematical activity: defining, proving, relating, example generating, and problem solving. Coordination of these two categorizations provides a framework that proves useful in providing finer-grained analyses of students’ conceptions and the potential value and/or limitations of such conceptions in certain contexts. 相似文献
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Yourim Yoon Yong-Hyuk Kim Byung-Ro Moon 《Journal of Mathematical Modelling and Algorithms》2011,10(3):269-276
We analyze two essential problems arising from edge-based graph partitioning. We show that one of them is an NP-hard problem
but the other is in P, presenting a novel methodology that links linear algebra theory to the graph problems as a part of
proving the facts. This is a significant trial in that linear algebra, which has been mostly adopted as a theoretical analysis
tool, is practically applied to solving actual graph problems. As a result of the linear algebraic manipulation, we could
devise a linear-time algorithm for the problem in P. 相似文献
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论述线性代数教材改革的必要性及其指导思想,主张线性代数教材建设应以矩阵为主线,以教育部颁发的硕士研究生人学考试线性代数部分的大纲为纲,结合教学现代化的现状,强调几何与代数相结合,重视代数学思想方法的渗透,化抽象为具体,同时在教学中还要引入数学建模的思想. 相似文献
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This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge. 相似文献
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Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices. 相似文献
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On soft equality 总被引:4,自引:0,他引:4
Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we deal with the algebraic structure of soft sets. The lattice structures of soft sets are constructed. The concept of soft equality is introduced and some related properties are derived. It is proved that soft equality is a congruence relation with respect to some operations and the soft quotient algebra is established. 相似文献
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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. 相似文献
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Clifford 代数,几何计算和几何推理 总被引:8,自引:0,他引:8
Clifford代数是一种深深根植于几何学之中的代数系统,被它的创始人称为几何代数.历史上,E.Cartan,R.Brauer,H.Weyl,C.Chevalley等数学大师都曾研究和应用过Clifford代数,对它的发展起了重要作用.近年来,Clifford代数在微分几何、理论物理、经典分析等方面取得了辉煌的成就,是现代理论数学和物理的一个核心工具,并在现代科技的各个领域,如机器人学、信号处理、计算机视觉、计算生物学、量子计算等方面有广泛的应用.本文主要介绍Clifford代数在几何计算和几何推理中的应用.作为一种优秀的描述和计算几何问题的代数语言,Clifford代数对于几何体,几何关系和几何变换有不依赖于坐标的、易于计算的多种表示,因而应用它进行几何自动推理,不仅使困难定理的证明往往变得极为简单,而且能够解决一些著名的公开问题,目前在国际上,几何自动推理已经成为Clifford代数的一个重要应用领域。 相似文献