向量组线性相关性理论的一种解释

向量组的线性相关性理论由于概念比较抽象、定理难以理解,因此是线性代数教学环节中的一项难点。通过将向量组形象地描述为"班级",将向量组中的每个向量看成是"班级中的每个同学",定义"班级地域相关","地域表示"等概念,有利于学生对向量组的线性相关、线性表示、极大线性无关组和秩等概念的学习和理解.     

一道线性代数习题的教学处理

一道线性代数习题的教学处理樊启斌（武汉测绘科技大学，武汉４３００７０）一、引言研究向量组的线性相关性或线性无关性，是线性代数课程的重要内容之一．由于它的概念高度抽象、基本理论相对集中，因而也是理工科线性代数的教学难点之一．为了进一步巩固和强化课堂教...     

一道线性代数习题的教学处理

研究向量组的线性相关性或线性无关性，是线性代数课程的重要内容之一．由于它的概念高度抽象、基本理论相对集中，因而也是理工科线性代数的教学难点之一，为了进一步巩固和强化课堂教学效果，帮助学生深刻理解基本概念，熟练掌握基本理论，提高抽象思维能力，笔者利用习题课这一教学环节，组织了一次别开生面的专题教学活动，收效显著，反响颇佳。     

极大线性无关向量组与初等变换

<正> 向量组的线性相关性是线性代数中的一个重要基本概念。假设已给m个n元向量a_1,a_2…a_m,这里a_i=(ai_1,ai_2,…ai_n),i=1,2…m。判别向量组a_1,a_2…a_m的线性相关性;求此向量组的极大线性无关组和向量组的秩;将此向量组中任一向量用极大线性无关组线性     

对同济大学编《线性代数》的一点看法

<正> 我院线性代数课程采用同济大学数学教研室编的《线性代数》为教材。在教学过程中,发现该书有一些值得商讨的地方,下面是我的一点看法,不妥之处欢迎批评指正。该书第三章“向量组的线性相关性与矩阵的秋”是全书份量最重的一章,概念较抽象,理论性也比较     

范例集锦

<正> 行列式的计算.逆矩阵的计算,矩阵秩的计算,线性方程组求解、二次型化为标准型、二次型正定性的判别等是线性代数中的重要计算问题,其中初等变换、高斯消去法起着重要作用,线性代数中有关基本理论和基本概念的证明,如向量组的线性相关性、秩的等式与不等式,判别线性空间等等,都涉及到很强的概念,下面我们通过例题,给出这些计算和证明的基本方法.     

线性相关性教学浅谈

<正> 线性代数中向量组的线性相关性这部分内容,是历届学生最感困难的一部分,是教学中的一大难关,原因是概念、理论既抽象又比较多且集中出现,学生不易消化和巩固,造成理论串疤,作业无法下手的困境。我从以下几方面下功夫,收到一定成效。一、课时划分(以华东六省工科数学教材为蓝本)注意突出重点,分散难点的原则,因此不能按教材中的自     

浅谈线性代数微课设计——以向量组的最大无关组与秩为例

微课的本质是通过对一个知识点的教学过程进行充分的设计,以"短而全、简而精"的形式呈现给学习者.基于线性代数抽象、难懂的特点,以《向量组的最大线性无关组与秩》的微课设计为例,通过理论与具体设计思路的阐述,分析探讨"如何对线性代数进行微课设计",以浅显易懂的形式呈现给学习者,进而实现高效教学.     

巧妙设计线性代数习题

通过具体实例的设计与求解，深化对向量组的最大线性无关组、最大线性无关组的不唯一性、向量组的线性表示等知识的理解，同时展示这部分知识在实际中的一种应用。增强学生应用所学知识解决实际问题的能力。     

向量空间中一些概念与结果

向量空间是《线性代数》中的一个教学难点,难教难学,其中一些概念和性质不好掌握,容易错用．一些新编教材和涉及此章的教改文章在这方面作了不少工作,很多有独到之处,但有些地方尚有错漏和需要改进之处．下面分述之．一、过渡矩阵线性代数中的过渡矩阵是这样来定义的...     

A note on marginal and conditional independence

Some statistical models imply that two random vectors are marginally independent as well as being conditionally independent with respect to another random vector. When the joint distribution of the vectors is normal, canonical correlation analysis may lead to relevant simplifications of the dependence structure. A similar application can be found in elliptical models, where linear independence does not imply statistical independence. Implications for Bayes analysis of the general linear model are discussed.     

常系数线性方程组的一种新解法

对于常系数线性微分方程组:dx/dt=Ax(A是n阶实常数矩阵)通过特征根λ和对应的特征行向量K:K~T(A-λE)=0将微分方程组化为线性方程组:1°当有n个互异的特征根λ_1,λ_2,…,λ_n,对应的线性无关的特征行向量为K_1,K_2,…,K_n,若记K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),则有方程组:(n∑i=1 k_ix_i)′=λ_j(n∑i=1 k_ix_I)(j=1,2,…,n);2°当有不同的特征根λ_1,λ_2,…,λ_m其重数分别为n_1,n_2,…,n_m,n_1+n_2+…+n_m=n,对应的线性无关的特征行向量为K_i=(k_1,K_2,…,k_n)(i=1,2,…,m),则有方程组:(n∑i=1 k_rx_r)′=λ_k(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)′=λ_j(n∑i=1k_rx_r)+c_(n_i)e~(λ_jt)((A-λ_kE)x_(i-1)=Ex_i,i=2,…,n_i).     

t-clique ideal and t-independence ideal of a graph

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover, some families of graphs for which the quotient ring of their clique ideal is Cohen–Macaulay are introduced and some homological invariants of the clique ideal of a graph G, which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs are studied.     

Best arm identification in generalized linear bandits

We consider the best-arm identification problem in generalized linear bandits: each arm has a vector of covariates, there is an unknown vector of parameters that is common across the arms, and a generalized linear model captures the dependence of rewards on the covariate and parameter vectors. The goal is to identify a near-optimal arm with high probability while minimizing the number of arm pulls (i.e., the sampling budget). We propose the first algorithm for this problem and provide theoretical guarantees on its accuracy and sampling efficiency.     

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

Supports of Locally Linearly Independent M-Refinable Functions,Attractors of Iterated Function Systems and Tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L ₂ solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.     

正态向量的函数与线性型之间的独立性

本文通过讨论正态向量X的函数f(X)与线性型AX之间的独立性，把已知的正态向量的二次型与线性型的独立性问题作了推广，并得出了一系列推论。     

Verification of Linear (In)Dependence in Finite Precision Arithmetic

We present theoretical background for verification of linear dependence/independence of columns of a matrix by means of finite precision arithmetic.     

A Test of Multivariate Independence Based on a Single Factor Model

A test of the independence of two sets of variables is developed to have high power against a special family of dependence. In this each set of variables has the structure of a single factor model and the dependence is solely via the correlation γ between the underlying latent variables. This is a model with only one nonzero canonical correlation. It is shown that a test based on the maximum likelihood estimate of γ is appreciably more powerful than that based on r₁, the largest sample canonical correlation. If, however, the model is used, not just as a family of alternatives but as the basis for interpretation, and if substantial cross-correlation is present then the procedure is essentially equivalent to the use of r₁.     

Nonlinearity creates linear independence

Any number of linearly independent plane waves can be obtained by scaling, shifting and/or rotating a plane wave created from an activation function. We investigate the condition which ensures linear independence of the plane waves. In the case all the three procedures are combined, the linear independence can be proved by ad hoc methods for most of commonly used activation functions. The result can be used for proving structural uniqueness of neural networks and for implementing finite mapping by neural networks.This paper was supported in part by Grant 07252213 from Ministry of Education and Culture, Japan.