Lecture 7 Linear Algebra Algorithms
linear algebra
The parametric form of this equation is ⃗ s + c1⃗ v1 + c2⃗ v2 + · · · + cp⃗ vp . Example: Solve the equation A⃗ x =⃗ b, where 1 −2 0 1 1 (A ⃗ b) ∼ 0 0 1 8 −1 . 0 0 0 0 0 Again, the free variables are x2 , x4 and further x1 −1 − 8x4 . So x1 1 + 2x2 − x4 1 x2 0 x2 + x2 = = x3 −1 − 8x4 −1 0 x4 x4 = 1 − 2x2 − x4 , x3 = 2 1 + x4 0 0
2
Homogeneous Matrix Equation(齐次矩阵方 程组)
Definition 1 (Homogeneous(齐次的)). A system of linear equation is said to be homogeneous if it can be written as A⃗ x =⃗ 0, where ⃗ 0 is the zero vector m ⃗ x = 0 is called a homogeneous (matrix) equation (齐次矩 (零向量) in R . A⃗ 阵方程). Obviously, the zero vector is a solution to the homogeneous linear system and this solution is called trivial solution (平凡解). If there is more solutions rather than the trivial solution to the system, we call the homogeneous equation A⃗ x =⃗ 0 has a nontrivial solution (非平凡解) 1
线性代数合同的英语
线性代数合同的英语Linear algebra is a fundamental branch of mathematics that has numerous applications in various fields such as physics, engineering, computer science, and economics. One of the key aspects of linear algebra is the concept of contracts which are used to represent and manipulate linear relationships between variables. In this essay, we will delve into the intricacies of linear algebra contracts and explore their significance in various contexts.At the core of linear algebra are matrices, which are rectangular arrays of numbers or other mathematical objects. These matrices can be used to represent linear transformations, systems of linear equations, and other mathematical structures. A linear algebra contract is a way of expressing these linear relationships in a concise and organized manner.One of the primary uses of linear algebra contracts is in the representation of systems of linear equations. Consider a set of n linear equations in m variables, where each equation can be expressed in the form:a11x1 + a12x2 + ... + a1mxm = b1a21x1 + a22x2 + ... + a2mxm = b2...an1x1 + an2x2 + ... + anmxm = bnThese equations can be represented using a matrix equation of the form Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the vector of constants. The linear algebra contract in this case would be the matrix A, which encapsulates the linear relationships between the variables and the constants.Another important application of linear algebra contracts is in the study of linear transformations. A linear transformation is a function that maps vectors in one space to vectors in another space, while preserving the linear structure of the vectors. These transformations can be represented using matrices, and the matrix itself can be considered a linear algebra contract that defines the transformation.For example, consider a linear transformation T: R^n -> R^m, which maps vectors in R^n to vectors in R^m. This transformation can be represented by a matrix A of size m x n, where the (i,j)th element of A represents the coefficient of the jth variable in the ith equation of the transformation. The linear algebra contract in this case would be the matrix A, which fully defines the linear transformation.Linear algebra contracts also play a crucial role in optimization problems, where the goal is to find the optimal solution to a problem subject to a set of constraints. These constraints can often be expressed as linear equations or inequalities, and the linear algebra contracts involved in the problem formulation are essential for solving the optimization problem efficiently.In the field of machine learning, linear algebra contracts are extensively used in the development of various algorithms and models. For instance, in linear regression, the goal is to find a linear relationship between a set of input variables and a target variable. The linear algebra contract in this case would be the coefficient matrix that defines the linear relationship.Similarly, in principal component analysis (PCA), a widely used dimensionality reduction technique, the linear algebra contract is the covariance matrix of the input data, which is used to identify the principal components that capture the most significant variations in the data.The applications of linear algebra contracts extend beyond the academic realm and into the realm of industry and commerce. In the field of finance, linear algebra contracts are used in the pricing and risk management of financial instruments, such as bonds, derivatives,and portfolio optimization.In the engineering field, linear algebra contracts are crucial for the design and analysis of complex systems, such as electrical circuits, mechanical structures, and control systems. These contracts are used to model the relationships between various components and variables, allowing engineers to optimize the performance and reliability of their designs.In conclusion, linear algebra contracts are a fundamental concept in the field of linear algebra, with widespread applications across various domains. These contracts provide a concise and organized way of representing and manipulating linear relationships, enabling researchers, engineers, and decision-makers to solve complex problems efficiently and effectively. As the world becomes increasingly data-driven, the importance of linear algebra contracts will only continue to grow, making it an essential tool for understanding and shaping the world around us.。
LinearAlgebra(I)_02
Rank
Matrix and vector
The ij th entry of A, the entry on the ith row and j th column, is usually denoted by aij . A is a zero matrix of size m × n if all entries of A are zero, that is, aij = 0, 1 ≤ i ≤ m, 1 ≤ j ≤ n. We use 0 to denote a zero matrix of any size. If aij = 0 for i = j , A is a diagonal matrix and is sometimes written as A = diag(a11 , a22 , · · · , ann ). If A is diagonal and aii = 1 for each i, A is an identity matrix. We use I to denote an identity matrix of any size.
NCTU ICN5502
Fall 2017
2-6
Outline
Matrix
Linear function
Vector space
Basis and dimension
Null space and range
Rank
Some rules of matrix operations
For A, B, C of compatible sizes and scalar α, (a) A + B = B + A. (b) α(A + B ) = αA + αB (c) A + (B + C ) = (A + B ) + C . (d) C (A + B ) = CA + CB . (e) (A + B )C = AC + BC . (f) A(BC ) = (AB )C . But usually AB = BA.
常用线性代数程序库简单介绍
BLAS（Basic Linear Algebra Subprograms 基础线性代数程序集）是一个应用程序接口（API）标准，用以规范发布基础线性代数操作的数值库（如矢量或矩阵乘法）。
该程序集最初发布于1979年，并用于建立更大的数值程序包（如LAPACK）。
在高性能计算领域，BLAS被广泛使用。
例如，LINPACK的运算成绩则很大程度上取决于BLAS中子程序DGEMM的表现。
为提高性能，各硬件厂商（如Intel）则针对其硬件对BLAS接口实现进行高度优化。
BLAS按照功能被分为三个级别：Level 1：矢量-矢量运算Level 2：矩阵-矢量运算Level 3：矩阵-矩阵运算实现Parallel Basic Linear Algebra Subprograms (PBLAS) is an implementation of Level 2 and3 BLAS intended for distributed memory architectures. It provides a computational backbonefor ScaLAPACK, a parallel implementation of LAPACK. It depends on Level 1 sequential BLAS operations for local computation and BLACS for communication between nodes.PLAPACK (Parallel Linear Algebra Package) is an infrastructure for coding parallel linear algebra algorithms at a high level of abstraction.The ScaLAPACK (or Scalable LAPACK) library includes a subset of LAPACK routines redesigned for distributed memory MIMD parallel computers. It is currently written in a Single-Program-Multiple-Data style using explicit message passing for interprocessor communication. It assumes matrices are laid out in a two-dimensional block cyclic decomposition.ScaLAPACK is designed for heterogeneous computing and is portable on any computer that supports MPI or PVM.ScaLAPACK depends on PBLAS operations in the same way LAPACK depends on BLAS.。
Introduction to Linear Algebra
»a = 5 a= 5
A vector is a mathematical quantity that is completely described by its magnitude and direction. An example of a three dimensional column vector might be 4 b= 3 5 uld easily assign bT to another variable c, as follows:
»c = b' c= 4 3 5
A matrix is a rectangular array of scalars, or in some instances, algebraic expressions which evaluate to scalars. Matrices are said to be m by n, where m is the number of rows in the matrix and n is the number of columns. A 3 by 4 matrix is shown here 2 A= 7 5 5 3 2 3 2 0 6 1 3 (3)
»a = 5;
Here we have used the semicolon operator to suppress the echo of the result. Without this semicolon MATLAB would display the result of the assignment:
»A(2,4) ans = 1
The transpose operator “flips” a matrix along its diagonal elements, creating a new matrix with the ith row being equal to the jth column of the original matrix, e.g. T A = 2 5 3 6 7 3 2 1 5 2 0 3
线性代数—Linear Algebra
λ1 =1
λ2 =1/2
其中λ1, λ2 是矩陣A的特徵值，我們令det(A-λI)=0求出λ1, 與λ2，而特徵向量x1屬於A-I的nullspace ，而特徵向量x2屬
所有對應同樣特徵值的特徵向量形成一個子空間。
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6.1 Introduction to Eigenvalues : Good News, Bad News
Elimination does not preserve theλ’s .
1 A= 1 1 1 λ= 0 and λ= 2 1 U = 0 1 0
When a matrix is shifted by I, each λ is shifted by 1. 但是特徵向量不變， x1=(1, 1) 與 x2 =(1, -1)
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6.1 Introduction to Eigenvalues
範例：排序矩陣的特徵值λ滿足|λ|=1
上頁映射矩陣R也是排序矩陣(列對調)，其特徵值λ為1,-1 0 1 R= 1 0 但是P4的特徵值λ為±1, ± i有複數，其對應之特徵向量 (1, ±1, 1, ±1)與 (1,ction to Eigenvalues
範例：映射矩陣R的特徵值λ為1,-1
0 R= 1 1 0
(R2的特徵值是λ2=1 ,特徵向量不變)
R=2P- I , R2 = I
if Px= λx then 2Px=2λx , (2P- I) x=(2λ-1)x=Rx
0 1 P4 = 0 0
2011/6/18
0 0 1 0 0 0 1 0 0 0 1 0
introduction to linear algebra 每章开头方框-概述说明以及解释
introduction to linear algebra 每章开头方框-概述说明以及解释1.引言1.1 概述线性代数是数学中的一个重要分支，主要研究向量空间和线性变换的性质及其应用。
它作为一门基础学科，在多个领域如物理学、计算机科学以及工程学等都有广泛的应用。
线性代数的研究对象包括向量、向量空间、矩阵、线性方程组等，通过对其性质和运算法则的研究，可以解决诸如解线性方程组、求特征值与特征向量等问题。
线性代数的基本概念包括向量、向量空间和线性变换。
向量是指在空间中具有大小和方向的量，可以表示为一组有序的实数或复数。
向量空间是一组满足一定条件的向量的集合，对于向量空间中的任意向量，我们可以进行加法和数乘运算，得到的结果仍然属于该向量空间。
线性变换是指将一个向量空间映射到另一个向量空间的运算。
线性方程组与矩阵是线性代数中的重要内容。
在实际问题中，常常需要解决多个线性方程组，而矩阵的运算和性质可以帮助我们有效地解决这些问题。
通过将线性方程组转化为矩阵形式，可以利用矩阵的特殊性质进行求解。
线性方程组的解可以具有唯一解、无解或者有无穷多解等情况，而矩阵的行列式和秩等性质能够帮助我们判断线性方程组的解的情况。
向量空间与线性变换是线性代数的核心内容。
向量空间的性质研究可以帮助我们理解向量的运算和性质，以及解释向量空间的几何意义。
线性变换是一种将一个向量空间映射到另一个向量空间的运算，通过线性变换可以将复杂的向量运算问题转化为简单的矩阵运算问题。
在线性变换中，我们需要关注其核、像以及变换的特征等性质，这些性质可以帮助我们理解线性变换的本质和作用。
综上所述，本章节将逐步介绍线性代数的基本概念、线性方程组与矩阵、向量空间与线性变换的相关内容。
通过深入学习和理解这些内容，我们能够掌握线性代数的基本原理和应用，为进一步研究更高级的线性代数问题打下坚实的基础。
1.2文章结构在文章结构部分，我们将介绍本文的组织结构和各章节的内容概述。
线性代数（linearalgebra）
线性代数（linear algebra）Linear algebra (Linear Algebra) is a branch of mathematics. Its research objects are vectors, vector spaces (or linear spaces), linear transformations and finite dimensional linear equations. Vector space is an important subject in modern mathematics. Therefore, linear algebra is widely used in abstract algebra and functional analysis. Linear algebra can be expressed concretely by analytic geometry. The theory of linear algebra has been generalized to operator theory. Since nonlinear models in scientific research can often be approximated as linear models, linear algebra has been widely applied to natural and social sciences.The development of linear algebraBecause the work of Descartes and Fermat, linear algebra basically appeared in seventeenth Century. Until the late eighteenth Century, the field of linear algebra was confined to planes and spaces. The first half of nineteenth Century to complete the transition matrix to the n-dimensional vector space theory begins with Kailai in the second half of nineteenth Century, because if when work reached its culmination in.1888, Peano axiomatically defined finite or infinite dimensional vector space. Toeplitz will be the main theorem is generalized to arbitrary body linear algebra on the general vector space. The concept of linear mapping can in most cases get rid of matrix computation directed to the inherent reasoning, that is not dependent on the selection of the base. Do not exchange and exchange or not with the ring as the operator domain, this concept to die, this concept very significantly extended vector space theory and re organize the nineteenth Century Instituteof the.The word "algebra" appeared relatively late in China, in the Qing Dynasty when the incoming China, it was translated into "Alj Bala", until 1859, the Qing Dynasty famous mathematician, translator Li Shanlan translated it as "algebra", still in use.The status of linear algebraLinear algebra is a subject that discusses matrix theory and finite dimensional vector spaces combined with matrices and their linear transformation theory.The main theory is mature in nineteenth Century, and the first cornerstone (the solution of two or three Yuan linear equations) appeared as early as two thousand years ago (see in our ancient mathematical masterpiece "nine chapters arithmetic").The linear algebra has many important applications in mathematics, mechanics, physics and technology, so it has important place in various branches of algebra;In the computer today, computer graphics, computer aided design, cryptography, virtual reality and so on are all part of the theory and algorithm of linear algebra;.Between geometric and algebraic methods embodied in the concept of the subject of the connection from the axiomatic method on the abstract concept and rigorous logic reasoning, cleverly summed up, to strengthen people's training in mathematics, science and intelligent gain is very useful;And with the development of science, we should not only study the relationship between the individual variables, but also further study the relationship between multiple variables, all kinds of practical problems in most cases can be linearized, and because of the development of the computer, the linearized problem can be calculated, linear algebra is a powerful tool to solve these problems.Basic introduction to linear algebraLinear algebra originated from the study of two-dimensional and three-dimensional Cartesian coordinate systems. Here, a vector is a line segment with a direction that is represented by both length and direction. Thus vectors can be used to represent physical quantities, such as force, or to add and multiply scalar quantities. This is the first example of a real vector space.Modern linear algebra has been extended to study arbitrary or infinite dimensional spaces. A vector space of dimension n is called n-dimensional space. In two-dimensional andthree-dimensional space, most useful conclusions can be extended to these high-dimensional spaces. Although many people do not easily imagine vectors in n-dimensional space, such vectors (i.e., n tuples) are very useful for representing data. Since n is a tuple, and the vector is an ordered list of n elements, most people can effectively generalize and manipulate data in this framework. For example, in economics, 8 dimensional vectors can be used to represent the gross national product (GNP) of 8 countries. When all the nationalorder (such as scheduled, China, the United States, Britain, France, Germany, Spain, India, Australia), you can use the vector (V1, V2, V3, V4, V5, V6, V7, V8) showed that these countries a year each GNP. Here, each country's GNP are in their respective positions.As a purely abstract concept used in proving theorems, vector spaces (linear spaces) are part of abstract algebra and have been well integrated into this field. Some notable examples are: irreversible linear maps or groups of matrices, rings of linear mappings in vector spaces. Linear algebra also plays an important role in mathematical analysis,Especially in vector analysis, higher order derivatives are described, and tensor product and commutative mapping are studied.A vector space is defined on a domain, such as a real or complex domain. Linear operators map the elements of a linear space into another linear space (or in the same linear space), and maintain the consistency of addition and scalar multiplication in the vector space. The set of all such transformations is itself a vector space. If a basis of linear space is determined, all linear transformations can be expressed as a table, called matrix. Further studies of matrix properties and matrix algorithms (including determinants and eigenvectors) are also considered part of linear algebra.We can simply say that the linear problems in Mathematics - those that exhibit linear problems - are most likely to be solved. For example, differential calculus studies the problemof linear approximation of functions. In practice, the difference between a nonlinear problem and a nonlinear one is very important.The linear algebra method refers to the problem of using a linear viewpoint to describe it and to describe it in the language of linear algebra and to solve it (when necessary) by using matrix operations. This is one of the most important applications in mathematics and engineering.Some useful theoremsEvery linear space has a base.The nonzero matrix n for a row of N rows A, if there is a matrix B that makes AB = BA = I (I is the unit matrix), then A is nonsingular matrix.A matrix is nonsingular if and only if its determinant is not zero.A matrix is nonsingular if and only if the linear transformation it represents is a automorphism.A matrix is semi positive if and only if each of its eigenvalues is greater than or equal to zero.A matrix is positive if and only if each of its eigenvalues is greater than zero.Generalizations and related topicsLinear algebra is a successful theory, and its method has been applied to other branches of mathematics.The theory of modulus is to study the substitution of scalar domains in linear algebra by ring substitution.Multilinear algebra transforms the "multivariable" problem of mapping into the problem of each variable, resulting in the concept of tensor.In the spectral theory of operators, by using mathematical analysis, infinite dimensional matrices can be controlled.All of these areas have very large technical difficulties.Basic contents of linear algebra in Chinese UniversitiesFirst, the nature and tasks of the courseThe course of linear algebra is an important basic theory course required by students of science and Engineering in universities and colleges. It is widely used in every field of science and technology. Especially today, with the development and popularization of computer, linear algebra has become the basic theory knowledge and important mathematical tool for engineering students. Linear algebra is to train thehigh-quality specialized personnel needed for the socialist modernization construction of our country. Through the study of this course, we should make students get:1 determinant2, matrix3. The correlation of vectors and the rank of matrices4 、 linear equations5, similar matrix and two typeAnd other basic concepts, basic theories and basic operational skills, and lay the necessary mathematical foundation for further courses and further knowledge of mathematics.While imparting knowledge through various teaching links gradually cultivate students with abstract thinking ability, logical reasoning ability, spatial imagination ability and self-learning ability, but also pay special attention to cultivate students with good operation ability and comprehensive use of the knowledge to the ability to analyze and solve problems.Two, the content of the course teaching, basic requirements and class allocation(1) teaching content1 determinant(1) definition of order n determinant(2) the nature of determinant(3) the calculation of the determinant is carried out in rows (columns)(4) the Clem rule for solving linear equations2, matrix(1) the concept of matrix, unit matrix, diagonal matrix, symmetric matrix(2) linear operations, multiplication operations, transpose operations and laws of matrices(3) inverse matrix concept and its properties, and inverse matrix with adjoint matrix(4) the operation of partitioned matrices3 vector(1) the concept of n-dimensional vectors(2) the linear correlation, linear independence definition and related theorems of vector groups, and the judgement of linear correlation(3) the maximal independent group of vectors and the rank of vectors(4) the concept of rank of matrix(5) elementary transformation of matrix, rank and inverse matrix of matrix by elementary transformation(6) n-dimensional vector spaces and subspaces, bases, dimensions, coordinates of vectors4 、 linear equations(1) the necessary and sufficient conditions for the existence of nonzero solutions of homogeneous linear equations and the necessary and sufficient conditions for the existence of solutions of nonhomogeneous linear equations(2) the fundamental solution, the general solution and the solution structure of the system of linear equations(3) the condition and judgement of the solution of nonhomogeneous linear equations and the solution of the system of equations(4) finding the general solution of linear equations by elementary row transformation5, similar matrix and two type(1) eigenvalues and eigenvectors of matrices and their solutions(2) similarity matrix and its properties(3) the necessary and sufficient conditions and methods of diagonalization of matrices(4) similar diagonal matrices of real symmetric matrices(5) two type and its matrix representation(6) the method of linearly independent vector group orthogonal normalization(7) the concept and property of orthogonal transformation and orthogonal matrix(8) orthogonal transformation is used as the standard shape of the two type(9) the canonical form of quadratic form and two form of two type are formulated by formula(10) the inertia theorem, the rank of the two type, the positive definite of the two type and their discrimination(two) basic requirements1, understand the definition of order n determinant, will use the definition of simple determinant calculation2, master the basic calculation methods and properties of determinant3, master Clem's law4. Understand the definition of a matrix5, master the matrix operation method and inverse matrix method6. Understanding the concept of vector dependency defines the relevance of the vector by definition7, grasp the method of finding the rank of the matrix, and understand the relation between the rank of the matrix and the correlation of the vector group8, understand the concept of vector space, will seek vector coordinates9. Master the matrix rank and inverse matrix with elementary transformation, and solve the system of linear equations10, master the method of solving linear equations, and know the simple application of linear equations11. Master the method of matrix eigenvalue and eigenvector12. Grasp the concept of similar matrices and the concept of diagonalization of matrices13, master the orthogonal transformation of two times for standard type method14, understand the inertia theorem of the two type, and use thematching method to find the sum of squares of the two type15. Grasp the concept and application of the positive definiteness of the two typeMATLABIt is a programming language and can be used as a teaching software for engineering linear algebra. It has been introduced into many university textbooks at home and abroad.。
LINEAR ALGEBRA 线性代数 课文 翻译
4 LINEAR ALGEBRA 线性代数“Linear algebra” is the study of linear sets of equations and their transformation properties. 线性代数是研究方程的线性几何以及他们的变换性质的Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.线性代数也研究空间旋转的分析，最小二乘拟合，耦合微分方程的解，确立通过三个已知点的一个圆以及在数学、物理和机械工程上的其他问题The matrix and determinant are extremely useful tools of linear algebra.矩阵和行列式是线性代数极为有用的工具One central problem of linear algebra is the solution of the matrix equation Ax = b for x. 线性代数的一个中心问题是矩阵方程Ax=b关于x的解While this can, in theory, be solved using a matrix inverse x = A−1b，other techniques such as Gaussian elimination are numerically more robust.理论上，他们可以用矩阵的逆x=A-1b求解，其他的做法例如高斯消去法在数值上更鲁棒。
大学数学专业英语教材
大学数学专业英语教材IntroductionMathematics plays a crucial role in various fields and industries, and studying mathematics at the university level requires a solid foundation in both the subject itself and the English language. A well-designed mathematics textbook for university students in the field of mathematics can effectively integrate mathematical concepts with English language learning. In this article, we will explore the essential features and requirements of a comprehensive English textbook for mathematics students at the university level.Chapter 1: Fundamental ConceptsThe first chapter of the textbook should cover the fundamental concepts of mathematics, introducing students to the basic principles that underpin the subject. It should provide concise explanations and definitions, supplemented with examples and illustrations to aid comprehension. Additionally, this chapter should include exercises to reinforce learning and promote critical thinking.Chapter 2: AlgebraAlgebra is a cornerstone of mathematics, and this chapter should delve into its key theories and principles. It should cover topics such as equations, inequalities, functions, and matrices. The textbook should present clear explanations of concepts, accompanied by real-life applications to demonstrate the practical relevance of algebra.Chapter 3: CalculusCalculus is essential for advanced mathematics and the study of other disciplines such as physics and engineering. The textbook should guide students through both differential and integral calculus, ensuring a thorough understanding of concepts like limits, derivatives, and integrals. Practical examples and exercises should be incorporated to enhance students' problem-solving skills.Chapter 4: Probability and StatisticsIn this chapter, the textbook should introduce students to probability theory and statistical analysis. The content should cover topics such as probability distributions, hypothesis testing, and regression analysis. The inclusion of real-world data sets and case studies can foster students' ability to apply statistical methods effectively.Chapter 5: Discrete MathematicsDiscrete mathematics is vital in areas like computer science and cryptography. This chapter should explore concepts such as set theory, logic, graph theory, and combinatorics. The textbook should present clear explanations of these topics, accompanied by relevant examples and exercises to consolidate understanding.Chapter 6: Linear AlgebraLinear algebra is widely applicable in various fields, including computer science and physics. This chapter should cover vector spaces, linear transformations, and eigenvalues. Emphasis should be placed on theconnections between linear algebra and other mathematical disciplines, demonstrating its practical significance.Chapter 7: Number TheoryNumber theory explores the properties and relationships of numbers, and it forms the basis for cryptographic algorithms and computer security systems. This chapter should introduce students to prime numbers, modular arithmetic, and cryptographic algorithms. Examples and exercises should be given to develop students' problem-solving skills in the realm of number theory.Chapter 8: Numerical AnalysisNumerical analysis involves using algorithms to solve mathematical problems on computers. This chapter should cover topics such as interpolation, numerical integration, and numerical solutions of equations. The textbook should provide step-by-step guidance on implementing numerical algorithms, allowing students to develop practical coding skills.ConclusionA comprehensive English textbook for university-level mathematics students should provide a solid foundation in mathematical concepts while simultaneously enhancing students' English language proficiency. By incorporating clear explanations, practical examples, and engaging exercises, this textbook can foster a deep understanding of mathematics within an English language learning context. Such a resource will empower students to pursue further studies in mathematics and apply their skills in various professional domains.。
VERYCD上目前所有的 哈佛耶鲁等世界名校 网络公开课程视频下载资源
VERYCD上目前已有的一些开放课程（1）麻省理工学院《麻省理工开放课程：微积分重点》(Highlights of Calculus)《麻省理工开放课程：单变量微积分》(Single Variable Calculus)《麻省理工开放课程：多变量微积分》(Multivariable Calculus)《麻省理工开放课程：微分方程》(Differential Equations,Spring,2004)《麻省理工开放课程：线性代数》(Linear Algebra)《麻省理工开放课程：经典力学》(Classical MEchanics)《麻省理工开放课程: 物理学I》(Physics I)《麻省理工开放课程：电磁学》(Electricity & Magnetism)《麻省理工开放课程: 振动与波》(Vibrations and Waves)《麻省理工开放课程：航天系统工程学》(Aircraft Systems Engineering)《麻省理工开放课程：算法导论》(Introduction to Algorithms)《麻省理工开放课程：计算机科学及编程导论》(MIT Introduction to Computer Science and Programming)《麻省理工开放课程：计算机程序设计与解释》(Structure and Interpretation of Computer Programs)《麻省理工开放课程：固态化学导论》(Introduction to Solid State Chemistry)《麻省理工开放课程：生物学》(Biology)《麻省理工开放课程：生物学导论》(Introduction to Biology)《麻省理工开放课程：生物工程学导论》(Introduction to Bioengineering)《麻省理工开放课程：西方世界的爱情哲学》(Philosophy of Love in the Western World)《麻省理工开放课程：哥德尔，埃舍尔，巴赫：一次心灵太空漫游》(Gödel, Escher, Bach: A Mental Space Odyssey)《麻省理工开放课程：建筑设计：地景中的建筑》(Architecture Studio : Building in Landscapes)《麻省理工开放课程：电影哲学》(Philosophy of Film)《麻省理工开放课程：艺术、科学和技术中的情感和想象》(Feeling and Imagination in Art, Science, and Technology)《麻省理工开放课程：心理学导论》(Introduction to Psychology)《麻省理工开放课程：西班牙语学习》(Learn Spanish)（2）斯坦福大学《斯坦福大学开放课程：编程范式》(Programming Paradigms )《斯坦福大学开放课程：抽象编程》(Programming Abstractions)《斯坦福大学开放课程：iPhone开发教程》(Phone Application Programming)《斯坦福大学开放课程：编程模式(C和C++)》(Introduction to Computer Science - Programming Abstractions)《斯坦福大学开放课程：编程方法》(Programming Methodology)《斯坦福大学开放课程：人机交互研讨》(Human-Computer Interaction Seminar)《斯坦福大学开放课程：机器学习》(Engineering Everywhere - Machine Learning)《斯坦福大学开放课程：机器人学》(Introduction to robotics)《斯坦福大学开放课程：傅立叶变换及应用》(The Fourier Transform and Its Applications )《斯坦福大学开放课程：近现代物理专题课程-宇宙学》(Modern Physics - Cosmology)《斯坦福大学开放课程：近现代物理专题课程-经典力学》(Modern Physics - Classical Mechanics)《斯坦福大学开放课程：近现代物理专题课程-统计力学》(Modern Physics - Statistic Mechanics)[《斯坦福大学开放课程：近现代物理专题课程-量子力学》(Modern Physics - Quantum Mechanics)《斯坦福大学开放课程：近现代物理专题课程-量子纠缠-part1》(Modern Physics - Quantum Entanglement, Part 1)《斯坦福大学开放课程：近现代物理专题课程-量子纠缠-part3》(Modern Physics - Quantum Entanglement, Part 3)《斯坦福大学开放课程：近现代物理专题课程-广义相对论》(Modern Physics - Einstein's Theory)《斯坦福大学开放课程：近现代物理专题课程-狭义相对论》(Modern Physics - Special Relativity)《斯坦福大学开放课程：线性动力系统绪论》(Introduction to Linear Dynamical Systems)《斯坦福大学开放课程：经济学》(Economics)《斯坦福大学开放课程：商业领袖和企业家》(Business Leaders and Entrepreneurs)《斯坦福大学开放课程：法律学》(Law)《斯坦福大学开放课程：达尔文的遗产》(Darwin's Legacy)《斯坦福大学开放课程：人类健康的未来：七个颠覆你思想的演讲》(The Future of Human Health: 7 Very Short Talks That Will Blow Your Mind)《斯坦福大学开放课程：迷你医学课堂：医学、健康及科技前沿》(Mini Med School：Medicine, Human Health, and the Frontiers of Science)《斯坦福大学开放课程：迷你医学课堂：人类健康之动态》(Mini Med School : The Dynamics of Human Health)（3）耶鲁大学《耶鲁大学开放课程：基础物理》(Fundamentals of Physics)《耶鲁大学开放课程：天体物理学之探索和争议》(Frontiers and Controversies in Astrophysics)《耶鲁大学开放课程：新生有机化学》(Freshman Organic Chemistry )《耶鲁大学开放课程：生物医学工程探索》(Frontiers of Biomedical Engineering)《耶鲁大学开放课程：博弈论》(Game Theory)《耶鲁大学开放课程：金融市场》(Financial Markets )《耶鲁大学开放课程：文学理论导论》(Introduction to Theory of Literature )《耶鲁大学开放课程：现代诗歌》(Modern Poetry)《耶鲁大学开放课程：1945年后的美国小说》(The American Novel Since 1945)《耶鲁大学开放课程: 弥尔顿》(Milton)《耶鲁大学开放课程：欧洲文明》(European Civilization )《耶鲁大学开放课程：旧约全书导论》(Introduction to the Old Testament (Hebrew Bible) )《耶鲁大学开放课程：新约及其历史背景》(Introduction to New Testament History and Literature)《耶鲁大学开放课程：1871年后的法国》(France Since 1871)《耶鲁大学开放课程：古希腊历史简介》(Introduction to Ancient Greek History )《耶鲁大学开放课程：美国内战与重建，1845-1877》(The Civil War and Reconstruction Era，1845-1877)《耶鲁大学开放课程：全球人口增长问题》(Global Problems of Population Growth)《耶鲁大学开放课程：进化，生态和行为原理》(Principles of Evolution, Ecology, and Behavior )《耶鲁大学开放课程：哲学：死亡》(Philosophy：Death)《耶鲁大学开放课程：政治哲学导论》(Introduction to Political Philosophy)《耶鲁大学开放课程：有关食物的心理学，生物学和政治学》(The Psychology, Biology and Politics of Food) 《耶鲁大学开放课程：心理学导论》(Introduction to Psychology)《耶鲁大学开放课程：罗马建筑》(Roman Architecture)《耶鲁大学开放课程：聆听音乐》(Listening to Music)（4）哈佛大学《哈佛大学开放课程：哈佛幸福课》(Positive Psychology at Harvard)《哈佛大学开放课程：公正：该如何做是好？》(Justice: What's the Right Thing to Do? )《哈佛大学开放课程：构设动态网站》(Building Dynamic Websites）（5）牛津大学《牛津大学开放课程：尼采的心灵与自然》(Nietzsche on Mind and Nature)《牛津大学开放课程：哲学概论》(General Philosophy)《牛津大学开放课程：哲学入门》(Philosophy for Beginners)《牛津大学开放课程：批判性推理入门》(Critical Reasoning for Beginners)（6）其它名校《普林斯顿大学开放课程：人性》(InnerCore)《普林斯顿大学开放课程：自由意志定理》(The Free Will Theorem)《剑桥大学开放课程：人类学》(Anthropology)《沃顿商学院开放课程：沃顿知识在线》(Knowledge@Wharton)《哥伦比亚大学开放课程：房地产金融学I》(Real Estate Finance I)附录二部分英美名校开放课程网站美国1. 麻省理工学院/index.htm2. 卡内基梅隆大学/openlearning/forstudents/freecourses3. 约翰霍普金斯大学彭博公共卫生学院/4. 斯坦福大学/5. 圣母大学/courselist6. 杜克大学法律中心的公共领域/cspd/lectures7. 哈佛医学院/public/8. 普林斯顿大学/main/index.php9. 耶鲁大学/10. 加州大学伯克利分校英国1. 牛津大学的文字资料馆2. Greshem学院/default.asp3. 格拉斯哥大学/downloads.html4. 萨里大学/Teaching/5. 诺丁汉大学/6. 剑桥大学播客/main/Podcasts.html参考资料：/thread-42142-1-1.html。
Linear_Algebra_and_It_Applications_Global_5e_01_07
SETS OF ONE ORቤተ መጻሕፍቲ ባይዱTWO VECTORS
A set containing only one vector – say, v – is linearly independent if and only if v is not the zero vector. This is because the vector equation x1v 0 has only the trivial solution when v 0. The zero vector is linearly dependent because x1 0 0 has many nontrivial solutions.
Slide 1.7- 6
© 2016 Pearson Education, Ltd.
LINEAR INDEPENDENCE
Substitute these values into equation (1) and obtain the equation below.
10v1 5v 2 5v3 0
If j 1, then c1v1 0 , which is impossible because v1 0 .
© 2016 Pearson Education, Ltd.
Slide 1.7- 13
SETS OF TWO OR MORE VECTORS
So j 1, and
© 2016 Pearson Education, Ltd.
Slide 1.7- 2
LINEAR INDEPENDENCE
Equation (2) is called a linear dependence relation among v1, …, vp when the weights are not all zero. An indexed set is linearly dependent if and only if it is not linearly independent.
LINEAR ALGEBRA 线性代数 课文 翻译
4 LINEAR ALGEBRA 线性代数“Linear algebra” is the study of linear sets of equations and their transformation properties. 线性代数是研究方程的线性几何以及他们的变换性质的Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.线性代数也研究空间旋转的分析，最小二乘拟合，耦合微分方程的解，确立通过三个已知点的一个圆以及在数学、物理和机械工程上的其他问题The matrix and determinant are extremely useful tools of linear algebra.矩阵和行列式是线性代数极为有用的工具One central problem of linear algebra is the solution of the matrix equation Ax = b for x. 线性代数的一个中心问题是矩阵方程Ax=b关于x的解While this can, in theory, be solved using a matrix inverse x = A−1b，other techniques such as Gaussian elimination are numerically more robust.理论上，他们可以用矩阵的逆x=A-1b求解，其他的做法例如高斯消去法在数值上更鲁棒。
线性规划
1984年印度的Karmarkar提出“投影梯度法”
线性规划是研究线性不等式组的理论，或者 说是研究（高维空间中）凸多面体的理论，是线 性代数的应用和发展。
1-1
线性规划基本概念
生产计划问题
如何合理使用有限的人力，物力 和资金，使得收到最好的经济效益。 如何合理使用有限的人力，物力 和资金，以达到最经济的方式，完 成生产计划的要求。
用于成功决策的实例：
魁北克水利部门关于用哪几个水库来满足每 天电力需求决策 农场信贷系统联邦土地银行关于如何支付到 期债券和应发售多少新债券以获取资金（每年 共计60亿美圆）来维持发展决策 北美长途运输公司关于每周如何调度数千辆 货车的决策 埃克森炼油厂关于调节冶炼能力去适应关于 无铅燃料生产的法律更改的决策
……………………………
b =
b2
……….
am1 am2 …. amn
bn
c1 c2 C= …… cn X=
x1 x2 0= …… xn
0 0 ….. 0
如何将一般问题化为标准型：
•若目标函数是求最小值 Min S = CX 令 S’= - S, 则 Max S’= - CX
如何将一般问题化为标准型：
第一章
线性规划与单纯形方法
1-0
线性规划（概论）
线性规划（Linear Programming)创始人： 1947年美国人G.B.丹齐克（Dantzing）
线性规划（概论）
线性规划（Linear Programming)创始人： 1947年美国人G.B.丹齐克（Dantzing） 1951年提出单纯形算法（Simpler）
各种食物的营养成分表
序 号
食 品 名 称 热 量 蛋 白 质
introduction to linear algebra 5th edition 中译
introduction to linear algebra 5th edition 中译"Introduction to Linear Algebra 5th Edition 中译"主题文章第一步：介绍《Linear Algebra 5th Edition 中译》及其重要性《Linear Algebra 5th Edition 中译》是一本介绍线性代数的教材。
线性代数是数学中的一个重要分支，涉及向量空间、线性方程组、特征值和特征向量等概念。
这些概念在数学、物理学、计算机科学、经济学和工程学等领域都有广泛应用。
本书旨在为读者提供系统、全面且易于理解的线性代数基础知识，帮助读者建立起与今后的学习和研究相结合的数学基础。
此次中文翻译对于那些使用中文学习的学生和研究者，特别是那些需要掌握线性代数以支持他们的学术和职业发展的人来说，具有重要意义。
第二步：介绍线性代数的基本概念和应用线性代数是一门研究线性映射和向量空间的学科。
它涉及将线性方程组转化为矩阵，解决线性方程组的问题，以及在现实世界中描述和解决各种问题的能力。
线性代数也是许多其他学科的基础，例如机器学习、图像处理和优化问题。
通过学习线性代数，读者可以理解向量、矩阵、空间、线性变换、特征值和特征向量等概念，培养抽象思维、推理能力和解决实际问题的能力。
第三步：介绍《Linear Algebra 5th Edition》的特点和优势《Linear Algebra 5th Edition 中译》具有以下特点和优势：1.系统全面：本书从基本概念开始介绍线性代数，逐步展开。
读者可以完整地掌握线性代数的核心概念和理论。
2.易于理解：作者使用简洁明了的语言和示例，帮助读者深入理解抽象的线性代数概念。
此外，作者还提供了大量的习题和解答，以巩固和测试理解。
3.应用广泛：本书覆盖了线性代数的许多应用领域，如物理学、计算机科学和经济学。
读者可以从实际问题出发，将所学的线性代数知识应用到实际中。
数学线性代数教学课件PPT
01
用克莱姆法则求解方程组实 际上相当于用逆矩阵的方法 求解线性方程组，克莱姆法
则常用于理论证明。
02
03
当方程组有解时，将其中单 位列向量对应的未知量取为 非自由未知量，其余的未知
量取为自由未知量。
04
克莱姆法则。用克莱姆法则 求解方程组有两个前提，一 是方程的个数要等于未知量 的个数。二是系数矩阵的行
线性代数定义
01
矩阵的行数和列数可以不一 样，行列式的行数与列数一 致。只能乘以行列式的一行。
02
03
行列式相等，就是值相等， 行和列数目不必相等，数据
也不必相等。
04
矩阵是一个数表，行列式是 一个数值，n阶的方阵。 矩 阵是用括号表示的，行列式
是用双竖线表示的。
一个数乘以矩阵，矩阵的 每个元素都要乘上这个数。 两个矩阵相等是指对应元素
现代线性代数已经扩展到研究任意 或无限维空间。一个维数为 n 的向 量空间叫做n 维空间。
线性算子将线性空间的元素映射到 另一个线性空间，保持向量空间上 加法和标量乘法的一致性。
每一个线性空间都有一 个基。矩阵非奇异（可 逆）当且仅当它的行列 式不为零。矩阵非奇异 当且仅当它代表的线性 变换是个自同构。
对一个 n 行 n 列的非 零矩阵 A，如果存在一 个矩阵 B 使 AB = BA =E（E是单位矩阵）， 则 A 为非奇异矩阵（ 或称可逆矩阵），B为 A的逆阵。
矩阵半正定当且仅当它 的每个特征值大于或等 于零。矩阵正定当且仅 当它的每个特征值都大 于零。解线性方程组的 克拉默法则。
Real battle
列式要不等于零。
矩阵消元法。将线性方程 组的增广矩阵变换化为行简 化阶梯形矩阵 ，简化阶梯 形矩阵为增广矩阵的线性方
线性代数linear algebra
线性代数linear algebra
人们日益依赖互联网，而线性代数作为一种重要的数学工具，也在影响着我们
的网络技术的发展。
线性代数是一门概括矩阵和向量的数学课程，利用矩阵和向量描述系统和关系。
首先，线性代数在机器学习中起着核心作用。
教程和论文中的多数重要的概念，都源自于线性代数，比如梯度下降，矩阵运算，矩阵分解等等，这些都会帮助机器学习模型更好，更快地运行。
此外，线性代数在计算机图形学中仍起着重要作用。
从把复杂的三维模型投影
到屏幕上，到实现全息分发系统，线性代数中提供了一些经典的算法。
即使在现代3D游戏中，也仍然使用到矩阵乘法进行视角转换，并在算法中把矩阵作为一种重
要的数据结构。
最后，线性代数在网络和安全技术中也有重要的作用。
总体而言，矩阵乘法可
以用来加密敏感的通信信息，并在消息发送和接收时实现加密和解密。
例如，RSA
加密算法就是基于矩阵乘法和模数运算，用来确保最终用户数据的安全。
总之，线性代数是一个互联网的重要部分。
从数据分析到计算机图形学，再到
网络和安全，矩阵和向量的数学计算技术，都不断影响着互联网的发展。
只有有效的利用线性代数的知识，才能有效地提高互联网的性能和安全。
LinearAlgebraandItsApplications第五版教学设计
Linear Algebra and Its Applications 第五版教学设计一、概述《Linear Algebra and Its Applications》是一本由Gilbert Strang编写的线性代数教材，其第五版于2016年出版。
在本教学设计中，我们将探讨如何利用这本教材来进行有效的线性代数教学。
二、目标本教学设计的主要目标是使学生能够理解线性代数的基本概念和应用。
具体目标包括：•理解矩阵的概念及其运算规律。
•理解向量空间的概念及其性质。
•学会求解线性方程组。
•理解线性变换及其矩阵表示。
•了解特征值和特征向量的概念及其应用。
三、教学内容1. 矩阵与向量•矩阵的定义和表示。
•线性运算规律：加法、数乘、矩阵乘法。
•矩阵的转置、逆、行列式。
•向量的定义和运算规律。
2. 向量空间•向量空间的定义和性质。
•子空间、基、维数。
•线性变换。
3. 线性方程组•高斯消元法、矩阵消元法。
•对角化、特征值和特征向量。
4. 线性变换•线性变换的定义及其矩阵表示。
•线性变换的性质和应用。
•特征值和特征向量的概念及其应用。
四、教学方法1. 授课采用小班授课的方式，每周3次，每次2个小时。
授课内容以各章节的概念和定理为主，带入一些具体的例子来帮助学生理解。
2. 课后作业每章节都有相应的练习题，学生需要完成每个章节的练习题目，并在下次课堂上提交作业。
老师会在课后指导学生进行作业的讲解，并针对作业中出现的错误加以纠正。
3. 课程设计针对目标和教学内容，设计学生小组项目，例如：使用Python编写线性代数相关的程序实现一些具体的应用场景。
学生可以自行组建小组，任选自己的课题或从老师提供的课题中进行选择。
每个小组需要在学期末提交自己的成果。
4. 确认知识在授课的同时，老师需要利用课程中途来巩固、确认学生的知识水平。
可以采用小测验、小测试、参加讨论等方式。
五、教学评估1. 学习笔记和作业学生需要书写每堂课的笔记，及时整理上课内容，理清思路。
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Level 3
C=C+AB
1
BLAS for Performance
BLAS for Performance
Intel Pentium 4 w/SSE2 1.7 GHz 2000 1500
Mflop/s
IBM RS/6000-590 (66 MHz, 264 Mflop/s Peak)
Level 3 BLAS
• return a matrix or a vector; • O(n 2 ) operations
° saxpy
° sgemv: matrix-vector multiply
• y(i) = a * x(i) + y(i), for i=1 to n. • s stands for single precision, daxpy is for double precision, caxpy for complex, and zaxpy for double complex,
Level 2 BLAS Level 1 BLAS
° Development of blocked algorithms important for performance
7 8
BLAS for Performance
Fast linear algebra kernels: BLAS
° Simple linear algebra kernels such as matrix matrix multiply Level 3 BLAS ° More complicated algorithms can be built from these basic kernels. ° The interfaces of these kernels have been standardized as the Basic Linear Algebra Subroutines (BLAS). ° Early agreement on standard interface (~1980) ° Led to portable libraries for vector and shared memory parallel machines. ° On distributed memory, there is a less -standard interface called the PBLAS
• BLAS1 (1970s): - vector operations: dot product, saxpy(y= α *x+y), etc - m=2*n, f=2*n, q ~1 or less • BLAS2 (mid 1980s) - matrix-vector operations: matrix vector multiply, etc - m=n2 , f=2*n2 , q~2, less overhead - somewhat faster than BLAS1 • BLAS3 (late 1980s) - matrix-matrix operations: matrix matrix multiply, etc - m >= 4n2 , f=O(n3 ), so q can possibly be as large as n, so BLAS3 is potentially much faster than BLAS2
10
Alpha EV 5/6 500MHz (1Gflop/s peak)
700 600 500 400 300 200 100 0 10 100 200 300 400 500
Ordeop/s
Level 2 BLAS Level 1 BLAS
BLAS 3 (n- by-n matrix matrix multiply) vs BLAS 2 (n- by-n matrix vector multiply) vs BLAS 1 ( saxpy of n vectors) 9
Slides are adapted from Jim Demmel, UCB’s Lecture on Linear Algebra Algorithms
2
Memory Hierarchy
° Key to high performance in effective use of memory hierarchy ° True on all architectures
• super scalar • pipelining
° sgemm: Matrix-matrix multiplication
• C = C +A*B, • where C is m-by-n, A is m-by-k, and B is k-by-n
° Complicated compiler interactions ° Hard to do by hand (but you’ll try) ° Automatic optimization an active research area
Level 1, 2 and 3 BLAS
Registers L 1 Cache L 2 Cache Local Memory Remote Memory Secondary Memory
° Level 1 BLAS Vector-Vector operations ° Level 2 BLAS Matrix Vector operations ° Level 3 BLAS Matrix Matrix operations
• y = y + A*x • where A is m-by-n, x is n-by-1 and y is m-by-1. • • • • A = A + y* xT, i.e., A(i,j) = A(i,j)+y(i)*x(j) where A is m-by-n, y is m-by-1, x is n-by-1, strsv: triangular solve solves y=T*x for x, where T is triangular
Level 1 BLAS ° Operate on vectors or pairs of vectors
• perform O(n) operations; • return either a vector or a scalar.
Level 2 BLAS ° Operate on a matrix and a vector;
• solves Y = T*X for X, • where T is a triangular matrix, and X is a rectangular matrix.
13
BLAS -- References
BLAS Papers
° C. Lawson, R. Hanson, D. Kincaid, and F. Krogh, Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions on Mathematical Software, 5:308--325, 1979. ° J. Dongarra, J. Du Croz, S. Hammarling , and R. Hanson, An Extended Set of Fortran Basic Linear Algebra Subprograms , ACM Transactions on Mathematical Software, 14(1):1 --32, 1988. ° J. Dongarra, J. Du Croz, I. Duff, S. Hammarling , A Set of Level 3 Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, 16(1):1--17, 1990.
Level 1
y = y +α x
2n 2 n2 2 n3
Flops/ Memory Refs 2/3 2 n/2
Registers L 1 Cache L 2 Cache Local Memory Remote Memory Secondary Memory 6
Level 2
y=y+Ax
° Good algorithms used BLAS3 when possible (LAPACK) ° /blas, /lapack
+ *
*
+
*
3
4
More on BLAS (Basic Linear Algebra Subroutines)
° Industry standard interface(evolving) ° Vendors, others supply optimized implementations ° History
Jack Dongarra, U of Tennessee
° Performance: manufacturers will provide tuned machine -specific BLAS, ° Program portability: machine dependencies are confined to the BLAS
12
° sger: rank-one update
° sscal y = a * x, for scalar a and vectors x,y ° sdot computes s = S
n i=1
x(i)*y(i)
11
2
Level 3 BLAS ° Operate on pairs or triples of matrices
• PHIPAC: /~bilmes/phipac • /~iyer/asci_slides.ps • ATLAS: /atlas/index.html