Linear Algebra and Its Applications (pastfinal)
国外优秀的高等数学教材
国外优秀的高等数学教材高等数学是大多数理工科学生必修的一门课程，它涵盖了微积分、线性代数、概率统计等多个重要概念和技巧。
为了提高学生的数学素养和应用能力，选择一本优秀的高等数学教材至关重要。
在国外，有很多备受推崇的高等数学教材，它们以其严谨的理论体系、易于理解的讲解方式和丰富的例题，成为了学生们学习和研究的宝贵资源。
本文将介绍几本国外优秀的高等数学教材，希望能为学生们在学习高等数学时提供参考和借鉴。
一、《Calculus: Early Transcendentals》《Calculus: Early Transcendentals》是由美国数学家詹姆斯·斯图尔特（James Stewart）所著的一本高等数学教材。
这本教材几乎成为了全球许多大学的高等数学教材标准教材，并且荣获了多个数学教育奖项。
其主要特点包括：1. 结构清晰：教材按照章节和节的结构编排，便于学生系统地学习和复习微积分的各个概念。
2. 知识严谨：该教材注重理论证明和逻辑推导，帮助学生深入理解微积分的原理和定理。
3. 真实应用: 《Calculus: Early Transcendentals》在理论讲解之外，还提供了大量真实世界中的应用例题，帮助学生理解微积分在物理、工程等领域的相关应用。
二、《Linear Algebra and Its Applications》《Linear Algebra and Its Applications》由美国数学家大卫·莱(David C. Lay)所著，是一本系统全面讲解线性代数的经典教材。
其主要特点包括：1. 清晰易懂：教材注重讲解线性代数的基本概念、定理和相关技巧，以简明易懂的语言指导学生。
2. 应用广泛：该教材将线性代数与现实生活中的问题相结合，以应用为导向，帮助学生更好地理解并应用线性代数的概念。
3. 丰富例题：《Linear Algebra and Its Applications》提供了大量的例题和习题，旨在让学生通过实战来加深对线性代数知识的理解和掌握。
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。
哈工大选修课 LINEAR ALGEBRA 试卷及答案
LINEAR ALGEBRAANDITS APPLICATIONS 姓名：易学号：成绩：1. Definitions(1) Pivot position in a matrix; (2) Echelon Form; (3) Elementary operations;(4) Onto mapping and one-to-one mapping; (5) Linearly independence.2. Describe the row reduction algorithm which produces a matrix in reduced echelon form.3. Find the 33⨯ matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90︒, and finally a translation that adds (-0.5, 2) to each point of a figure.4. Find a basis for the null space of the matrix361171223124584A ---⎡⎤⎢⎥=--⎢⎥⎢⎥--⎣⎦5. Find a basis for Col A of the matrix1332-9-2-22-822307134-111-8A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦6. Let a and b be positive numbers. Find the area of the region bounded by the ellipse whose equation is22221x y ab+=7. Provide twenty statements for the invertible matrix theorem. 8. Show and prove the Gram-Schmidt process. 9. Show and prove the diagonalization theorem.10. Prove that the eigenvectors corresponding to distinct eigenvalues are linearly independent.Answers:1. Definitions(1) Pivot position in a matrix:A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.(2) Echelon Form:A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:1.All nonzero rows are above any rows of all zeros.2.Each leading entry of a row is in a column to the right of the leading entry of the row above it.3.All entries in a column below a leading entry are zeros.If a matrix in a echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):4.The leading entry in each nonzero row is 1.5.Each leading 1 is the only nonzero entry in its column.(3)Elementary operations:Elementary operations can refer to elementary row operations or elementary column operations.There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):1.(Replacement) Replace one row by the sum of itself anda multiple of another row.2.(Interchange) Interchange two rows.3.(scaling) Multiply all entries in a row by a nonzero constant.(4)Onto mapping and one-to-one mapping:A mapping T : n →m is said to be onto m if each b in m is the image of at least one x in n.A mapping T : n →m is said to be one-to-one if each b in m is the image of at most one x in n.(5)Linearly independence:An indexed set of vectors {V1, . . . ,V p} in n is said to be linearly independent if the vector equationx 1v 1+x 2v 2+ . . . +x p v p = 0Has only the trivial solution. The set {V 1, . . . ,V p } is said to be linearly dependent if there exist weights c 1, . . . ,c p , not all zero, such that c 1v 1+c 2v 2+ . . . +c p v p = 02. Describe the row reduction algorithm which produces a matrix in reduced echelon form. Solution: Step 1:Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top. Step 2:Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position. Step 3:Use row replacement operations to create zeros in all positions below the pivot. Step 4:Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there all no more nonzero rows to modify. Step 5:Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by scaling operation.3. Find the 33⨯ matrix that corresponds to the composite transformation of a scaling by 0.3, a rotation of 90︒, and finally a translation that adds (-0.5, 2) to each point of a figure. Solution:If ψ=π/2, then sin ψ=1 and cos ψ=0. Then we have ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−→−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡110003.00003.01y x y x scale⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-−−→−110003.00003.0100001010y x R o t a t e⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-−−−→−110003.00003.0100001010125.0010001y x T r a n s l a t eThe matrix for the composite transformation is ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-10003.00003.0100001010125.0010001⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--=10003.00003.0125.0001010⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--=125.0003.003.004. Find a basis for the null space of the matrix 361171223124584A ---⎡⎤⎢⎥=--⎢⎥⎢⎥--⎣⎦Solution:First, write the solution of A X=0 in parametric vector form: A ~ ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡---00002302101000201, x 1-2x 2 -x 4+3x 5=0 x 3+2x 4-2x 5=0 0=0The general solution is x 1=2x 2+x 4-3x 5, x 3=-2x 4+2x 5, with x 2, x 4, and x 5 free. ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡-+⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡-+⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡+--+=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡10203012010001222325425454254254321x x x x x x x x x x x x x x x xu v w=x 2u+x 4v+x 5w (1)Equation (1) shows that Nul A coincides with the set of all linear conbinations of u, v and w . That is, {u, v, w}generates Nul A. In fact, this construction of u, v and w automatically makes them linearly independent, because (1) shows that 0=x 2u+x 4v+x 5w only if the weights x 2, x 4, and x 5 are all zero.So {u, v , w} is a basis for Nul A.5. Find a basis for Col A of the matrix 1332-9-2-22-822307134-111-8A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦Solution: A ~ ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡---07490012002300130001, so the rank of A is 3. Then we have a basis for Col A of the matrix: U = ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡0001, v = ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡0013and w = ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡--07496. Let a and b be positive numbers. Find the area of the region bounded by the ellipse whose equation is22221x y ab+=Solution:We claim that E is the image of the unit disk D under the linear transformation Tdetermined by the matrix A=⎥⎦⎤⎢⎣⎡b a 00, because if u= ⎥⎦⎤⎢⎣⎡21u u , x=⎥⎦⎤⎢⎣⎡21x x , and x = Au, then u 1 =ax 1 and u 2 =bx 2It follows that u is in the unit disk, with 12221≤+u u , if and only if x is in E , with1)()(2221≤+b x a x . Then we have{area of ellipse} = {area of T (D )} = |det A| {area of D} = ab π(1)2= πab7. Provide twenty statements for the invertible matrix theorem.Let A be a square n n ⨯ matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or false. a. A is an invertible matrix.b. A is row equivalent to the n n ⨯ identity matrix.c. A has n pivot positions.d. The equation Ax = 0 has only the trivial solution.e. The columns of A form a linearly independent set.f. The linear transformation x → Ax is one-to-one.g. The equation Ax = b has at least one solution for each b in n.h. The columns of A spann.i. The linear transformation x → Ax maps nonton.j. There is an n n ⨯ matrix C such that CA = I. k. There is an n n ⨯ matrix D such that AD = I. l. A T is an invertible matrix. m. If 0A ≠, then ()()T11T A A --=n. If A, B are all invertible, then (AB)* = B *A *o. T**T )(A )(A =p. If 0A ≠, then ()()*11*A A --=q. ()*1n *A 1)(A --=-r. If 0A ≠, then ()()L11L A A --= ( L is a natural number )s. ()*1n *A K)(KA --=-t. If 0A ≠, then *1A A1A =-8. Show and prove the Gram-Schmidt process.Solution:The Gram-Schmidt process:Given a basis {x 1, . . . , x p } for a subspace W of n, define11x v = 1112222v v v v x x v ⋅⋅-=222231111333v v v v x v v v v x x v ⋅⋅-⋅⋅-=. ..1p 1p 1p 1p p 2222p 1111p p p v v v v x v v v v x v v v v x x v ----⋅-⋅⋅⋅-⋅⋅-⋅⋅-=Then {v 1, . . . , v p } is an orthogonal basis for W. In additionSpan {v 1, . . . , v p } = {x 1, . . . , x p } for p k ≤≤1 PROOFFor p k ≤≤1, let W k = Span {v 1, . . . , v p }. Set 11x v =, so that Span {v 1} = Span {x 1}.Suppose, for some k < p, we have constructed v 1, . . . , v k so that {v 1, . . . , v k } is an orthogonal basis for W k . Define1k w1k 1k x p r o j x v k+++-= By the Orthogonal Decomposition Theorem, v k+1 is orthogonal to W k . Note that proj Wk x k+1 is in W k and hence also in W k+1. Since x k+1 is in W k+1, so is v k+1 (because W k+1 is a subspace and is closed under subtraction). Furthermore, 0v 1k ≠+ because x k+1 is not in W k = Span {x 1, . . . , x p }. Hence {v 1, . . . , v k } is an orthogonal set of nonzero vectors in the (k+1)-dismensional space W k+1. By the Basis Theorem, this set is an orthogonal basis for W k+1. Hence W k+1 = Span {v 1, . . . , v k+1}. When k + 1 = p, the process stops.9. Show and prove the diagonalization theorem. Solution:diagonalization theorem:If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal. PROOFLet v 1 and v 2 be eigenvectors that correspond to distinct eigenvalues, say, 1λand 2λ. T o show that 0v v 21=⋅, compute2T 12T 11211v )(A v v )v (λv v λ==⋅ Since v 1 is an eigenvector ()()2T12T T1Avv v A v ==)(221v v Tλ=2122T12v v λv v λ⋅==Hence ()0v v λλ2121=⋅-, but ()0λλ21≠-, so 0v v 21=⋅10. Prove that the eigenvectors corresponding to distinct eigenvalues are linearly independent. Solution:If v 1, . . . , v r are eigenvectors that correspond to distinct eignvalues λ1, . . . , λr of an n n ⨯ matrix A.Suppose {v 1, . . . , v r } is linearly dependent. Since v 1 is nonzero, Theorem, Characterization of Linearly Dependent Sets, says that one of the vectors in the set is linear combination of the preceding vectors. Let p be the least index such that v p +1 is a linear combination of he preceding (linearly independent) vectors. Then there exist scalars c 1, . . . ,c p such that 1p p p 11v v c v c +=+⋅⋅⋅+ (1) Multiplying both sides of (1) by A and using the fact that Av k = λk v k for each k, we obtain 111+=+⋅⋅⋅+p p p Av Av c Av c11111++=+⋅⋅⋅+p p p p p v v c v c λλλ (2) Multiplying both sides of (1) by 1+p λ and subtracting the result from (2), we have0)()(11111=-+⋅⋅⋅+-++p p p p c v c λλλλ (3) Since {v 1, . . . , v p } is linearly independent, the weights in (3) are all zero. But none of the factors 1+-p i λλ are zero, because the eigenvalues are distinct. Hence 0=i c for i = 1, . . . ,p. But when (1) says that 01=+p v , which is impossible. Hence {v 1, . . . , v r } cannot be linearly dependent and therefore must be linearly independent.。
Linear Algebra and its Applications
Linear Algebra and its Applications432(2010)2089–2099Contents lists available at ScienceDirect Linear Algebra and its Applications j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/l aaIntegrating learning theories and application-based modules in teaching linear algebraୋWilliam Martin a,∗,Sergio Loch b,Laurel Cooley c,Scott Dexter d,Draga Vidakovic ea Department of Mathematics and School of Education,210F Family Life Center,NDSU Department#2625,P.O.Box6050,Fargo ND 58105-6050,United Statesb Department of Mathematics,Grand View University,1200Grandview Avenue,Des Moines,IA50316,United Statesc Department of Mathematics,CUNY Graduate Center and Brooklyn College,2900Bedford Avenue,Brooklyn,New York11210, United Statesd Department of Computer and Information Science,CUNY Brooklyn College,2900Bedford Avenue Brooklyn,NY11210,United Statese Department of Mathematics and Statistics,Georgia State University,University Plaza,Atlanta,GA30303,United StatesA R T I C L E I N F O AB S T R AC TArticle history:Received2October2008Accepted29August2009Available online30September2009 Submitted by L.Verde-StarAMS classification:Primary:97H60Secondary:97C30Keywords:Linear algebraLearning theoryCurriculumPedagogyConstructivist theoriesAPOS–Action–Process–Object–Schema Theoretical frameworkEncapsulated process The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a)linear algebra and(b)learning theory as applied to the study of mathematics with an emphasis on linear algebra.The purpose of the ongoing research,partially funded by the National Science Foundation,is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some students,a rich understanding of both domains and that had a mutually reinforcing effect.Furthermore,there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and,consequently,better learning by their students.The courses developed were appropriate for mathematics majors,pre-service secondary mathematics teachers, and practicing mathematics teachers.The learning seminar focused most heavily on constructivist theories,although it also examinedThe work reported in this paper was partially supported by funding from the National Science Foundation(DUE CCLI 0442574).∗Corresponding author.Address:NDSU School of Education,NDSU Department of Mathematics,210F Family Life Center, NDSU Department#2625,P.O.Box6050,Fargo ND58105-6050,United States.Tel.:+17012317104;fax:+17012317416.E-mail addresses:william.martin@(W.Martin),sloch@(S.Loch),LCooley@ (L.Cooley),SDexter@(S.Dexter),dvidakovic@(D.Vidakovic).0024-3795/$-see front matter©2009Elsevier Inc.All rights reserved.doi:10.1016/a.2009.08.0302090W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099Thematicized schema Triad–intraInterTransGenetic decomposition Vector additionMatrixMatrix multiplication Matrix representation BasisColumn spaceRow spaceNull space Eigenspace Transformation socio-cultural and historical perspectives.A particular theory, Action–Process–Object–Schema(APOS)[10],was emphasized and examined through the lens of studying linear algebra.APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics.The linear algebra courses include the standard set of undergraduate topics.This paper reports the re-sults of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.©2009Elsevier Inc.All rights reserved.1.Research rationaleThe research team of the Linear Algebra Project(LAP)developed and implemented a curriculum and a pedagogy for parallel courses in linear algebra and learning theory as applied to the study of math-ematics with an emphasis on linear algebra.The purpose of the research,which was partially funded by the National Science Foundation(DUE CCLI0442574),was to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of high school mathematics teachers,in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some teachers,a richer understanding of both domains that had a mutually reinforcing effect and affected their thinking about their identities and practices as teachers.It has been observed that linear algebra courses often are viewed by students as a collection of definitions and procedures to be learned by rote.Scanning the table of contents of many commonly used undergraduate textbooks will provide a common list of terms such as listed here(based on linear algebra texts by Strang[1]and Lang[2]).Vector space Kernel GaussianIndependence Image TriangularLinear combination Inverse Gram–SchmidtSpan Transpose EigenvectorBasis Orthogonal Singular valueSubspace Operator DecompositionProjection Diagonalization LU formMatrix Normal form NormDimension Eignvalue ConditionLinear transformation Similarity IsomorphismRank Diagonalize DeterminantThis is not something unique to linear algebra–a similar situation holds for many undergraduate mathematics courses.Certainly the authors of undergraduate texts do not share this student view of mathematics.In fact,the variety ways in which different authors organize their texts reflects the individual ways in which they have conceptualized introductory linear algebra courses.The wide vari-ability that can be seen in a perusal of the many linear algebra texts that are used is a reflection the many ways that mathematicians think about linear algebra and their beliefs about how students can come to make sense of the content.Instruction in a course is based on considerations of content,pedagogy, resources(texts and other materials),and beliefs about teaching and learning of mathematics.The interplay of these ideas shaped our research project.We deliberately mention two authors with clearly differing perspectives on an undergraduate linear algebra course:Strang’s organization of the material takes an applied or application perspective,while Lang views the material from more of a“pure mathematics”perspective.A review of the wide variety of textbooks to classify and categorize the different views of the subject would reveal a broad variety of perspectives on the teaching of the subject.We have taken a view that seeks to go beyond the mathe-matical content to integrate current theoretical perspectives on the teaching and learning of undergrad-uate mathematics.Our project used integration of mathematical content,applications,and learningW.Martin et al./Linear Algebra and its Applications432(2010)2089–20992091 theories to provide enhanced learning experiences using rich content,student meta cognition,and their own experience and intuition.The project also used co-teaching and collaboration among faculty with expertise in a variety of areas including mathematics,computer science and mathematics education.If one moves beyond the organization of the content of textbooks wefind that at their heart they do cover a common core of the key ideas of linear algebra–all including fundamental concepts such as vector space and linear transformation.These observations lead to our key question“How is one to think about this task of organizing instruction to optimize learning?”In our work we focus on the conception of linear algebra that is developed by the student and its relationship with what we reveal about our own understanding of the subject.It seems that even in cases where researchers consciously study the teaching and learning of linear algebra(or other mathematics topics)the questions are“What does it mean to understand linear algebra?”and“How do I organize instruction so that students develop that conception as fully as possible?”In broadest terms, our work involves(a)simultaneous study of linear algebra and learning theories,(b)having students connect learning theories to their study of linear algebra,and(c)the use of parallel mathematics and education courses and integrated workshops.As students simultaneously study mathematics and learning theory related to the study of mathe-matics,we expect that reflection or meta cognition on their own learning will enable them to construct deeper and more meaningful understanding in both domains.We chose linear algebra for several reasons:It has not been the focus of as much instructional research as calculus,it involves abstraction and proof,and it is taken by many students in different programs for a variety of reasons.It seems to us to involve important mathematical content along with rich applications,with abstraction that builds on experience and intuition.In our pilot study we taught parallel courses:The regular upper division undergraduate linear algebra course and a seminar in learning theories in mathematics education.Early in the project we also organized an intensive three-day workshop for teachers and prospective teachers that included topics in linear algebra and examination of learning theory.In each case(two sets of parallel courses and the workshop)we had students reflect on their learning of linear algebra content and asked them to use their own learning experiences to reflect on the ideas about teaching and learning of mathematics.Students read articles–in the case of the workshop,this reading was in advance of the long weekend session–drawn from mathematics education sources including[3–10].APOS(Action,Process,Object,Schema)is a theoretical framework that has been used by many researchers who study the learning of undergraduate and graduate mathematics[10,11].We include a sketch of the structure of this framework and refer the reader to the literature for more detailed descriptions.More detailed and specific illustrations of its use are widely available[12].The APOS Theoretical Framework involves four levels of understanding that can be described for a wide variety of mathematical concepts such as function,vector space,linear transformation:Action,Process,Object (either an encapsulated process or a thematicized schema),Schema(Intra,inter,trans–triad stages of schema formation).Genetic decomposition is the analysis of a particular concept in which developing understanding is described as a dynamic process of mental constructions that continually develop, abstract,and enrich the structural organization of an individual’s knowledge.We believe that students’simultaneous study of linear algebra along with theoretical examination of teaching and learning–particularly on what it means to develop conceptual understanding in a domain –will promote learning and understanding in both domains.Fundamentally,this reflects our view that conceptual understanding in any domain involves rich mental connections that link important ideas or facts,increasing the individual’s ability to relate new situations and problems to that existing cognitive framework.This view of conceptual understanding of mathematics has been described by various prominent math education researchers such as Hiebert and Carpenter[6]and Hiebert and Lefevre[7].2.Action–Process–Object–Schema theory(APOS)APOS theory is a theoretical perspective of learning based on an interpretation of Piaget’s construc-tivism and poses descriptions of mental constructions that may occur in understanding a mathematical concept.These constructions are called Actions,Processes,Objects,and Schema.2092W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099 An action is a transformation of a mathematical object according to an explicit algorithm seen as externally driven.It may be a manipulation of objects or acting upon a memorized fact.When one reflects upon an action,constructing an internal operation for a transformation,the action begins to be interiorized.A process is this internal transformation of an object.Each step may be described or reflected upon without actually performing it.Processes may be transformed through reversal or coordination with other processes.There are two ways in which an individual may construct an object.A person may reflect on actions applied to a particular process and become aware of the process as a totality.One realizes that transformations(whether actions or processes)can act on the process,and is able to actually construct such transformations.At this point,the individual has reconstructed a process as a cognitive object. In this case we say that the process has been encapsulated into an object.One may also construct a cognitive object by reflecting on a schema,becoming aware of it as a totality.Thus,he or she is able to perform actions on it and we say the individual has thematized the schema into an object.With an object conception one is able to de-encapsulate that object back into the process from which it came, or,in the case of a thematized schema,unpack it into its various components.Piaget and Garcia[13] indicate that thematization has occurred when there is a change from usage or implicit application to consequent use and conceptualization.A schema is a collection of actions,processes,objects,and other previously constructed schemata which are coordinated and synthesized to form mathematical structures utilized in problem situations. Objects may be transformed by higher-level actions,leading to new processes,objects,and schemata. Hence,reconstruction continues in evolving schemata.To illustrate different conceptions of the APOS theory,imagine the following’teaching’scenario.We give students multi-part activities in a technology supported environment.In particular,we assume students are using Maple in the computer lab.The multi-part activities,focusing on vectors and operations,in Maple begin with a given Maple code and drawing.In case of scalar multiplication of the vector,students are asked to substitute one parameter in the Maple code,execute the code and observe what has happened.They are asked to repeat this activity with a different value of the parameter.Then students are asked to predict what will happen in a more general case and to explain their reasoning.Similarly,students may explore addition and subtraction of vectors.In the next part of activity students might be asked to investigate about the commutative property of vector addition.Based on APOS theory,in thefirst part of the activity–in which students are asked to perform certain operation and make observations–our intention is to induce each student’s action conception of that concept.By asking students to imagine what will happen if they make a certain change–but do not physically perform that change–we are hoping to induce a somewhat higher level of students’thinking, the process level.In order to predict what will happen students would have to imagine performing the action based on the actions they performed before(reflective abstraction).Activities designed to explore on vector addition properties require students to encapsulate the process of addition of two vectors into an object on which some other action could be performed.For example,in order for a student to conclude that u+v=v+u,he/she must encapsulate a process of adding two vectors u+v into an object(resulting vector)which can further be compared[action]with another vector representing the addition of v+u.As with all theories of learning,APOS has a limitation that researchers may only observe externally what one produces and discusses.While schemata are viewed as dynamic,the task is to attempt to take a snap shot of understanding at a point in time using a genetic decomposition.A genetic decomposition is a description by the researchers of specific mental constructions one may make in understanding a mathematical concept.As with most theories(economics,physics)that have restrictions,it can still be very useful in describing what is observed.3.Initial researchIn our preliminary study we investigated three research questions:•Do participants make connections between linear algebra content and learning theories?•Do participants reflect upon their own learning in terms of studied learning theories?W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992093•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?In addition to linear algebra course activities designed to engage students in explorations of concepts and discussions about learning theories and connections between the two domains,we had students construct concept maps and describe how they viewed the connections between the two subjects. We found that some participants saw significant connections and were able to apply APOS theory appropriately to their learning of linear algebra.For example,here is a sketch outline of how one participant described the elements of the APOS framework late in the semester.The student showed a reasonable understanding of the theoretical framework and then was able to provide an example from linear algebra to illustrate the model.The student’s description of the elements of APOS:Action:“Students’approach is to apply‘external’rules tofind solutions.The rules are said to be external because students do not have an internalized understanding of the concept or the procedure tofind a solution.”Process:“At the process level,students are able to solve problems using an internalized understand-ing of the algorithm.They do not need to write out an equation or draw a graph of a function,for example.They can look at a problem and understand what is going on and what the solution might look like.”Object level as performing actions on a process:“At the object level,students have an integrated understanding of the processes used to solve problems relating to a particular concept.They un-derstand how a process can be transformed by different actions.They understand how different processes,with regard to a particular mathematical concept,are related.If a problem does not conform to their particular action-level understanding,they can modify the procedures necessary tofind a solution.”Schema as a‘set’of knowledge that may be modified:“Schema–At the schema level,students possess a set of knowledge related to a particular concept.They are able to modify this set of knowledge as they gain more experience working with the concept and solving different kinds of problems.They see how the concept is related to other concepts and how processes within the concept relate to each other.”She used the ideas of determinant and basis to illustrate her understanding of the framework. (Another student also described how student recognition of the recursive relationship of computations of determinants of different orders corresponded to differing levels of understanding in the APOS framework.)Action conception of determinant:“A student at the action level can use an algorithm to calculate the determinant of a matrix.At this level(at least for me),the formula was complicated enough that I would always check that the determinant was correct byfinding the inverse and multiplying by the original matrix to check the solution.”Process conception of determinant:“The student knows different methods to use to calculate a determinant and can,in some cases,look at a matrix and determine its value without calculations.”Object conception:“At the object level,students see the determinant as a tool for understanding and describing matrices.They understand the implications of the value of the determinant of a matrix as a way to describe a matrix.They can use the determinant of a matrix(equal to or not equal to zero)to describe properties of the elements of a matrix.”Triad development of a schema(intra,inter,trans):“A singular concept–basis.There is a basis for a space.The student can describe a basis without calculation.The student canfind different types of bases(column space,row space,null space,eigenspace)and use these values to describe matrices.”The descriptions of components of APOS along with examples illustrate that this student was able to make valid connections between the theoretical framework and the content of linear algebra.While the2094W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099descriptions may not match those that would be given by scholars using APOS as a research framework, the student does demonstrate a recognition of and ability to provide examples of how understanding of linear algebra can be organized conceptually as more that a collection of facts.As would be expected,not all participants showed gains in either domain.We viewed the results of this study as a proof of concept,since there were some participants who clearly gained from the experience.We also recognized that there were problems associated with the implementation of our plan.To summarize ourfindings in relation to the research questions:•Do participants make connections between linear algebra content and learning theories?Yes,to widely varying degrees and levels of sophistication.•Do participants reflect upon their own learning in terms of studied learning theories?Yes,to the extent possible from their conception of the learning theories and understanding of linear algebra.•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?Participants describe how their experiences will shape their own teaching,but we did not visit their classes.Of the11students at one site who took the parallel courses,we identified three in our case studies (a detailed report of that study is presently under review)who demonstrated a significant ability to connect learning theories with their own learning of linear algebra.At another site,three teachers pursuing math education graduate studies were able to varying degrees to make these connections –two demonstrated strong ability to relate content to APOS and described important ways that the experience had affected their own thoughts about teaching mathematics.Participants in the workshop produced richer concept maps of linear algebra topics by the end of the weekend.Still,there were participants who showed little ability to connect material from linear algebra and APOS.A common misunderstanding of the APOS framework was that increasing levels cor-responded to increasing difficulty or complexity.For example,a student might suggest that computing the determinant of a2×2matrix was at the action level,while computation of a determinant in the 4×4case was at the object level because of the increased complexity of the computations.(Contrast this with the previously mentioned student who observed that the object conception was necessary to recognize that higher dimension determinants are computed recursively from lower dimension determinants.)We faced more significant problems than the extent to which students developed an understanding of the ideas that were presented.We found it very difficult to get students–especially undergraduates –to agree to take an additional course while studying linear algebra.Most of the participants in our pilot projects were either mathematics teachers or prospective mathematics teachers.Other students simply do not have the time in their schedules to pursue an elective seminar not directly related to their own area of interest.This problem led us to a new project in which we plan to integrate the material on learning theory–perhaps implicitly for the students–in the linear algebra course.Our focus will be on working with faculty teaching the course to ensure that they understand the theory and are able to help ensure that course activities reflect these ideas about learning.4.Continuing researchOur current Linear Algebra in New Environments(LINE)project focuses on having faculty work collaboratively to develop a series of modules that use applications to help students develop conceptual understanding of key linear algebra concepts.The project has three organizing concepts:•Promote enhanced learning of linear algebra through integrated study of mathematical content, applications,and the learning process.•Increase faculty understanding and application of mathematical learning theories in teaching linear algebra.•Promote and support improved instruction through co-teaching and collaboration among faculty with expertise in a variety of areas,such as education and STEM disciplines.W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992095 For example,computer and video graphics involve linear transformations.Students will complete a series of activities that use manipulation of graphical images to illustrate and help them move from action and process conceptions of linear transformations to object conceptions and the development of a linear transformation schema.Some of these ideas were inspired by material in Judith Cederberg’s geometry text[14]and some software developed by David Meel,both using matrix representations of geometric linear transformations.The modules will have these characteristics:•Embed learning theory in linear algebra course for both the instructor and the students.•Use applied modules to illustrate the organization of linear algebra concepts.•Applications draw on student intuitions to aid their mental constructions and organization of knowledge.•Consciously include meta-cognition in the course.To illustrate,we sketch the outline of a possible series of activities in a module on geometric linear transformations.The faculty team–including individuals with expertise in mathematics,education, and computer science–will develop a series of modules to engage students in activities that include reflection and meta cognition about their learning of linear algebra.(The Appendix contains a more detailed description of a module that includes these activities.)Task1:Use Photoshop or GIMP to manipulate images(rotate,scale,flip,shear tools).Describe and reflect on processes.This activity uses an ACTION conception of transformation.Task2:Devise rules to map one vector to another.Describe and reflect on process.This activity involves both ACTION and PROCESS conceptions.Task3:Use a matrix representation to map vectors.This requires both PROCESS and OBJECT conceptions.Task4:Compare transform of sum with sum of transforms for matrices in Task3as compared to other non-linear functions.This involves ACTION,PROCESS,and OBJECT conceptions.Task5:Compare pre-image and transformed image of rectangles in the plane–identify software tool that was used(from Task1)and how it might be represented in matrix form.This requires OBJECT and SCHEMA conceptions.Education,mathematics and computer science faculty participating in this project will work prior to the semester to gain familiarity with the APOS framework and to identify and sketch potential modules for the linear algebra course.During the semester,collaborative teams of faculty continue to develop and refine modules that reflect important concepts,interesting applications,and learning theory:Modules will present activities that help students develop important concepts rather than simply presenting important concepts for students to absorb.The researchers will study the impact of project activities on student learning:We expect that students will be able to describe their knowledge of linear algebra in a more conceptual(structured) way during and after the course.We also will study the impact of the project on faculty thinking about teaching and learning:As a result of this work,we expect that faculty will be able to describe both the important concepts of linear algebra and how those concepts are mentally developed and organized by students.Finally,we will study the impact on instructional practice:Participating faculty should continue to use instructional practices that focus both on important content and how students develop their understanding of that content.5.SummaryOur preliminary study demonstrated that prospective and practicing mathematics teachers were able to make connections between their concurrent study of linear algebra and of learning theories relating to mathematics education,specifically the APOS theoretical framework.In cases where the participants developed understanding in both domains,it was apparent that this connected learning strengthened understanding in both areas.Unfortunately,we were unable to encourage undergraduate students to consider studying both linear algebra and learning theory in separate,parallel courses. Consequently,we developed a new strategy that embeds the learning theory in the linear algebra。
相似变换矩阵p的求法
相似变换矩阵p的求法相似变换矩阵P的求法，可以通过以下步骤进行：1. 求解特征向量和特征值：对于给定的原始矩阵A，首先需要求解其特征向量和特征值。
特征向量是一个非零向量，其满足以下关系式：Av=λv，其中A是原始矩阵，v是特征向量，λ是特征值。
可以通过求解A的特征方程来得到特征值，然后通过求解(A-λI)v=0来得到特征向量，其中I是单位矩阵。
2. 构建相似变换矩阵P：得到特征向量后，将它们按列组成一个矩阵P。
这个矩阵P就是相似变换矩阵。
3. 检验相似性：将矩阵P应用于原始矩阵A上，得到P^-1AP，其中P^-1是P的逆矩阵。
如果P^-1AP可以化简为一个对角矩阵，即存在对角矩阵D使得P^-1AP=D，那么矩阵A和D是相似的。
相似变换矩阵的求法还可以通过以下参考内容进行进一步学习：1. 《线性代数及其应用》（Linear Algebra and Its Applications）：本书是Gilbert Strang编写的一本经典线性代数教材，对相似变换矩阵的求法有详细的介绍，并提供了相关的例题和习题来加深理解。
2. 《数学分析与线性代数》（Mathematical Analysis and Linear Algebra）：这本书由同济大学出版社出版，是一本针对工科类专业的线性代数入门教材。
其中包括了相似变换矩阵的求法，结合实际应用情况进行了讲解。
3. 相关的课程讲义和教学视频：可以搜索在线教育平台（如Coursera、edX、网络大学等）上的线性代数课程，其中会有相关的讲义和教学视频，可以更加形象地解释相似变换矩阵的求法。
4. 线性代数在线学习资源：线性代数的在线学习资源，如Khan Academy和MIT OpenCourseWare等，提供了许多免费的线性代数教材和视频，其中包括了相似变换矩阵的求法内容。
总之，相似变换矩阵P的求法涉及到求解特征向量和特征值，构建相似变换矩阵P，以及检验相似性。
Linear_Algebra_and_It_Applications_Global_5e_02_02
A is invertible and
This theorem says that a 2 2 matrix A is invertible if and only if det A 0 .
© 2016 Pearson Education, Ltd. Slide 2.2- 3
MATRIX OPERATIONS
Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A.
© 2016 Pearson Education, Ltd. Slide 2.2- 8
© 2016 Pearson Education, Ltd. Slide 2.2- 7
1 ( A )T. is invertible, and its inverse is
ELEMENTARY MATRICES
1 Example 5: Let E1 0 4 1 0 0 a E3 0 1 0 , A d g 0 0 5 0 0 0 1 0 1 0 , E2 1 0 0 , 0 1 0 0 1 b c e f h i
1
1
© 2016 Pearson Education, Ltd. Slide 2.2- 14
ALGORITHM FOR FINDING A
1
Example 2: Find the inverse of the matrix
1 0 0 A 1 0 3 , if it exists. 4 3 8
© 2016 Pearson Education, Ltd. Slide 2.2- 6
线性代数及应用学习指导
线性代数及应用学习指导线性代数是数学的一个重要分支，主要研究线性空间与线性映射的性质及其应用。
它广泛应用于数学、物理学、工程学以及计算机科学等领域。
以下是学习线性代数的指导和建议。
1. 巩固基础知识：学习线性代数前，要确保自己对基础数学知识，如数学分析、高等代数等有一定的了解和掌握。
这将有助于理解和应用线性代数的概念和方法。
2. 学习教材选择：选择一本系统、全面的线性代数教材进行学习。
推荐的经典教材包括《线性代数及其应用》(Linear Algebra and its Applications)、《线性代数导论》(A First Course in Linear Algebra)等。
这些教材内容丰富，例题和习题较多，学完后可以打下较扎实的线性代数基础。
3. 学习方法：线性代数的学习需要理论与实践相结合。
可以先通过阅读教材，理解概念、定理和证明过程。
然后，重点关注典型例题的解法和思路，尝试自己推导和求解。
最后，通过习题进行巩固和拓展。
练习不同类型的习题有助于培养解决实际问题的能力。
4. 注意直观理解：线性代数的概念较抽象，有时难以直接理解。
但依然需要努力培养直观理解能力。
例如，对于矩阵、向量等，可以通过几何直观去理解它们的性质和运算规则。
5. 多角度思考和应用：线性代数是一门非常广泛的学科，能够应用到各个领域。
学习线性代数时，可以尝试从不同的角度思考问题，如几何、物理、工程等，加深对知识的理解和应用。
6. 利用网络资源：线性代数涉及的知识点较多，可以利用网络资源去查找相关教学视频、学习资料和练习题。
高质量的线上课程，如Coursera、网易云课堂等，可以帮助学生更深入地理解和应用知识。
7. 培养编程能力：线性代数在计算机科学领域有着广泛的应用。
掌握编程语言，如Python、MATLAB等，可以通过程序实现仿真、数据分析等，加深对线性代数的理解和应用。
总之，学习线性代数需要掌握基本概念和方法，注重理论与实践的结合，多角度思考和应用。
美国高等数学最好的教材
美国高等数学最好的教材在美国高等数学教育领域，选择一本优秀的教材对学生的学习成果产生了重要的影响。
本文将介绍几本在美国广受好评的高等数学教材，分析它们的特点和优势，旨在帮助读者选择适合自己的教材。
1. "Calculus: Early Transcendentals" by James StewartJames Stewart的《微积分：早期超越函数》是一本备受赞誉的高等数学教材。
这本教材以清晰易懂的语言和详细的解释，全面覆盖了微积分的各个方面，包括函数、极限、导数和积分等内容。
它引入了实际应用和实例，帮助学生将数学理论与实际问题相结合。
此外，教材中还包含了丰富的练习题和解答，帮助学生巩固知识和提高解题能力。
2. "Linear Algebra and its Applications" by David C. LayDavid C. Lay的《线性代数及其应用》是一本经典的线性代数教材。
该教材以简明扼要的风格介绍了线性代数的基本原理和应用。
它提供了大量的例子和图表，帮助学生更好地理解抽象的数学概念。
此外，教材还特别注重应用，引入了线性代数在工程、经济学和计算机科学等领域的实际应用。
3. "Probability and Statistics for Engineers and Scientists" by Ronald E. WalpoleRonald E. Walpole的《工程与科学的概率与统计学》是一本广泛应用于工科和科学领域的概率与统计学教材。
该教材以问题解决的方法引导学生学习概率与统计学的基本理论和方法。
它以实际案例和环境中的应用为基础，将统计学与实际问题联系起来，帮助学生理解统计学的概念和应用技巧。
此外，教材中还提供了大量的练习题和答案，供学生巩固所学知识。
4. "Differential Equations and Linear Algebra" by Gilbert StrangGilbert Strang的《微分方程与线性代数》是一本以推导和解释为主导的教材。
美国大学高等数学教材pdf
美国大学高等数学教材pdf美国大学的高等数学教材一直以来都备受关注。
它们被认为是全球数学教育的重要参考资料之一。
为了方便学生学习和研究，很多高等数学教材都提供了PDF版本的电子书籍。
本文将介绍一些著名的美国大学高等数学教材PDF资源，并简单评述其特点和优势。
1. "Calculus: Early Transcendentals" （《微积分：早期超越函数》）这是一本经典的高等数学教材，由James Stewart编写。
它覆盖了微积分的各个方面，从导数和积分开始，到微分方程和多变量微积分。
这本教材有清晰的表达和详细的解释，让学生能够深入理解数学概念。
它的PDF版本可以在网上免费获取，方便学生自主学习。
2. "Linear Algebra and Its Applications" （《线性代数及其应用》）这本教材由David C. Lay编写，涵盖了线性代数的基本理论和应用。
它以简洁明了的方式介绍了向量、矩阵、线性变换和特征值等概念，并提供了大量实例和习题供学生练习。
既适用于初学者，也适合用作高等线性代数的参考资料。
这本教材的PDF版本可以从学校图书馆或学术网站上获取。
3. "Differential Equations and Their Applications" （《微分方程及其应用》）这本教材由Martin Braun撰写，主要介绍了常微分方程的理论和应用。
它包含丰富的实例和案例，帮助学生理解微分方程的解法和实际应用。
此外，这本教材还涵盖了偏微分方程和动力系统的基础知识。
学生可以通过学校图书馆或在线学术资源获取它的PDF版本。
4. "Probability and Statistics for Engineering and the Sciences" （《工程与科学中的概率与统计》）这本教材由Jay L. Devore编写，重点介绍了概率论和统计学在工程和科学领域的应用。
国外著名的高等数学教材
国外著名的高等数学教材高等数学作为理工科学生必修的一门重要课程，对于培养学生的数学思维和解决问题的能力起到了至关重要的作用。
在国外，许多著名的高等数学教材被广泛应用于大学教育和学术研究领域。
本文将介绍几部国外著名的高等数学教材，带领读者领略不同文化背景下的数学教育风貌。
一、《Calculus: Early Transcendentals》（《微积分：早期的超越》）《Calculus: Early Transcendentals》是由美国著名数学家James Stewart编写的一本高等数学教材。
该书突出了微积分的几何直观性和实际应用，并结合了丰富的例题和练习，帮助学生理解和掌握微积分的基本概念和技巧。
教材内详细介绍了微积分的基础知识，如函数、极限、导数和积分等，旨在培养学生的数学建模和问题解决能力。
二、《Linear Algebra and Its Applications》（《线性代数及其应用》）《Linear Algebra and Its Applications》是美国数学家David C. Lay等人合著的一本线性代数教材。
该教材系统地介绍了线性代数的理论和应用，内容包括向量空间、线性变换、特征值与特征向量等。
教材注重理论和实践相结合，融入了许多实际问题的例子，使学生更好地理解线性代数的概念和方法。
三、《Probability and Statistics for Engineers and Scientists》（《工程与科学的概率与统计》）《Probability and Statistics for Engineers and Scientists》是美国数学家Ronald E. Walpole等人编写的一本概率与统计教材。
该书着重介绍了概率与统计在工程和科学领域的应用，内容包括概率论、随机变量、概率分布、统计推断等。
教材将实际问题与数学模型相结合，引导学生从实践中掌握概率与统计的基本原理与方法。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
1 0 0 0
0 1 0 0
3 −1 0 0
−3 2 . 0 0
Thus the null space has basis (−3, 1, 1, 0), (3, −2, 0, 1). By Gram-Schmidt process, we get an orthogonal basis u1 = (−3, 1, 1, 0), u2 = (0, −1, 1, 1) of NulA. By the meaning of NulA, the space (NulA)⊥ is spanned by the row vectors of A. Thus (1, 0, 3, −3) and (0, 1, −1, 2) is a basis of (NulA)⊥ . We apply the Gram-Schmidt process in the reversed order to get an orthogonal basis v1 = (0, 1, −1, 2), v2 = (2, 3, 3, 0) of (NulA)⊥ . u1 , u2 , v1 , v2 is then an orthogonal basis of R4 . (6) 1) Take A and B to be diagonal matrices with respectively 1, 1, 1 and 2, 1, 1 on the diagonals. 2) If the columns of A are orthogonal, then multiplying some different numbers on the columns of A will still produce orthogonal columns, but will typically produce nonorthogonal rows. For example, 1 −1 1 2 has orthogonal columns (and orthogonal rows). But has orthogonal columns but 1 1 −1 2 1 2 0 not orthogonal rows. A 3 × 3 counterexample is given by −1 2 0 0 0 1 1 1 1 1 0 0 3) X = is not diagonalizable. However, X = + and each summand is 0 1 0 0 0 1 diagonalizable (because each has two distinct eigenvalues). (7) F; F; T; T; F; T; T; F; T; T; F; F; T; T; F; T; T; T. Final (1) (10 points) Consider 10 5 −4 0 . 10 1
OLD FINAL EXAMS (Answer not guaranteed to be correct) Final (1) (15 points) Consider 4 2 . 1 1
5 3 A= 0 0
−4 −2 0 0
−2 0 1 1
1) Find a basis of eigenvectors of A; 2) Use the resulting diagonalization to compute A4 − 3A3 + 2A2 . (2) (10 points) Consider polynomials p1 = x2 − x, p2 = 2x2 − 2x + 1, p3 = x2 − 2x.
4 4 4 −4
3) If {v1 , v2 , · · · , vn } span V , then dim V ≤ n; 4) If {v1 , v2 , · · · , vn } span V , then {v1 , v2 , · · · , vn , vn+1 , · · · , vn+m } also span V ; 5) If n < dim V , then {v1 , v2 , · · · , vn } are linearly independent; 6) If A is a 11 × 17 matrix, and the general solution of the Ax = 0 has 8 free variables, then rankA = 9. 7) An n × n matrix can have at most n eigenvalues; 8) If A = 0, then 0 is not an eigenvalue of A; 9) If A is diagonalizable, then AT is also diagonalizable; 10) If u is an eigenvector of A and B , then it is an eigenvector of A + B ; 11) If all eigenvalues of A are 1, then A = I ; 12) If {u1 , v1 , w1 } and {u2 , v2 , w2 } are orthogonal sets, then {u1 + u2 , v1 + v2 , w1 + w2 } is also an orthogonal set; 13) If {v1 , v2 , · · · , vn , vn+1 , · · · , vn+m } is orthonormal, then {v1 , v2 , · · · , vn } is also orthonormal; 14) If the columns of a square matrix A are orthonormal, then the rows of A are also orthonormal; 15) If {u1 , u2 } is a basis of W , then projW v = proju1 v + proju2 v ; 16) projW (u + v ) = projW u + projW v ; 17) If u ∈ V is orthogonal to all vectors in V , then u = 0; 18) If U is an orthogonal matrix, then U 2 is also orthogonal; Answer to Final (1) The chacteristic equation is λ(λ − 1)(λ − 2)2 = 0. For λ1 = 0, we get eigenvector u1 = (2, 4, −1, 1). For λ2 = 1, we get eigenvector u2 = (1, 1, 0, 0). For λ3 = 2, the eigenspace is 2-dimensional with basis u3 = (4, 3, 0, 0) and u4 = (2, 2, 1, 1). u1 , u2 , u3 , u4 is basis of eigenvectors of A. If P = [u1 , u2 , u3 , u4 ], then Λ = P −1 AP is a diagonal matrix with 0, 1, 2, 2 as diagonal entries. It follows from Λ4 − 3Λ3 + 2Λ2 = 0 that A4 − 3A3 + 2A2 = 0. (2) The coordinates of p1 , p2 , p3 with respect to the basis 1, x, x2 are u1 = (0, −1, 1), u2 = (1, −2, 2), u3 = (0, −2, 1). Since the matrix [u1 , u2 , u3 ] is invertible, u1 , u2 , u3 is a basis of R3 . Thus p1 , p2 , p3 is a basis of P2 . If 2x2 + 3x − 1 = c1 p1 + c2 p2 + c3 p3 , then c1 + 2c2 + c3 = 2, −c1 − 2c2 − 2c3 = 3, c2 = −1. The solution is (c1 , c2 , c3 ) = (9, −1, −5). (3) A can be reduced by row operations to 1 2 0 −1 0 1 1 2 0 0 0 3 . 0 0 0 0 0 0 0 0 Therefore rankA = 3, dimColA = 3, and dimNulA = 4 − 3 = 1. (4) By Gram-Schmidt process, we have v1 = u1 = (1, 0, 1, 0) 4 2 ·v 1 v2 = u2 − u . v1 ·v1 v1 = (2, 2, 2, 0) − 2 (1, 0, 1, 0) = (0, 2, 0, 0) u 3 ·v 1 u 3 ·v 2 6 v3 = u3 − v1 ·v1 v1 − v2 ·v2 v2 = (3, −3, 0, 3) − 6 (1 , 0 , 1 , 0) − (0 , 2 , 0 , 0) = (0 , 0 , 0 , 3) 2 4 Since u4 ∈ span{v1 , v2 , v3 } (one will get v4 = 0 by Gram-Schmidt process), v1 , v2 , v3 is an orthogonal basis of W . We have x · v1 x · v2 x · v3 projW (2, 1, 0, 1) = v1 + v2 + v3 = (1, 1, 1, 1). v1 · v1 v2 · v2 v3 · v3 √ Thus dist((2, 1, 0, 1), W ) = (2, 1, 0, 1)−projW (2, 1, 0, 1) = 2 , and dist((2, 1, 0, 1), W ⊥ ) = projW (2, 1, 0, 1) = 2.