Finite-Dimensional Vector Spaces - P. Halmos (Springer, 1987)
在学习代数学之余,值得一看的代数学书籍。里面介绍了更为丰富的代数学概念和结论。PREFACEMy purpose in this book is to treat linear transformations on finite-dimensional vector spaces by the methods of more general theories. Theidea is to emphasize the simple geometric notions common to many partsof mathematics and its applications, and to do so in a language that givesaway the trade secrets and tells the student what is in the back of the mindsof people proving theorems about integral equations and Hilbert spaces.The reader does not, however, have to share my prejudiced motivationExcept for an occasional reference to undergraduate mathematics the bookis self-contained and may be read by anyone who is trying to get a feelingfor the linear problems usually discussed in courses on matrix theory orhigher"algebra. The algebraic, coordinate-free methods do not lose powerand elegance by specialization to a finite number of dimensions, and theyare, in my belief, as elementary as the classical coordinatized treatmentI originally intended this book to contain a theorem if and only if aninfinite-dimensional generalization of it already exists, The temptingeasiness of some essentially finite-dimensional notions and results washowever, irresistible, and in the final result my initial intentions are justbarely visible. They are most clearly seen in the emphasis, throughout, ongeneralizable methods instead of sharpest possible results. The reader maysometimes see some obvious way of shortening the proofs i give In suchcases the chances are that the infinite-dimensional analogue of the shorterproof is either much longer or else non-existent.A preliminary edition of the book (Annals of Mathematics Studies,Number 7, first published by the Princeton University Press in 1942)hasbeen circulating for several years. In addition to some minor changes instyle and in order, the difference between the preceding version and thisone is that the latter contains the following new material:(1) a brief dis-cussion of fields, and, in the treatment of vector spaces with inner productsspecial attention to the real case.(2)a definition of determinants ininvariant terms, via the theory of multilinear forms. 3 ExercisesThe exercises(well over three hundred of them) constitute the mostsignificant addition; I hope that they will be found useful by both studentPREFACEand teacher. There are two things about them the reader should knowFirst, if an exercise is neither imperative "prove that.., )nor interrogtive("is it true that...?" )but merely declarative, then it is intendedas a challenge. For such exercises the reader is asked to discover if theassertion is true or false, prove it if true and construct a counterexample iffalse, and, most important of all, discuss such alterations of hypothesis andconclusion as will make the true ones false and the false ones true. Secondthe exercises, whatever their grammatical form, are not always placed 8oas to make their very position a hint to their solution. Frequently exer-cises are stated as soon as the statement makes sense, quite a bit beforemachinery for a quick solution has been developed. A reader who tries(even unsuccessfully) to solve such a"misplaced"exercise is likely to ap-preciate and to understand the subsequent developments much better forhis attempt. Having in mind possible future editions of the book, I askthe reader to let me know about errors in the exercises, and to suggest im-provements and additions. (Needless to say, the same goes for the text.)None of the theorems and only very few of the exercises are my discovery;most of them are known to most working mathematicians, and have beenknown for a long time. Although i do not give a detailed list of my sources,I am nevertheless deeply aware of my indebtedness to the books and papersfrom which I learned and to the friends and strangers who, before andafter the publication of the first version, gave me much valuable encourage-ment and criticism. Iam particularly grateful to three men: J. L. Dooband arlen Brown, who read the entire manuscript of the first and thesecond version, respectively, and made many useful suggestions, andJohn von Neumann, who was one of the originators of the modern spiritand methods that I have tried to present and whose teaching was theinspiration for this bookP、R.HCONTENTS的 FAPTERPAGRI SPACESI. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 55. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 108. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Calculus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dualspaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24;17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a directsum, 30; 20. Dual of a direct sum, 31; 21. Qguotient spaces, 33;22. Dimension of a quotient space, 34; 23. Bilinear forms, 3524. Tensor products, 38; 25. Product bases, 40 26. Permutations41; 27. Cycles,44; 28. Parity, 46; 29. Multilinear forms, 4830. Alternating formB, 50; 31. Alternating forms of maximal degree,52II. TRANSFORMATIONS32. Linear transformations, 55; 33. Transformations as vectors, 5634. Products, 58; 35. Polynomials, 59 36. Inverses, 61; 37. Mat-rices, 64; 38. Matrices of transformations, 67; 39. Invariance,7l;40. Reducibility, 72 41. Projections, 73 42. Combinations of pro-jections, 74; 43. Projections and invariance, 76; 44. Adjoints, 78;45. Adjoints of projections, 80; 46. Change of basis, 82 47. Similarity, 84; 48. Quotient transformations, 87; 49. Range and null-space, 88; 50. Rank and nullity, 90; 51. Transformations of rankone, 92 52. Tensor products of transformations, 95; 53. Determinants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Triangular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112III ORTHOGONALITY11859. Inner products, 118; 60. Complex inner products, 120; 61. Innerproduct spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124;64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127;CONTENTS66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gBCHAPTERtheses versus brackets, 13169. Natural isomorphisms, 138;70. Self-adjoint transformations, 135: 71. Polarization, 13872. Positive transformations, 139; 73. Isometries, 142; 74. Changeof orthonormal basis, 144; 75. Perpendicular projections, 14676. Combinations of perpendicular projections, 148; 77. Com-plexification, 150; 78. Characterization of spectra, 158; 79. Spec-ptral theorem, 155; 80. normal transformations, 159; 81. Orthogonaltransformations, 162; 82. Functions of transformations, 16583. Polar decomposition, 169; 84. Commutativity, 171; 85. Self-adjoint transformations of rank one, 172IV. ANALYSIS....17586. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions forthe norm, 178; 89. bounds of a self-adjoint transformation, 17990. Minimax principle, 181; 91. Convergence of linear transformations, 182 92. Ergodic theorem, 184 98. Power series, 186APPENDIX. HILBERT SPACERECOMMENDED READING, 195INDEX OF TERMS, 197INDEX OF SYMBOLS, 200CHAPTER ISPACES§L. FieldsIn what follows we shall have occasion to use various classes of numbers(such as the class of all real numbers or the class of all complex numbers)Because we should not at this early stage commit ourselves to any specificclass, we shall adopt the dodge of referring to numbers as scalars. Thereader will not lose anything essential if he consistently interprets scalarsas real numbers or as complex numbers in the examples that we shallstudy both classes will occur. To be specific(and also in order to operateat the proper level of generality) we proceed to list all the general factsabout scalars that we shall need to assume(A)To every pair, a and B, of scalars there corresponds a scalar a+called the sum of a and B, in such a way that(1) addition is commutative,a+β=β+a,(2)addition is associative, a+(8+y)=(a+B)+y(3 there exists a unique scalar o(called zero)such that a+0= a forevery scalar a, and(4)to every scalar a there corresponds a unique scalar -a such that十(0(B)To every pair, a and B, of scalars there corresponds a scalar aBcalled the product of a and B, in such a way that(1)multiplication is commutative, aB pa(2)multiplication is associative, a(Br)=(aB)Y,( )there exists a unique non-zero scalar 1 (called one)such that al afor every scalar a, and(4)to every non-zero scalar a there corresponds a unique scalar a-1or-such that aaSPACES(C)Multiplication is distributive with respect to addition, a(a+n)If addition and multiplication are defined within some set of objectsscalars) so that the conditions(A),B), and (c)are satisfied, then thatset(together with the given operations) is called a field. Thus, for examplethe set Q of all rational numbers(with the ordinary definitions of sumand product)is a field, and the same is true of the set of all real numberaand the set e of all complex numbersHHXERCISIS1. Almost all the laws of elementary arithmetic are consequences of the axiomsdefining a field. Prove, in particular, that if 5 is field and if a, and y belongto 5. then the following relations hold80+a=ab )Ifa+B=a+r, then p=yca+(B-a)=B (Here B-a=B+(a)(d)a0=0 c=0.(For clarity or emphasis we sometimes use the dot to indi-cate multiplication.()(-a)(-p)(g).If aB=0, then either a=0 or B=0(or both).2.(a)Is the set of all positive integers a field? (In familiar systems, such as theintegers, we shall almost always use the ordinary operations of addition and multi-lication. On the rare occasions when we depart from this convention, we shallgive ample warningAs for "positive, "by that word we mean, here and elsewherein this book, "greater than or equal to zero If 0 is to be excluded, we shall say"strictly positive(b)What about the set of all integers?(c) Can the answers to these questiong be changed by re-defining addition ormultiplication (or both)?3. Let m be an integer, m2 2, and let Zm be the set of all positive integers lessthan m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the leastpositive remainder obtained by dividing the(ordinary) sum of a and B by m, andproduct of a and B by m.(Example: if m= 12, then 3+11=2 and 3. 11=9)a) Prove that i is a field if and only if m is a prime.(b What is -1 in Z5?(c) What is囊izr?4. The example of Z, (where p is a prime)shows that not quite all the laws ofelementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove thatif is a field, then either the result of repeatedly adding 1 to itself is always dif-ferent from 0, or else the first time that it is equal to0 occurs when the numberof summands is a prime. (The characteristic of the field s is defined to be 0 in thefirst case and the crucial prime in the second)SEC. 2VECTOR SPACES35. Let Q(v2)be the set of all real numbers of the form a+Bv2, wherea and B are rational.(a)Ie(√2) a field?(b )What if a and B are required to be integer?6.(a)Does the set of all polynomials with integer coefficients form a feld?(b)What if the coeficients are allowed to be real numbers?7: Let g be the set of all(ordered) pairs(a, b)of real numbers(a) If addition and multiplication are defined by(a月)+(,6)=(a+y,B+6)and(a,B)(Y,8)=(ary,B6),does s become a field?(b )If addition and multiplication are defined by(α,月)+⑦,b)=(a+%,B+6)daB)(,b)=(ay-6a6+的y),is g a field then?(c)What happens (in both the preceding cases)if we consider ordered pairs ofcomplex numbers instead?§2. Vector spaceWe come now to the basic concept of this book. For the definitionthat follows we assume that we are given a particular field s; the scalarsto be used are to be elements of gDEFINITION. A vector space is a set o of elements called vectors satisfyingthe following axiomsQ (A)To every pair, a and g, of vectors in u there corresponds vectora t y, called the aum of a and y, in such a way that(1)& ddition is commutative,x十y=y十a(2)addition is associative, t+(y+2)=(+y)+a(3)there exists in V a unique vector 0(called the origin) such thata t0=s for every vector and(4)to every vector r in U there corresponds a unique vector -rthat c+(-x)=o(B)To every pair, a and E, where a is a scalar and a is a vector in u,there corresponds a vector at in 0, called the product of a and a, in sucha way that(1)multiplication by scalars is associative, a(Bx)=aB)=, and(2 lz a s for every vector xSPACESSFC B(C)(1)Multiplication by scalars is distributive with respect to vectorddition, a(+y=a+ ag, and2)multiplication by vectors is distributive with respect to scalar ad-dition, (a B )r s ac+ Bc.These axioms are not claimed to be logically independent; they aremerely a convenient characterization of the objects we wish to study. Therelation between a vector space V and the underlying field s is usuallydescribed by saying that v is a vector space over 5. If S is the field Rof real number, u is called a real vector space; similarly if s is Q or if gise, we speak of rational vector spaces or complex vector space§3. ExamplesBefore discussing the implications of the axioms, we give some examplesWe shall refer to these examples over and over again, and we shall use thenotation established here throughout the rest of our work.(1) Let e(= e)be the set of all complex numbers; if we interpretr+y and az as ordinary complex numerical addition and multiplicatione becomes a complex vector space2)Let o be the set of all polynomials, with complex coeficients, in avariable t. To make into a complex vector space, we interpret vectoraddition and scalar multiplication as the ordinary addition of two poly-nomials and the multiplication of a polynomial by a complex numberthe origin in o is the polynomial identically zeroExample(1)is too simple and example (2)is too complicated to betypical of the main contents of this book. We give now another exampleof complex vector spaces which(as we shall see later)is general enough forall our purposes.3)Let en,n= 1, 2,. be the set of all n-tuples of complex numbers.Ix=(1,…,轨)andy=(m1,…,n) are elements of e, we write,,bdefinitionz+y=〔1+叽,…十物m)0=(0,…,0),-inIt is easy to verify that all parts of our axioms(a),(B), and (C),52, aresatisfied, so that en is a complex vector space; it will be called n-dimenaionalcomplex coordinate space
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仿真软件saber说明书 很详细
介绍saber仿真的很好的书籍 建议按步骤学习SABER电气系统培训手册Saber ElectricalSystems Workshop练习指南北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册介绍Saber Electrical Systems介绍本课程的目的是使用户熟悉 Saber模拟器。课程为 Saber功能培训:这一部分通过相关的简单电路和系统集中讲解如何使用 Saber模拟器的各个功能。内容用户应从以下内容出发:如何应用 Saber改善电路和系统设计如何通过 Saber进行由上至下和由下至上设计如何完成不同类型的分析如何使用 Saber模型如何使用 Saber模型库如何查阅帮助如出错如何解决背景要求需具备基本的工程知识熟练的计算机操作·非必需不需要具备仿真经验北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册相关说明相关说明Saber book, Saber在线帮助系统,描述了 Saber sketch和 Cosmos Scope的特性以及 Saber的一般功能,比如菜单的使用和打印。还提供了每个 Saber命令和 Saber guide界面的详细信息Sorh8 ing Saberi讲解如何在 Saber sketch中生成电路设计图并如何应用 Saber分析Getting Started Using Saber with the Frameway integrations帮助您分析两个示范电路,并使您熟悉在 Cadence和 Mentor图表的框架环境下主要的使用过程。Analyzing Designs Using saber描述了如何应用 Saber获取图表、模拟设计和优化参数值Saber Design Esamples示范如何应用 Saber对设计图进行仿真和分析。北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册惯例Saber Electrical systems惯例培训手册采用以下惯例ButtonButton这样的字体用来在用户界面上突出显示按键的描述和编号。ComputerComputer”这样的字体用来突出屏幕上输入内容(即您在命令行或某区域输入内容)。Dialog Menu Form“ DialogMenu Forn”这样的字体表示对话框标题土级菜单标题及表格标题。DocnammeNonAme”这样的字体用来指示印刷手写体标题(同 Computer字样)FieldNameFieldName”这样的字体用来突出区域名称。FilenameFilename”这样的字体显示路径称或目录名称。Menuchoice“ Menu choice”这样的体用来突出菜单路径,如引导选择Fie>openDesign单击迅速地按动并松开鼠标键按键并保持按鼠标键,并不松开。双击连续两次快速按动松开鼠标键。北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册Saber Electrical SystemsLab#1-DC分析在第一个练习,您将对DC(工作点)分析进行基础的了解。这是你在大部分设计中将要作的基本分析。打开RLC设计UNIX用户1.找到 Saber_ Training_ Files/Saber_ Electrical_Training/ Feature Labs目录2.输入: sketchWindows用户:1. itf Start >Programs Synopsys >Saben)XX> Saber Sketch所有用户(从 Saber Sketch下拉菜单中1.选择Fie>Open> Design2左键单击并按住鼠标在菜单上打开下拉萊单。滑动光标选择相应子菜单并展开。点击下拉菜单中部可固定蕖单,即使松开鼠标键,菜单也不会弹回。3.浏览RLC自录(在 Saber_Training Files/Saber_ Electrical_TrainingFeature Labs路径下)。4.双击 Open Design中的exrl文件名(如果文件扩展名可见,选择带有 ai sch扩展名的文件)。RLC示意图显示如下:25mwimid Ivoutv pulse initial: 0ouselu北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册Lab#1-DC分析从此处,您开始在 Saber Sketch中设计。5.点击Show/ Hide Saber Guide按键,该按键使您可以进λ Saber guide仿真,在整个设计过程中都将用到该按键。(按键位于Saber sketch图标栏的右侧)。Saber guide图标栏出现。工作点分析1.点击 Operating Point按键,幸2.点击OK,接受默认。生成列表,确定工作点(在生成列表前如提示是否保存,回答yes)。如有错误,将有信息提示。点击 Simulation Transcript按键 Smdl,打开 Saber Guide transcrip窗口您可以通过 Saber guide Transcript窗口监控 Saber guide命令进程。也可显示完成一次分析的执行时间。3.当分析完成,从 Saber sketch下拉菜单中选择Results> Operating Point Report2.显示分析结果。4.单击OK接受工作点报告缺省值Report Tool弹出,显示分析结果。注意所有的显示值都为0。检查该结果是否正确,查看驱动滤波器源电压的初始值。该示意图显示电压初始值为0,脉冲值为1。表示在0时刻电源电压为0,所以该结果正确为得到DC分析的非零值,您可以改变电源的初始值,例如,您可以将初始值设为1,脉冲值置为0。这样您将得到棉反的波形改变输人电压并重新分析1.改变示例中电源的初始电压a,在 initial:0处的0附近单击左键b.通过箭头键将光标置于0的右侧(如果需要)c,单击删除键删除0,在该位置键入1d.单击 Return{ Enter)或鼠标左键2.改变示例中的脉冲值a.在puse:1处的0附近单击左键b.通过箭头键将光标置于1的右侧(如果需要)c.单击删除键删除1,在该位置键入0d.单击 Return( Enter)或鼠标左键北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册Saber Electrical Systems您已完成下面两项工作改变了示例中电源的初始电压及脉冲值。当您编辑列表并输入 Saber时,这些值会随之自动动态改变,这就意味着模拟器已经接收了新值,您不需要保存或重新生成网表(您可以在 Saber Guide Transcript窗口中看到这些值的变化)。现在重新启动DC分析,除了相应的变化,您可以看到 Saber执行分析时有很多选项。3.从 Saber Guide下拉菜单栏中,选择Analyses Operating Point DC Operating Point.*4.在 Operating Point分析中,分析结束后,执行以下操作,工作点报告可以自动显示在Saber guide transcriptt窗口内a.点击 Display After Analysis旁边的Yes按钮b.点击OKc.在 Saber Guide Transcript窗口内查看结果结果显示1V的输入电压产生0909V输出电压。你理解这个结果吗?为什么?为了便于结合给定电压与实际设计示意图进行分析,可按如下方法在 Saber sketch i中显示DC分析结果:标注分析结果1.从 Saber Sketch下拉菜单栏中选择 Results> Back Annotation2.在后注释栏点击OK注意示例中仿真电压如何出现关闭Lab#11.按以下步骤关闭报告及报告工具栏a.选择File>Cose关闭报告,关闭前不保存 de. rpt文件b,.选择File> Close Window关闭报告工具2.恢复电压源初始值,为下一练习作准备(即将初始值还原为0,脉冲还原为1)请告知培训人员您已经完成了第一部分的内容。北京才略科技有限公司TEL:010)82673952/82673953SABER电气系统培训手册Saber Electrical SystemsLab#2一时域分析在本节,您将对RLC滤波器进行时域(瞬态)分析,以确定其脉冲输入响应。继续采用您上一节建立的RLC电路。瞬态分析1.点击 Transient Analysis按钮,2.在时域瞬态分析中,输入数值:a. End Time: Jomb. Time Step: 0. Iu选择该值是因为脉冲升降时间是1毫秒,初始时间长应置为110脉冲时间)Run DC Analysis First: Yesd. Plot After Analysis: Yes-Open Onl3.点击OK执行分析完成瞬态分析, OsmoscOpe.信号管理器和图形窗们命自动打开Cosmos scope的使用当完成分析后, OsmoscOpe自动打开(因为分析中您选择了Yes. Open Only to Plot AfterAnalysis)。同样生成了两个窗口,如下所示信号管理器 Signal Manager:信号管理器显示当前激活状态下了图形文件名称(图形文件包含仿真数据在 OsmoscOpe中可见)。gnal ManagerFe Polle SionalsSignal FiterOpen ClothesPlotsClose Plotfdles( 1]ex die. ac ai plDisplay PottiesSetupMatch All图形文件窗口 Plot file window:图形文件窗口显示相应图形文件中的信号名称。北京才略科技有限公司TEL:010)82673952/82673953
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