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
- 2020-12-05下载
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程序员必读的职业生涯规划书【精品】.pdf
程序员是一种逻辑动物，只有当他们理解求职到底是一种什么行为以后，才能做出有意义的行动。 所以本书不讲职场、不灌鸡汤，从价值论开始，以独创的职业画布工具展开，从四大象限、九大方面为你讲解。 有果更有因、有理更有据，分析数据、简历模板、书写工具、技能树图一应俱全，实为程序员们应该人手一本的职业规划书。原理篇我之所以会去研究求职,以至于最后在人才这个方向创业,很大程度上是因为以前所有的求职指导都是经验论,缺乏逻辑性。而程序员是一种逻辑动物,只有当他们理解求职到底是一种什么行为以后,才能做出有意义的行动。我花了很久去寻找背后的理论支持,直到我重逢了价值论。原理篇从价值理论开始,讲述我们求职行为的本质。价值论人才的使用价值供需信息透明度交薪资换雇主价值自从中国转冋市场经济以后,市场规则就开始左右一切。虽然人才不完全等同于其他物品,但现在先让我们把人才也作为一类商品来看待。使用价值除了我们这些做人力资源相关行业的公司外,绝大部分公司购买人才都是为了使用,所以他们看中的是其使用价值。这个使用价值说得更直白一点,就是人才如何直接或者间接的为公司挣钱。使用价值不是独立存在的,而是相对于使用者存在。所以我们程序员自己的价值,也是相对于公司而言的。有同学说,我技术很好啊,又会机器学习又会编译原理,凭什么那些写 Javascript的薪水比我高一倍?谁让你在一家建站公司上班呢。对一家做网站的公司而言,机器学习和编译原理是不能为它带来收益的,而 Javascript写成的带有完美动画的交互组件却能实实在在的拉升公司产品的销售。而同样是这个人,如果他去一家以大数据分析为核心业务的公司工作,那么他的价值就不样了。所以你的价值,和你牛不牛无关,只和你能为你的雇主提供多少价值有关。(当然,大多数情况下,你能力越牛越能提供更多价值。)这是最根本的规则。增加自己的使用价值很简单,提升自己的业务能力就好。供需有个80后自嘲的段子是这么说的读小学时,大学不要钱;读大学时,小学不要钱;还没工作时,工作是分配的;可以工作时,得自谋职业;没挣钱时,房子是分配的;能挣钱时,发现一辈子的薪水也买不起房子。我不知道国内人才市场是什么时侯市场化的,就算成悲催的80后开始工作时吧。人才市场化意味着你有了选择公司的权利,同时也意味着公司有了选择你的权利。这个时候,交换价值就出来了。虽然交换价值以使用价值为基础,但它更容易受供需的影响简单的说,当企业的职位空缺远少于找工作的人数时,人才的价格就会下降;而当企业的职位空缺比找工作的人数更多时,人才的价格就会上升。在过去相当长时间里,我们都处于前一种情况。这意味着求职者要彼此竞争,而招聘方可以选择要价更低的候选人。这在大家大学毕业找工作时应该深有体会。幸运的是,供需也是可以调整的,技巧就在于选择更好的细分市场。因为需求的多样性是存在的,所以如果你能在一个大的需求中切入一个需大大大于求的细分市场,那么你就能得到远高于其他人的回报。举个例子,同样是管服务器,普通运维工程师和云计算运维工程师的薪资差异是非常大的。一个普通运维要变成云计算运维,需要补充的知识并不是特别多。所以你只要合理安排好自己的职业规划,比如以相对较低的薪资到类似新浪云这样的地方工作一到两年,你的能力和交换价值都会大幅度提升。信息透明度当人才市场很小的时候,信息是很透明的。因为很容易了解到各自的情况。但当信息量变大后,你就会发现虽然整个市场很大,但只有你接触到的才对你有意义。比如北京现在有100家公司都在招聘PHP,但你只知道其中3家,这个时候,其他97家公司的存在对你而言是没有意义的,即使这3家给你的薪资比其他公司低,你也只能被迫接受。这就是信息透明度对我们求职的影响。没有网络招聘的时候,我们很难对这些公司进行比较,折腾过几家公司后,就屈服了。有了网络招聘,求职者活得稍微好一些了,可以不岀门看到全国的招聘情况;但Jobboard形式的招聘站是为招聘方设计的,它们通过构造信息不对称,向求职者优先显示那些付费却未必最好的公司,迫使求职者以更低的薪资为这些可见的公司工作。(这无可厚非,所有中介体都是通过信息不对称来收费的)所以要想拿到足够好的薪资和获得足够多的机会,我们要学会和信息不对称进行抗争。定要在短时间内获取到大量的机会,这样才能「做选择题」而不是「做判断题」。关于如何改变信息不对称,操作篇中的「渠道」部分我们会详细说明。跳槽不是什么跳槽不是为了追求价值最大化虽然前边我们从纯商业的角度讨论了跳槽这件事,但实际上我们并不推荐求职者单纯地通过薪资的高低来选择雇主。原因有两个。工作是生活的一部分你每天有24小时,上班8小时,它是你三分之一的生命。如果你只是因为高薪选择了一家上班不开心的公司,那么接下来三分之一的日子你都会在痛苦中渡过。而你得到的,只是多出来的那么每月几千块钱而已。职业的不可逆性别的事情错了我们可以重来,而职业选择上,每一步都是好几年的时光。一旦走下去,就很难回头,因为一旦换职业方向、哪怕换个开发语言,都意味着你之前的积累很可能前功尽弃。所以眼光放长远点,不要只看眼前的高薪,更要看好未来的发展,方向错了,跑得越猛输得越狠。跳槽不是找工作跳槽不是找工作,而是换工作。所以它和你刚毕业时找工作不一样,它是有成本的到一家新的公司,需要放弃原来公司的期权,需要重新熟悉业务,重新和同事搞好人际关系,上班路上需要花更多的时间,甚至需要离开自己熟悉的城市。你并不是一无所有,虽然往往只有在你失去后才发现。所以跳槽之前要考虑好自己的付出和收益:衡量好得失:没事别裸辞。跳槽不是加薪的砝码不要以跳槽为砝码去谈加薪,这种迫胁性的谈薪资方式会使主管对你丧失信仼。不管涨薪成不成功,他都不得不为你的离职做好准备,而当准备完成后,你就可以走人了。类似的,当你提珧槽时,如果你的主管通过加薪来挽留你,最好也不要同意。程序员如何谈薪资如果你对现在公司很满意,只是觉得薪资大低,那么可以先和你的主管聊聊我一般这么谈加薪的首先,讲一讲自己最近在工作上的成长,看主管是否认同;●然后,从能力提升角度,向主管要一个更大的发展空间和更大的业务挑战;最后,问问当你的能力达到这个新的水平时,蒜资是否可以同步提升起来。这样谈有几个好处:它建立了一个谈话基础,那就是薪资应该和能力相匹配它不谈现在的薪资,而是谈未来,一方面表明你对现在的薪资还算满意(以减少主管顾虑),另一方面对主管来讲,他没有立刻给你涨薪的压力;·它体现了你对自己能力提升的渴求,而如果在谈话中出现任何不顺利,你可以把中途把对话结束,根本不提加薪的事情。当然,坏处就是你得隔段时间才能加薪了。跳槽到底为什么说了那么多跳槽不是什么,那跳槽到底为什么呢?求职的本质在跳槽之前,你想过我们为什么要工作么?其实并不是真的如我们父辈们认为的那样,人活着就是要有一份工作的。工作最重要的目的是累积金钱。这是我们在这个现实世界用以维持生计的必需资源。挣钱这件事,至少有四种形式:帮别人挣钱(打工);为自己挣钱(SOHO);雇别人挣钱(企业主);让钱自己挣钱(投资)。为什么我们总是选择第一种呢?因为第一种门槛和风险都最低。而求职只是我们在自己资源和能力不足以进入后几种挣钱形式时,通过出卖自己的劳动能力换取资源的阶段性行为当然,如果能在第一种形式下获得自己满意的物质回报和精神满足,你也可以永远不考虑其他形式,但其他形式在挣钱的效率和数量上是有优势的。这个我们在稍后的章节会细聊。跳槽的意义跳槽其实是打工这种挣钱形态下,我们进行自我调节的方式,通过合理地跳槽,我们可以寻找到市场需求和自我实现的最佳匹配,从而在金钱和成长上双丰收。「市场需求」描述了企业渴求的员工;「自我实现」描述了你想要的生活;当它们重合到起,梦想就照进了现实。这会带给你几年非常愉快的工作生活,但随着我们不断成长,职位和能力可能出现不匹配,这时候我们又需要通过跳槽来再次调节。很多时候,这也是很无奈的事情,不进则退,人在江湖身不由己。跳槽的原则我有一个很好用的跳槽原则,之前发到微博上被赞了上百次:
- 2021-05-07下载
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