数量经济学讲义数理经济学

1、 HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站LectureNote4.3LinearTransformationsand4.27.Definition.Weshallsaythataf : L HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站LectureNote4.3LinearTransformationsand4.27.Definition.Weshallsaythataf : L1 L2 is linear iff f (a,b F)(x,y L1): f(ax +by) =af(x)+bf(y)Linearfunctionsare

2、often calledlineartransformations,ratherthan4.28. 1. p Rn,f : Rn R by: 2.Aisan mn real matrix,andf : Rn Rm by: 13.ConsiderthespaceCdefinedin4.13,anddefineby (f) = f0Alinear function maps a linear combination of two points in the domain into the same linear combination of the images of those two poin

3、ts in the range space. samerelationshipholdsforanyfinitelinearcombinationofpointsinthe4.30. Suppose f : L1 L2 is a linear function, and S1,S2 subspaces of L ,respectively. Then f(S ) is a subspace L , f1(S ) is 122subspaceof L14.31.Definition.f : L is a linear function, then the is called f1kernel o

4、fthefunction,ortransformation,andisdenotedby kf 4.32.Proposition.f : L1 L2 isalinearfunction,thenfisone-to-oneiff kf =04.32b. Suppose L1 is a linear space with finite dimension. f : L1 L2 is a linear function.Then we have dim(kf )+dim(f(L1) =dim(L1) 1 HYPERLINK / 官方总站 HYPERLINK / 中华经济学习网 HYPERLINK /

5、 中华经济学习网 HYPERLINK / 官方总站4.33 Proposition. is both linear and one-to-one, and f HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站4.33 Proposition. is both linear and one-to-one, and f : L1 S = f(L ).Then the g = f1 :S islinear,one-one,andonto 1114.34. Definition. A linear function, f : L1 L2, which is also one-t

6、o-one and onto said to be an isomorphism; and if such a function exists, L1 is said to be to L2 (orwesimplysaytheyare4.35. Proposition. f : L1 L2 and g : L2 L3 are both isomorphism, then compositionh =g o f ,isalsoan4.36. IsomorphismasanequivalenceAny and all implications which can be deduced from l

7、inear space assumptions regarding a particular linear space, L, apply equally in any other linear space, L, which is an element of L.4.37. Theorem. If L is any real linear space of finite dimension, n, then L isomorphicto Rn 4.4 Normed Linear4.39. Definition. We say a function : L R+ is a norm iff,

8、for all x,y L , aF : 1.(x)=0 x =0; 2.(x +y)(x)+(y); 3.(ax)=|a |(x). A linear equippedwith anormis calleda normedlinear 4.40. 2 HYPERLINK / 官方总站 HYPERLINK / 中华经济学习网 HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站In Rn , the usual Euclidean normis a normby the above HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站In Rn , th

9、e usual Euclidean normis a normby the above On the space m (the space of bounded real sequences), (x) = sup | xn |n(f) = max | f(x) |.Onspace Ca,b , define isanormCa,b xa,b4.41. Definition. We shall say that a function, d :LL is a metric for Liff, all x,y,z L ,wed(x,y)=0iffx=y 4.42. Theorem. If L is

10、 a normed linear space, with norm , and if we d : LL R+ by d(x,y) =(x y). then d is a metric for L. Furthermore, d thefollowingtwoadditionalconditions:forallx,y,zinLandainF,we(homogeneity):d(ax,ay)=|a|(translationinvariance):d(x+z,4.43. interiorpointofX; X is open;a point of closure of closed set. (

11、As was the case in Rn , it can be shown that X is closed iff it is equal to its closure)4.44. 1. CauchyIt is easy to show that if a sequence converges, then it is a Cauchy sequence. 3 HYPERLINK / 官方总站 HYPERLINK / 中华经济学习网 HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站the other hand, in an arbitrary normed line

12、ar space, a Cauchy sequence may not be convergent.However, if it is the case that every Cauchy sequence in a linear space, L, converges to a point in L, we will say that L is complete. In particular, a normed HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站the other hand, in an arbitrary normed linear space, a

13、Cauchy sequence may not be convergent.However, if it is the case that every Cauchy sequence in a linear space, L, converges to a point in L, we will say that L is complete. In particular, a normedlinearspace iscalled aBanach space.RnisExample: Linear f xR |(mN)(n m):x 0. nxn where x =(1, ,., ,0,0,.)

14、.ItisaCauchysequencebutnotn4.45. Continuousatapointx;continuousUniformlycontinuousonaset4.46. 1. Let L be the space Ca,b , and let x in a,b be fixed. (f) = max | f(x) |. xa,bthen define : L R by (f)= f(x). Then this function is uniformly on2. Let Lbe any real normed linear space, with norm |.|, let

15、Abe a non-empty of L, and define the function (x,A) =inf |x y |, for x L . Then yuniformlycontinuouson(ComparedwithThm3.20and4.47.Theorem.Suppose f :L1 L2,whereLsarenormedlinearspacewithi ThenthefollowingconditionsaremutuallyThefunctionfisForeachopensubset,U,of L , f1(U) isopenin L 21Foreachclosedsu

16、bset,C,L f1(C) isclosedin L 214 HYPERLINK / 官方总站 HYPERLINK / 中华经济学习网 HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站4.5InnerProduct4.51 Definition If L is a real HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站4.5InnerProduct4.51 Definition If L is a real linear space, we say that a f : LL R is inner product for Liff, wri

17、ting xy in place of f(x,y), we have for all x,y,z in L, allainxx=0,and xx=0iffxy=y3. (ax)y=a(xy),x(y+z)=xy+x4.52 Definitions A real linear space, L, equipped with an inner product, is called an inner product space. If an inner product space is also complete, it is called a Hilbert space. (In the rem

18、ainder of this section, we will always take L to be an inner productspace,butnonecessarily4.53 n1. In Rn ,thefamiliardefinition: xy x (Wecandefine | x |=(xx)1/2i 2. In l2 ,define xy xi yi 4.54. Theorem (Cauchy-Schwarz Inequality) Suppose L is an inner product Define | x |=(x x)1/2.Thenforallx,yinL,w

19、ehave |x y |x | y | 4.55.PropositionThefunction|.|definedonLby | x |=(x x)1/2 isanormfor4.56 Proposition (Bi-continuous) Suppose L is inner product space. x*,y* L,and 0, 0,suchx,y 5 HYPERLINK / 官方总站 HYPERLINK / 中华经济学习网 HYPERLINK / 中华经济学习网 HYPERLINK / 官方总站max| x x* |,| y HYPERLINK / 中华经济学习网 HYPERLINK

20、 / 官方总站max| x x* |,| y y* | |x.y x*.y* |4.57Proposition | x +y |2 +| x y|2=2 | x|2 +2 | y|4.58 Definition. In an inner product space, we say x and y are orthogonal iff xandwrite x 4.59.PropositionIfxisorthogonaltoy,then | x +y |2=| x |2 +| y|4.60. Definitions A set of vectors, X xa | xa 0,aA, in L i

21、s said to be orthogonalsystemifffor each a,bA suchthat a b,wehave xa xb 0.IfXadditionsatisfies: (aA): xa xb 1,thenXissaidtobeanorthonormal4.61PropositionAnorthogonalsystemislinearly4.62Definition IfS isa linear subspaceof L,a subset,X, of Lwhichis ansystemiscalledanorthogonalbasisforSiff4.63 Definition If S is a linear subspace of L, we define S, the ScomplementofalinearsubspaceS,xL|yS :x y 4.64Proposition. If S isa linearsubspace, is a linear subspace of Las SAnd S S =4.65 T