Solutions Manual for Introductory Linear Algebra

Transcription

1 جايعت انقذس ان فتىحت QUDS OPEN UNIVESITY Solutions Mnul for Introdutor iner lger uthordrmhmoud l-bei

2 l-quds Open Universit SolutionsMnul forintrodutor iner lger uthordrmhmoud l-bei Evlutor DrMuhtr l-shellh

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4 Introdution The SolutionsMnul for Introdutor iner lger hs een prepredto e omplementr to the tetoo to help students to understnd the sientifi mteril In this Mnul there re mn eerises nd solutions on eh setion of eh units well s the glossr hs een prepred in oth ri nd English lnguges in ddition to the tle of ontents whih lso shows therequired mteril from the tetoo In ddition to this mnul nd the tetoo there is prtil Mnulfor Mthd nd QSBprogrms DrMhmoud l-bei pril

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6 Tetoo Introdutor iner lger with pplitions Contents equired Mteril from The Tetoo Chpteriner Equtions nd Mtries Eerises 9 Eerises Eerises Eerises Eerises 6nswers to Eerises8 7Glossr 9 ChpterDeterminnts Eerises Eerises nswers to Eerises8 Glossr ChpterVetors nd Vetor Spes Eerises 9 Eerises Eerises Eerises

7 6Eerises 7Eerises 8nswers to Eerises 9 Glossr69 Chpteriner Trnsformtions nd Mtries Eerises 7 Eerises 7 Eerises 7 nswers to Eerises77 Glossr89 ChpterEigenvlues nd Eigenvetors Eerises 9 Eerises 9 nswers to Eerises96 Glossr 6 Chpter6iner Progrmming 6Eerises 6nswers to Eerises7 6 Glossr 6

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10 Eerises Solve the given liner sstems the method of elimintion -=8 += ++= ++ =8 -+ = += - = + =7 n oil refiner proesses low-sulfur nd high-sulfur fueleh ton of low-sulfur fuel requires minutes in the lending plnt nd minutes in the refining plnteh ton of high sulfer fuel requires minutes in the plending plnt nd minutes in the refining plnt If the plending plnt is ville for houresnd the refining plnt is ville for houres how mn tons of eh tp of fuel should e mnuftured? dietiin is prepring mel onsisting of foods B nd CEh oune of food ontins units of protien units offt nd units of rohdrteeh unit of food B ontins units of protein units of ft nd unit of rohdrte Eh unit of food C ontins units of protein units of ft nd unit of rohdrteif the mel must provide etl units of protein units of ft nd units of rohdrtehow mn ounes of eh tpe of food should e used? 9

11 Eerises et C nd B If possile ompute B T C+ If d nd find d d Consider the following liner sstem 7 w Find the oeffiient mtri Write the liner sstem in mtri form Find the ugmented mtri Write the liner sstem with ugmented mtri? mnufturer mes two inds of produtsp nd Qt eh of two plnts nd YIn ming these produts the pollutnts sulfurdioidenitri oidend prtiulte mtter re produedthe mounts of pollutnts produed re given in ilogrms the mtri

12 SulfurNitriPrtiulte Dioideoidemtter ProdutP ProdutQ Sttend federl ordinnes require tht these pollutntse removedthe dil ost of removing eh ilogrm of pollutnt is given in dollrs the mtri Plnt Plnt Y 8 B 7 Sulfur 9 Nitri Prtiulte dioide Oide mtter Wht do the entries in the mtri ProdutB tell the mnufturer? 6 photogrph usiness hs store in eh of the following ities New YorDenvernd os ngelespriulr me of mer is ville in utomtisemiutomindnonutomtimodelsmoreovereh mer hs mthed flsh unit nd mer is usull sold together with the orresponding flsh unitthe selling pries of the mers nd flsh units re given in dollrs the mtri utomtisemiutomtinonutomti Cmer Flsh unit The numer of sets mer nd flshunit ville t eh store is given the mtri

13 New Yor Denver os ngeles B 8 utomti Semiutomti Noutomti Wht is the totl vlue of the Cmer in New Yor? Wht is the totl vlue of the flsh units in os ngeles? Eerises If nd B show tht B= 6 If nd B nd C show tht B = C Consider two qui food ompniesm nd NEh er ompnm eeps of its ustomerswhile swith to NEh er N eeps of its ustomerswhile swith to MSuppose tht the initil distriution of the mret is given Find the distriution of the mret fter er Find the stle distriution of the mret

14 et nd B e smmetri mtries Show tht +B is smmetri Show tht B is smmetri if nd onl ifb=b If is nnn mtri prove tht T nd T re smmetri + T is smmetri - T is sew smmetri Eerises Whih of the following mtries re in redued row ehelon form? B C D E F

15 G H et Find the mtries otined performing the followingelements row opertions on Interhnging the seond nd fourth rows Multipling the third row dding - times the first row to the fourth row If Find mtri Cin redued row ehelon form tht is row equivlent to Find ll solutions to the given liner sstem Find ll vlues of for whih the resulting of the following liner sstemhs

16 No solution unique solution Infinit mn solutions 6 Solve the liner sstem with thefollowing given ugmented mtri 7 furniture mnufturer mes hirs offee tles nd diningroom tles Eh hir requires minutes of snding 6 minutes of stining nd minutes of vrnishing Eh offee tle requires minutes of snding 8 minutesof stiningnd minutes of vrnishingeh dining room tle requires minutes of snding minutesof stiningnd 8 minutes of vrnishing The snding enh is ville 6 hours per wee the stining enh hours per wee nd the vrnishing enh 8 hours per wee How mn per wee of eh tpe of furniture should e mde? Eerises Show tht is non-singulr Show tht

17 6 is singulr Find the inverse of the following mtri if possile Whih of the following liner sstems hve nontrivil solution? Consider n industril proess whose mtri is Find the input mtri for the following output mtri C B

18 6For wht vlues of does the homogeneous sstem hve nontrivil solution? 7 Suppose tht nd B re squre mtries nd B=If B is nonsingulr find 8 et e nnmtriprove tht if is non-singulr then the homogeneous sstem =O hs trivil solution 7

19 8 8 8 se / 8 is solution The eqution ond the to dd Then eqution the Multipl Mens 6 8 / / 6 nswers to Eerises Eerises

20 / 6 + = / Suppose tht low- sulfur high- sulfurnd note tht hours = 6= minutes nd hours = 6=8 minutes 8

21 ton 6 / 8 ton Suppose tht food food B food C 9 / 7 7 / 9 Eerises 8 7 C B B T T 6 8 C B T

22 / / d d d d The oeffiient mtri is Theliner sstem in mtri form w = 7 C The ugmented mtri 7

23 7 6 8 For eh produt P or Q the dil ost of pollution ontrol t plnt or t Plnt Y 6 $ $6 Eerises 6 B

24 C B C B The informtion n edispled in mtri form s The distriution of the mret fter er is 9 9 o The stle distriution of the mret is += 7 7 +B is smmetri mtri if +B=+B T

25 +B T = T + B T But = T nd B = B T euse nd B re smmetri mtries so +B T = T + B T =+B - Suppose tht B=B B is smmetri mtri if B=B T B T =B T T =B=B B is smmetri mtri - Suppose thtb is smmetri mtri Te B=B T =B T T =B s oth nd B re smmetri mtries B B T issmmetri if T = T T T T = T T T = T Similrl for T + T is smmetri if + T =+ T T + T T = T T + T = + T - T is sew smmetri if - T = - - T T - - T T = - T - T T =- T +=- T Eerises EG

26 6 6 7 = =667 =-667

27 6 No solution unique solution Infinit mn solutions 6 =- = = - 7 et =numer of hirs =numer of offee tles = numer of dining room tles /

28 7 = = = Eerises is non-singulr mtri euse there eist mtri B suh tht B=B=I Suppose tht Is n inverse of Then Equting orresponding elements of these two mtries we otined the liner sstem d B I d d d B /

29 8 So this liner sstem hs no solution Hene there is no suh mtri B nd is singulr mtri = - B 6 C 6 B solution nontrivil hs sstem The Mtri Singulr solution nontrivil hs sstem The Mtri Singulr 7 7 7

30 7 B O B B O B O O 8 O O O 7Glossr يصفىفت Mtri يصفىفت صفريت mtri Zero يصفىفت يربعت Mtri Squre انشكم انصفي ان يز form edued ow Ehelon ان صفىفت انقطريت Mtri Digonl يصفىفت انىحذة Mtri Identit يصفىفت يثهثيت Mtri Tringulr يصفىفت ان عايالث Mtri Coeffiient ان صفىفت ان تذة Mtri ugmented ان ظاو انخطي Sstem iner ان ظير انضربي نه صفىفت يعكىس ان صفىفت Mtri Inverse of ي قىل ان صفىفت Mtri Trnspose of رتبت ان صفىفت Mtri n of ع هياث انصف انبسيظ Opertions Elementr ow 9

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34 Eerises Find the numer of inversions in eh of following permuttions of s Determinte whether eh of the following permuttions of s Is even or odd Compute thefollowing determinnts If

35 Find the determinnts of the following mtries nd Evlute the following determinnts using the properties of this Setion B C D 6 8 B C

36 6 If nd B re mtries with Clulte nd B d B B T B e 7 Show tht if = - then 8Show tht if T = then 9 Show tht This determinnt is lled Vndermonde determinnt Eerises et Compute ll the oftors Compute the determinnts of the following mtries

37 6 et Find dj Compute Compute dj Compute the inverse of the following mtri Use theorem to determine whih of the following mtries re non-singulr B B 7 C

38 d D 6 Use Corollr to find out whether the following homogeneous sstems hve nontrivil solutions +-= ++= -+= +++w= -+-w= +++w= --+w= 7 Solve the following liner sstem Crmer's rule ++-w=- ++w= +-+w= -+w= 8 Prove tht dj n 7

39 nswers to Eerises Eerises five inversions Seven inversions Fourinversions evenn= Odd n= Even n= --=7 -=- B C 8 D

40 9 6 7 B B B B B B C

41 7 Eerises 7 7 B B B B B d T T e 9 8 T T 9 7

42 7 B

43 9 6 C 9 T C dj I dj 8 6 6

44 C T C dj dj Mtri Singulr Mtri Singulr B Mtri Singulr C Mtri gulr Non D d sin Solution Nontrivil Hs

45 7 8 Solution Trivil Hs B 6 w n n n n dj I dj I dj I dj

46 Glossr ان حذد ان ت ى نهع صر The Minor of ij ij ان تعايم Coftor ان حذد Determinnt قري ان صفىفت Mtri djoint of قاعذة كراي ر Crmer's ule

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50 Eerises Determine the hed of the vetor whose til is t Determine the til of the vetor whose hed is t et = Y=-Z=nd U=-Find nd so tht Z= U Y Z U Find the length of the following vetors -- Find the distne etween pirs of points -- 6 Find unit vetor in the diretion of = 9

51 =- 7 Find Y =-Y= = Y = - 8 Write eh of the following vetors in terms of i nd j -- 9 ship is eing pushed tugot with fore of pounds long the negtive -is while nother tugot is pushing long the negtive -is with fore of ponds Find the mgnitude nd seth the diretion of the resultnt fore Eerises Find +Y-Y nd -Y if =-Y=- et =-Y=--Z=-nd U=uFind nd u so tht Z+Y= Z+U=Y Find the length of the following vetors - Find the distne etween the following pirs of points - -6 Find Y =Y=-

52 =-Y=-- 6Whih of the following vetors =6-8 =--- =--- = =- 6 =-re Orthogonl? Prllel? In the sme diretion? 7 Find unit vetor in the diretion of = - = 8Write eh of the following vetorsin in terms of ijnd - - 9Write eh of the following vetorsin s mtri i+j- i+j lrge steel mnufturer who hs emploees lists eh emploee's slr s omponent of vetor S in If n 8 perent ross-the-ord slr inrese hs een pproved find n epression involving S giving ll the new slries Eerises Determine whether thegiven set together with the given opertions is vetor spe If it is not vetor spe list the properties of Definition tht fil to hold The set of ll ordered triples of rel numers with the opertions nd

53 The set of ll ordered triples of rel numers of the form with the opertions nd The set of ll positive rel numers with the opertions nd Whih of the following susets of re suspes of? The set of ll vetors of the form dwhere -= dwhere =+ nd d=- dwhere = nd = -d Whih of the following susets of the vetor spe of ll mtries under the usul opertions of mtri ddition nd slr multiplition re suspes? d where d e where nd f e d f Eerises Whih of the following vetors re liner omintions of =- =-nd =--? Whih of the following vetors re liner omintions of P t=t +t+ p t=t + p t=t-? t +t+

54 -t +t- Whih of the following sets of vetors spn? d- Do the polnomils t +t+t -t+t +-t +t -t+ spn P? Whih of the following sets of vetors in re linerl dependent? For those tht re epress one vetor s liner omintion of the rest = =6- =- =- = = = = =66 d = = = 6Eerises Whih of the following vetors re ses for? Whih of the following vetors re ses for P? t +t +t t + 6t +8t +6t+ t +t +t+ t +t +t+ t +t +t+ t +t +t+ t +t +t+ et W e the suspe of P spnned t t t t t tt t t

55 Find sis for Wht is the dimension of W? Find sis for tht inludes the vetors nd - Find the dimension of the solution spe of 7Eerises Find sis for the suspe V of spnned the vetors Find the row nd olumn rns of Is linerl independent set of vetors in? Determine whih of the liner sstems hve solution ompring the rns of the oeffiient nd ugmented mtries nd 8 7 S

56 8nswers to Eerises Eerises = +m=+-= = +n=+=7 The hed of the vetor is 7 = -m=-=- = -n=-=- The til of the vetor is -- == = 8 6

57 6 6 ˆ ˆ 7 Y = +-= Y= +-= 8 iˆ ˆj iˆ ˆj 9 Eerises +Y=-+-=- -Y=---=- 6

58 =-6 -Y=---= = - --+=- -= = + = = -+u= -- +u-+= -- +=- =-6 u-=- u= +== = = --- = = - 6 nd nd 6 nd nd 6 nd 6 = 6 = = 6 = 6 = nd euse C None ˆ 6 euse ˆ ˆ ˆ ˆ 7 i j i j

59 S Eerises et = Y = Z nd d in Not vetor Spe d nd f do not hold Vetor Spe Vetor Spe et = - = Y = - = ˆ j i ˆ ˆ ˆ j i ˆ ˆ ˆ Y Y Z Y Z Y d d d d d d d d

60 9 +Y= = = - = So W is not suspe of is Suspe of issuspe of et W is suspe Suspe Eerises = =-- W Y W d d Y d d Y W Y d W /

61 6 To show tht spns we let =d e n vetor in We now see onstnts nd suh tht K =d K =d-+ K =- K =- K =- So spns To show tht-spns we let =d e n vetor in We now see onstnts nd suh tht K + -+ =d t t t P t P t P t t t P t P t P d d

62 6 To show tht spns we let =d e n vetor in We now see onstnts nd suh tht K =d So the vetor is not spnned the vetors not do vetors these - ; ; spns vetors the spnned not is vetor the so d d d 6 6 d d 6 6

63 6 These vetors do not spns dto show tht - spns we let =d e n vetor in We now see onstnts nd nd suh tht K =d K + + = K + + = = K + + =d So the vetors - spns To show tht t +t+t -t+t +-t +t -t+ spns Pwe let =t +t +t+d e n vetor in P We now see onstnts nd suh tht K t +t++ t -t+t ++ t ++ -t +t -t+=t +t +t+d K =d d d d d d d d d d d d d d d

64 6 So vetor lie t +t +t+ is not spnned the vetorst +t+t - t+t +-t +t -t+ The vetors t +t+t -t+t +-t +t -t+ do not spns P To show tht = =6- =- re linerl depenent we see onstnts nd suh tht K = + + = +6 - = K - + = Te To show tht =- = re linerl independent we see onstnts nd suh tht K -+ = The vetors =- = re linerl independent t t t t 6 dependent linerl re vetors given The

65 6 To show tht = = = =66 re linerl dependent we see onstnts nd suh tht K = K + + = K = + +6 = et = K = =- = - The vetors = = = =66 re linerl dependent d To show tht = = = re linerl independent we see onstnts nd suh tht K + + = K + + = + = + + = 6 6

66 6 K = = = The vetors = = = re linerl independent 6Eerises To show tht - is linerl independent we form the Eqution + =O + - =O + - = So - is linerl independent To show tht - spns we let = e n vetor in We now see onstnts nd suh tht K + -== + - =

67 66 So spns nd is sis for To show tht - is linerl independent weform the eqution + + = O = = + + = - + This sstem hs infinit mn solutions s Showing tht is linerl dependent Hene S does not spn nd is not sis for is sis for Possile nswer dimw= Possile nswer S S t t t t

68 67 The solution set is The dimension is ero 7Eerises Possile nswer The row nd olumn rn re inerl dependent S S

69 68 The rn of the oeffiient nd ugmented mtries re So the liner sstem hs solution The rn of the oeffiient mtri is The rn of the ugmented mtri oeffiient mtri is So the liner sstem hs no solution

70 9Glossr يتجه Vetor رو بعذ N-dimensionln فضاء خطي Spe iner فضاء جزئي Suspe يج ىعت يىنذة Set Spnning يستقهت خطيا Independent inerl يرتبطت خطيا dependent inerl كثير حذود Polnomil تركيبت خطيت Comintion iner أساس Bsis بعذ Dimension 69

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73 7 Eerises Whih of the following re liner trnsformtion? Whih of the following re liner trnsformtion? et e liner trnsformtion for whih we now tht Wht is Wht is nd??

74 7 et p p e liner trnsformtion for whih we now tht = t =t nd t =t +t Find t -t+ Find t +t+ Eerises et e defined? Find er Is one to one? Is onto? et e defined? Find sis for er Find sis for rnge Verif Theorem 7 et 6 e iner trnsformtion? If dim er = wht is dim rnge? If dim rnge = wht is dim er? Determine the dimension of the solution spe to the homogeneous sstem = for the given mtri

75 7 = Eerises et S = e sis for Find the oordinte vetors of the following vetors with respet to S et e sis for P Find the oordinte vetors of the following vetors with respet to S t + t et e defined et S nd T e the nturl ses for nd respetivel lso et 7 7 t t t S T nd S

76 7 e ses for nd respetivel Find the mtri of with respet to S nd T Compute et e defined et S nd T e the nturl ses for nd respetivel lso let e ses for nd respetivel Find the representtion of with respet to S nd T Compute using oth representtions ete liner trnsformtion Suppose tht themtri of with respet to the sis is sin nd in otined mtries the g u T nd S T nd S T nd S S

77 76 where Compute Compute Compute 6 et e defined Find the mtri of with respet to the nturl sis S for Using the definition of nd lso using the mtri otined in \ Find nd S S nd nd

78 77 nswers to Eerises Eerises et Then lso if is rel numer then Hene is liner trnsformtion et Y Y Y Y Y Y Y Y

79 78 Hene is not liner trnsformtion et lso if is rel numer then Hene is liner trnsformtion Hene is not liner trnsformtion Y Y Y Y Y Y 9

80 79 We n solve this question using 9

81 8 t -t+=t -t+ t -t+= t +t-t + = t -t +t+ t +t+=t +t+ t +t+= t +t+t + = t +t +t+ Eerises = + = = == is one - to one euse Given n re n rel numers n we find in so tht we re seeing solution to the liner sstem er er nd where in Y? Y Y

82 8 We re seeing solution to the liner sstem The redued row ehelon form of the ugmented mtri is Thus solution eists onl for nd so is not onto

83 8 dimer +dimrnge =dim V t t t t t t s s s t s t t t t t s t er for sis posssile rnge for sis posssile

84 8 + = dimrnge =-= dimer =-= The solution spe The dimension of the solution spe to the homogeneous sstem = for the given mtri is Eerises The oordinte vetor of the vetor with respet to S 7 s s s s s s 7 7 s s s s

85 8 The oordinte vetor of the vetor with respet to S t + = t ++ t-+ t= t + + t+ - = - = = - + = = - = The oordinte vetor of the vetor t + with respet To S The mtri of with respet to S nd T 7 7

86 8 The mtri of with respet to S` nd T ` - For - For The representtion of with respet to S nd T 7 8 7

87 86 S T T T T

88 87 The representtion of with respet to - For - For S T s s T nd S 7

89 88 6 -Using the definition of - Using the mtri otined in * * * * * * 8 * *

90 Glossr تحىيم خطي Trnsformtion iner ان ىاة Kernel ان ذي nge واحذ نىاحذ One One to شايم Onto ا عكاس efletion دورا ottion تركيب انتحىيالث انخطيت Trnsformtion Composition of iner ان صفىفت ان اقهت Mtri Trnsition ت ثيم انتحىيم انخطي ب صفىفت Trnsformtion Mtri epresenttion of iner 89

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94 9 Eerises Find the hrteristi polnomil of eh mtri Find the hrteristi polnomileigenvlues nd eigenvetors of eh mtri Find whih of the mtries re digonlile

95 9 Find for eh mtri if possile non-singulr mtri Psuh tht P - P is digonl Prove tht if nd B re similr mtries the hve the sme hrteristi polnomils nd hene the sme eigenvlues Eerises Verif tht P = Is n orthogonl mtri Digonlile eh given mtri nd find n orthogonl mtri P suh tht P - P is digonl Digonlile eh given mtri

96 9 Show tht if is n orthogonl mtri then Show tht if is n orthogonl mtri then is orthogonl

97 96 nswers to Eerises Eerises 7 I 6 9 I f I s et s s s s I s s s I

98 97 s et s s I 6 f I s et s s I s et s s I digonlile not is mtri the theorem g u B Eigenves Eigenvls sin

99 98 So nd B hve the sme hrteristi polnomils nd hene the sme eigenvlues Eerises Digonlile is mtri the theorem g u Eigenves Eigenvls sin digonlile not is mtri the theorem g u Eigenves Eigenvls sin sin P or P theorem g U Digonlile Eigenves Eigenvls I B I I P I P P I P P I P P P IP P P P I B I P P B P P T

100 99 P Is orthogonlmtri eigenves eigenvls P D eigenves eigenvls P D sin D theorem g U eigenvls I

101 sin D theorem g U eigenvls I * * * I

102 sin D theorem g U eigenvls if is n orthogonl mtri then - = t Need to show tht - - = - T Suppose tht is n orthogonl mtri B ting the trnspose oth sides of equtionwe get - T = T T == = - T So T T mtri orthogonl n is

103 Glossr قي ت ي يزة Vlue Eigen يتجه ي يز Vetor Eigen كثير انحذود ان يز Polnomil Chrteristi ان عادنت ان يزة Eqution Chrteristi انفضاء ان يز Spe Eigen انتشابه Similrit يصفىفاث يتشابهت mtries Similr ان صفىفت انقابهت نهتحىيم ان يصفىفت قطريت Mtri Digonlile

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106 6Eerises Steel produer mes two tpes of steel regulr nd speil ton of regulr steel requires hours in the open herth furne nd hours in the soing pit ; ton of speil steel requires hours in the openherth furne nd hours in the soing pit The open- herth furne is ville 8 hours per d nd the soing pit is ville hours per d The profit on tonof regulr steel is$ nd it is $ on ton of speil steeldetermine how mn tons of eh tpe of steel should e mde to mimie the profit television produer designs progrm sed on omedin nd time for ommerils The dvertiser insists on t lest minutes of dvertising time the sttion insists on no more thn minutes of dvertising time nd the omedin insists on t lest minutes of the omed progrm lso the totl time llotted for the dvertising nd omed portions of the progrm nnot eeed minutes If it hs een determined tht eh minute of dvertisingver retive ttrts viewers nd eh minute of the omed progrm ttrts viewers how should the time e divided etween dvertising nd progrmming to mimie the numer of viewerminutes? The protein Diet Clu serves lunheon onsisting of two dishes nd B Suppose tht eh unit of hs grm of ft grm of rohdrte nd grms of protein wheres eh unit of B hs grms of ft grm of rohdrte nd 6 grms of protein If the dietiin plnning the lunheon wnts to provide no more thn

107 6 grms of ft or more thn 7 grms of rohdrte how mn units of nd how mn units of B should e served to mimie the mount of protein onsumed? Seth the set of points stisfing the given sstem of inequlities Minimie = - Sujet to 6 Formulte the following liner progrmming prolem s new prolem with sl vriles Mimie = sujet to

108 6 nswers to Eerises 6 Eerises Supposetht is the mount of regulr steel nd is the mount of speil steel Mimie Z= + Sujet to Suppose tht is dvertising time nd is the time of the omed progrm Mimie Z= + Sujet to 8 7

109 Suppose tht is the mount units of tpe nd is the the mount units of tpe B Mimie Z = +6 Sujet to 7 8

110 The point 8 9 Z

111 min Mimie = + +7 Sujet to

112 6 Glossr ان تباي ت انخطيت Inequlit iner اقترا انهذف Funtion Ojetive قيىد شروط Constrints

113 The End

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