Tải bản đầy đủ - 0 (trang)
10 Linear Models in Business, Science, and Engineering

# 10 Linear Models in Business, Science, and Engineering

Tải bản đầy đủ - 0trang

1.10 Linear Models in Business, Science, and Engineering

81

TABLE 1

Amounts(g)Suppliedper100 gofIngredient

Nonfatmilk

Soyﬂour

Whey

Amounts(g)Suppliedby

CambridgeDietinOneDay

Protein

36

51

13

33

Carbohydrate

52

34

74

45

0

7

Nutrient

Fat

1.1

3

SOLUTION Let x1 , x2 , and x3 , respectively, denotethenumberofunits(100g)of

thesefoodstuffs. Oneapproachtotheproblemistoderiveequationsforeachnutrient

separately. Forinstance, theproduct

x1 unitsof

nonfatmilk

proteinperunit

ofnonfatmilk

givestheamountofproteinsuppliedby x1 unitsofnonfatmilk. Tothisamount, we

eachnutrient.

A moreefﬁcientmethod, andonethatisconceptuallysimpler, istoconsidera

“nutrientvector”foreachfoodstuffandbuildjustonevectorequation. Theamountof

nutrientssuppliedby x1 unitsofnonfatmilkisthescalarmultiple

Scalar

x1 unitsof

nonfatmilk

Vector

nutrientsperunit

D x1 a1

ofnonfatmilk

(1)

where a1 istheﬁrstcolumninTable1. Let a2 and a3 bethecorrespondingvectors

forsoyﬂourandwhey, respectively, andlet b bethevectorthatliststhetotalnutrients

required(thelastcolumnofthetable). Then x2 a2 and x3 a3 givethenutrientssupplied

by x2 unitsofsoyﬂourand x3 unitsofwhey, respectively. Sotherelevantequationis

x1 a1 C x2 a2 C x3 a3 D b

(2)

Rowreductionoftheaugmentedmatrixforthecorrespondingsystemofequations

showsthat

2

3

2

3

36 51

13 33

1 0 0 :277

4 52

40

34

74 45 5

1 0 :392 5

0

7

1.1 3

0 0 1 :233

Tothreesigniﬁcantdigits, thedietrequires.277unitsofnonfatmilk, .392unitsof

soyﬂour, and.233unitsofwheyinordertoprovidethedesiredamountsofprotein,

carbohydrate, andfat.

Itisimportantthatthevaluesof x1 , x2 , and x3 foundabovearenonnegative. Thisis

necessaryforthesolutiontobephysicallyfeasible. (Howcouldyouuse :233 unitsof

whey, forinstance?) Withalargenumberofnutrientrequirements, itmaybenecessary

tousealargernumberoffoodstuffsinordertoproduceasystemofequationswitha

“nonnegative”solution. Thusmany, manydifferentcombinationsoffoodstuffsmay

needtobeexaminedinordertoﬁndasystemofequationswithsuchasolution. In

fact, themanufactureroftheCambridgeDietwasabletosupply31nutrientsinprecise

amountsusingonly33ingredients.

ofnutrientssuppliedbyeachfoodstuffcanbewrittenasascalarmultipleofavector, as

in(1). Thatis, thenutrientssuppliedbyafoodstuffare proportional totheamountof

82

CHAPTER 1

Linear Equations in Linear Algebra

theamountsfromthevariousfoodstuffs.

Problemsofformulatingspecializeddietsforhumansandlivestockoccurfrequently. Usuallytheyaretreatedbylinearprogrammingtechniques. Ourmethodof

Linear Equations and Electrical Networks

WEB

Currentﬂowinasimpleelectricalnetworkcanbedescribedbyasystemoflinear

equations. A voltagesourcesuchasabatteryforcesacurrentofelectronstoﬂow

throughthenetwork. Whenthecurrentpassesthrougharesistor(suchasalightbulbor

motor), someofthevoltageis“usedup”; byOhm’slaw, this“voltagedrop”acrossa

resistorisgivenby

V D RI

wherethevoltage V ismeasuredin volts, theresistance R in ohms (denotedby ), and

thecurrentﬂow I in amperes (amps, forshort).

ThenetworkinFig. 1containsthreeclosedloops. Thecurrentsﬂowinginloops1,

2, and3aredenotedby I1 ; I2 , and I3 , respectively. Thedesignateddirectionsofsuch

loopcurrents arearbitrary. Ifacurrentturnsouttobenegative, thentheactualdirection

ofcurrentﬂowisoppositetothatchosenintheﬁgure. Ifthecurrentdirectionshownis

awayfromthepositive(longer)sideofabattery( )aroundtothenegative(shorter)

side, thevoltageispositive; otherwise, thevoltageisnegative.

Currentﬂowinaloopisgovernedbythefollowingrule.

KIRCHHOFF'S VOLTAGE LAW

Thealgebraicsumofthe RI voltagedropsinonedirectionaroundaloopequals

thealgebraicsumofthevoltagesourcesinthesamedirectionaroundtheloop.

EXAMPLE 2 DeterminetheloopcurrentsinthenetworkinFig. 1.

30 volts

A

I1

B

C

5 volts

I2

I3

D

SOLUTION Forloop1, thecurrent I1 ﬂowsthroughthreeresistors, andthesumofthe

RI voltagedropsis

4I1 C 4I1 C 3I1 D .4 C 4 C 3/I1 D 11I1

Currentfromloop2alsoﬂowsinpartofloop1, throughtheshort branch between A

and B . Theassociated RI dropthereis 3I2 volts. However, thecurrentdirectionfor

thebranch AB inloop1isoppositetothatchosenfortheﬂowinloop2, sothealgebraic

sumofall RI dropsforloop1is 11I1 3I2 . Sincethevoltageinloop1is C30 volts,

Kirchhoff’svoltagelawimpliesthat

11I1

20 volts

FIGURE 1

Theequationforloop2is

3I2 D 30

3I1 C 6I2

I3 D 5

Theterm 3I1 comesfromtheﬂowoftheloop-1currentthroughthebranch AB (with

anegativevoltagedropbecausethecurrentﬂowthereisoppositetotheﬂowinloop2).

Theterm 6I2 isthesumofallresistancesinloop2, multipliedbytheloopcurrent. The

term I3 D 1 I3 comesfromtheloop-3currentﬂowingthroughthe1-ohmresistor

inbranch CD, inthedirectionoppositetotheﬂowinloop2. Theloop-3equationis

I2 C 3I3 D

25

1.10 Linear Models in Business, Science, and Engineering

83

Notethatthe5-voltbatteryinbranch CD iscountedaspartofbothloop2andloop3,

butitis 5 voltsforloop3becauseofthedirectionchosenforthecurrentinloop3.

The20-voltbatteryisnegativeforthesamereason.

Theloopcurrentsarefoundbysolvingthesystem

11I1 3I2

D

3I1 C 6I2

I3 D

I2 C 3I3 D

30

5

25

(3)

Rowoperationsontheaugmentedmatrixleadtothesolution: I1 D 3amps, I2 D

1amp, and I3 D 8 amps. Thenegativevalueof I3 indicatesthattheactualcurrent

inloop3ﬂowsinthedirectionoppositetothatshowninFig. 1.

Itisinstructivetolookatsystem(3)asavectorequation:

2

3

2

3

2

3 2

3

11

3

0

30

I1 4 3 5 C I2 4 6 5 C I3 4 1 5 D 4 5 5

0

1

3

25

r1

r2

r3

(4)

v

Theﬁrstentryofeachvectorconcernstheﬁrstloop, andsimilarlyforthesecondand

thirdentries. Theﬁrstresistorvector r1 liststheresistanceinthevariousloopsthrough

whichcurrent I1 ﬂows. A resistanceiswrittennegativelywhen I1 ﬂowsagainstthe

ﬂowdirectioninanotherloop. ExamineFig. 1andseehowtocomputetheentriesin

r1 ; thendothesamefor r2 and r3 . Thematrixformofequation(4),

2 3

I1

Ri D v; where R D Œ r1 r2 r3  and i D 4 I2 5

I3

providesamatrixversionofOhm’slaw. Ifallloopcurrentsarechoseninthesame

direction(say, counterclockwise), thenallentriesoffthemaindiagonalof R willbe

negative.

Thematrixequation Ri D v makesthelinearityofthismodeleasytoseeataglance.

Forinstance, ifthevoltagevectorisdoubled, thenthecurrentvectormustdouble. Also,

a superpositionprinciple holds. Thatis, thesolutionofequation(4)isthesumofthe

solutionsoftheequations

2 3

2 3

2

3

30

0

0

Ri D 4 0 5;

Ri D 4 5 5; and Ri D 4 0 5

0

0

25

Eachequationherecorrespondstothecircuitwithonlyonevoltagesource(theother

sourcesbeingreplacedbywiresthatcloseeachloop). Themodelforcurrentﬂowis

linear preciselybecauseOhm’slawandKirchhoff’slawarelinear: Thevoltagedrop

acrossaresistoris proportional tothecurrentﬂowingthroughit(Ohm), andthe sum of

thevoltagedropsinaloopequalsthesumofthevoltagesourcesintheloop(Kirchhoff).

Loopcurrentsinanetworkcanbeusedtodeterminethecurrentinanybranchof

thenetwork. Ifonlyoneloopcurrentpassesthroughabranch, suchasfrom B to D

inFig. 1, thebranchcurrentequalstheloopcurrent. Ifmorethanoneloopcurrent

passesthroughabranch, suchasfrom A to B , thebranchcurrentisthealgebraicsum

oftheloopcurrentsinthebranch(Kirchhoff’scurrentlaw). Forinstance, thecurrentin

branch AB is I1 I2 D 3 1 D 2amps, inthedirectionof I1 . Thecurrentinbranch

CD is I2 I3 D 9amps.

84

CHAPTER 1

Linear Equations in Linear Algebra

Difference Equations

Inmanyﬁeldssuchasecology, economics, andengineering, aneedarisestomodel

areeachmeasuredatdiscretetimeintervals, producingasequenceofvectors x0 , x1 ,

x2 ; : : : : Theentriesin xk provideinformationaboutthe state ofthesystematthetime

ofthe k thmeasurement.

Ifthereisamatrix A suchthat x1 D Ax0 , x2 D Ax1 , and, ingeneral,

xk C1 D Axk

for k D 0; 1; 2; : : :

(5)

then(5)iscalleda lineardifferenceequation (or recurrencerelation). Givensuch

anequation, onecancompute x1 , x2 , andsoon, provided x0 isknown. Sections4.8

and4.9, andseveralsectionsinChapter 5, willdevelopformulasfor xk anddescribe

whatcanhappento xk as k increasesindeﬁnitely. Thediscussionbelowillustrateshow

A subjectofinteresttodemographersisthemovementofpopulationsorgroupsof

peoplefromoneregiontoanother. Thesimplemodelhereconsidersthechangesinthe

populationofacertaincityanditssurroundingsuburbsoveraperiodofyears.

Fixaninitialyear—say, 2000—anddenotethepopulationsofthecityandsuburbs

thatyearby r0 and s0 , respectively. Let x0 bethepopulationvector

Ä

r

Citypopulation, 2000

x0 D 0

s0

Suburbanpopulation, 2000

For2001andsubsequentyears, denotethepopulationsofthecityandsuburbsbythe

vectors

Ä

Ä

Ä

r

r

r

x3 D 3 ; : : :

x2 D 2 ;

x1 D 1 ;

s3

s2

s1

Ourgoalistodescribemathematicallyhowthesevectorsmightberelated.

movestothesuburbs(and95%remainsinthecity), while3%ofthesuburbanpopulation

movestothecity(and97%remainsinthesuburbs). SeeFig. 2.

City

Suburbs

.05

.95

.97

.03

FIGURE 2 Annualpercentagemigrationbetweencityandsuburbs.

After1year, theoriginal r0 personsinthecityarenowdistributedbetweencityand

suburbsas

Ä

Ä

Remainincity

:95r0

:95

(6)

D r0

Movetosuburbs

:05r0

:05

The s0 personsinthesuburbsin2000aredistributed1yearlateras

Ä

:03

Movetocity

s0

:97

Remaininsuburbs

(7)

1.10 Linear Models in Business, Science, and Engineering

85

Thevectorsin(6)and(7)accountforallofthepopulationin2001.³ Thus

Ä

Ä

Ä

Ä

Ä

r1

:95

:03

:95 :03 r0

D r0

C s0

D

s1

:05

:97

:05 :97 s0

Thatis,

x1 D M x0

(8)

where M isthe migrationmatrix determinedbythefollowingtable:

From:

City Suburbs

To:

:95

:05

City

Suburbs

Ä

:03

:97

Equation(8)describeshowthepopulationchangesfrom2000to2001. Ifthemigration

percentagesremainconstant, thenthechangefrom2001to2002isgivenby

x2 D M x1

andsimilarlyfor2002to2003andsubsequentyears. Ingeneral,

xk C1 D M xk

for k D 0; 1; 2; : : :

(9)

Thesequenceofvectors fx0 ; x1 ; x2 ; : : :g describesthepopulationofthecity/suburban

regionoveraperiodofyears.

EXAMPLE 3 Computethepopulationoftheregionjustdescribedfortheyears

2001and2002, giventhatthepopulationin2000was600,000inthecityand400,000

inthesuburbs.

Ä

600;000

. For2001,

SOLUTION Theinitialpopulationin2000is x0 D

400;000

Ä

Ä

Ä

582;000

:95 :03 600;000

D

x1 D

418;000

:05 :97 400;000

For2002,

x2 D M x1 D

Ä

:95

:05

:03

:97

Ä

582;000

418;000

D

Ä

565;440

434;560

Themodelforpopulationmovementin(9)is linear becausethecorrespondence

xk 7! xk C1 isalineartransformation. Thelinearitydependsontwofacts: thenumber

ofpeoplewhochosetomovefromoneareatoanotheris proportional tothenumberof

peopleinthatarea, asshownin(6)and(7), andthecumulativeeffectofthesechoices

PRACTICE PROBLEM

Findamatrix A andvectors x and b suchthattheprobleminExample1amountsto

solvingtheequation Ax D b.

³ Forsimplicity, weignoreotherinﬂuencesonthepopulationsuchasbirths, deaths, andmigrationintoand

outofthecity/suburbanregion.

86

CHAPTER 1

Linear Equations in Linear Algebra

1.10 EXERCISES

1. Thecontainerofabreakfastcerealusuallyliststhenumber

ofcaloriesandtheamountsofprotein, carbohydrate, and

fatcontainedinoneservingofthecereal. Theamountsfor

twocommoncerealsaregivenbelow. Supposeamixtureof

thesetwocerealsistobepreparedthatcontainsexactly295

calories, 9 gofprotein, 48 gofcarbohydrate, and8 goffat.

a. Setupavectorequationforthisproblem. Includeastatementofwhatthevariablesinyourequationrepresent.

b. Writeanequivalentmatrixequation, andthendetermine

ifthedesiredmixtureofthetwocerealscanbeprepared.

NutritionInformationperServing

GeneralMills

Quaker®

Nutrient

Cheerios®

100%NaturalCereal

Calories

110

130

Protein(g)

4

3

Carbohydrate(g)

20

18

Fat(g)

2

5

2. OneservingofShreddedWheatsupplies160calories, 5 gof

protein, 6 gofﬁber, and1 goffat. OneservingofCrispix®

supplies110calories, 2 gofprotein, .1 gofﬁber, and.4 gof

fat.

a. Setupamatrix B andavector u suchthat B u givesthe

amountsofcalories, protein, ﬁber, andfatcontainedin

amixtureofthreeservingsofShreddedWheatandtwo

servingsofCrispix.

classical Mac and Cheese to Annie’s® Whole Wheat

ShellsandWhiteCheddar. Whatproportionsofservings

ofeachfoodshouldsheusetomeetthesamegoalsasin

part(a)?

4. TheCambridgeDietsupplies.8 gofcalciumperday, in

1. Theamountsofcalciumperunit(100 g)suppliedbythe

threeingredientsintheCambridgeDietareasfollows: 1.26 g

fromnonfatmilk, .19 gfromsoyﬂour, and.8 gfromwhey.

Anotheringredientinthedietmixtureisisolatedsoyprotein,

whichprovidesthefollowingnutrientsineachunit: 80 gof

protein, 0 gofcarbohydrate, 3.4 goffat, and.18 gofcalcium.

a. Setupamatrixequationwhosesolutiondeterminesthe

amountsofnonfatmilk, soyﬂour, whey, andisolated

soyproteinnecessarytosupplythepreciseamountsof

protein, carbohydrate, fat, andcalciumintheCambridge

Diet. Statewhatthevariablesintheequationrepresent.

InExercises5–8, writeamatrixequationthatdeterminestheloop

currents. [M] IfMATLAB oranothermatrixprogramisavailable,

solvethesystemfortheloopcurrents.

5.

20 V

b. [M] Supposethatyouwantacerealwithmoreﬁberthan

CrispixbutfewercaloriesthanShreddedWheat. Isit

possibleforamixtureofthetwocerealstosupply130

calories, 3.20 gofprotein, 2.46 gofﬁber, and.64 gof

fat? Ifso, whatisthemixture?

a. [M] Ifshewantstolimitherlunchto400caloriesbut

get30 gofproteinand10 gofﬁber, whatproportionsof

servingsofMacandCheese, broccoli, andchickenshould

sheuse?

b. [M] Shefoundthattherewastoomuchbroccoliinthe

proportionsfrompart(a), soshedecidedtoswitchfrom

I2

20 V

I2

40 V

I3

10 V

I4

30 V

I2

I1

40 V

20 V

I4

I3

10 V

7.

I1

30 V

30 V

10 V

3. Aftertakinganutritionclass, abigAnnie’s® MacandCheese

fan decidesto improvethelevelsofproteinandﬁberin

Thenutritionalinformationforthefoodsreferredtointhis

exercisearegiveninthetablebelow.

NutritionInformationperServing

Nutrient MacandCheese Broccoli Chicken Shells

Calories

270

51

70

260

Protein(g)

10

5.4

15

9

Fiber(g)

2

5.2

0

5

I1

6.

I4

10 V

I3

20 V

1.10 Linear Models in Business, Science, and Engineering

8.

50 V

40 V

I1

I4

I5

I2

I3

30 V

12. [M] Budget® RentA CarinWichita, Kansashasaﬂeetof

maybereturnedtoanyofthethreelocations. Thevarious

fractionsofcarsreturnedtothethreelocationsareshownin

thematrixbelow. SupposethatonMondaythereare295cars

attheairport(orrentedfromthere), 55carsattheeastside

ofﬁce, and150carsatthewestsideofﬁce. Whatwillbethe

approximatedistributionofcarsonWednesday?

CarsRentedFrom:

Airport East

West

2

3

:97

:05

:10

4:00

:90

:055

:03

:05

:85

ReturnedTo:

Airport

East

West

20 V

suburbanpopulationmovesintothecity. In2010, therewere

800,000residentsinthecityand500,000inthesuburbs.

where x0 istheinitialpopulationin2010. Thenestimate

the populations in the city and in the suburbs two years

later, in2012. (Ignoreotherfactorsthatmightinﬂuencethe

populationsizes.)

suburbanpopulationmovesintothecity. In2010, therewere

10,000,000residentsinthecityand800,000inthesuburbs.

where x0 istheinitialpopulationin2010. Thenestimatethe

populationsinthecityandinthesuburbstwoyearslater, in

2012.

11. In1994, thepopulationofCaliforniawas31,524,000, and

thepopulationlivingintheUnitedStatesbut outside Californiawas228,680,000. Duringtheyear, itisestimatedthat

516,100personsmovedfromCaliforniatoelsewhereinthe

UnitedStates, while381,262personsmovedtoCalifornia

fromelsewhereintheUnitedStates.4

a. Setupthemigrationmatrixforthissituation, usingﬁve

decimalplacesforthemigrationratesintoandoutof

California. Letyourworkshowhowyouproducedthe

migrationmatrix.

b. [M] Computetheprojectedpopulationsintheyear2000

forCaliforniaandelsewhereintheUnitedStates, assumingthatthemigrationratesdidnotchangeduringthe6yearperiod. (Thesecalculationsdonottakeintoaccount

births, deaths, orthesubstantialmigrationofpersonsinto

CaliforniaandelsewhereintheUnitedStatesfromother

countries.)

4 MigrationdatasuppliedbytheDemographicResearchUnitofthe

CaliforniaStateDepartmentofFinance.

87

13. [M] Let M and x0 beasinExample3.

a. Computethepopulationvectors xk for k D 1; : : : ; 20.

Discusswhatyouﬁnd.

b. Repeatpart(a)withaninitialpopulationof350,000in

thecityand650,000inthesuburbs. Whatdoyouﬁnd?

14. [M] Studyhowchangesinboundarytemperaturesonasteel

plateaffectthetemperaturesatinteriorpointsontheplate.

a. Beginbyestimatingthetemperatures T1 , T2 , T3 , T4 at

eachofthesetsoffourpointsonthesteelplateshownin

theﬁgure. Ineachcase, thevalueof Tk isapproximated

bytheaverageofthetemperaturesatthefourclosest

points. SeeExercises33and34inSection1.1, where

thevalues(indegrees)turnouttobe .20; 27:5; 30; 22:5/.

Howisthislistofvaluesrelatedtoyourresultsforthe

pointsinset(a)andset(b)?

b. Without making any computations, guess the interior

temperaturesin(a)whentheboundarytemperaturesare

allmultipledby3. Checkyourguess.

listoffourinteriortemperatures.

Plate A

Plate B

20º

20º

1

2

4

3

20º

20º

(a)

10º

10º

1

2

4

3

10º

10º

(b)

40º

40º

88

CHAPTER 1

Linear Equations in Linear Algebra

SOLUTION TO PRACTICE PROBLEM

2

36

A D 4 52

0

3

13

74 5;

1:1

51

34

7

2

3

x1

x D 4 x2 5;

x3

2

3

33

b D 4 45 5

3

CHAPTER 1 SUPPLEMENTARY EXERCISES

true, citeappropriatefactsortheorems. Iffalse, explainwhy

orgiveacounterexamplethatshowswhythestatementisnot

trueineverycase.

a. Everymatrixisrowequivalenttoauniquematrixin

echelonform.

b. Anysystemof n linearequationsin n variableshasat

most n solutions.

c.

Ifasystemoflinearequationshastwodifferentsolutions, itmusthaveinﬁnitelymanysolutions.

d. Ifasystemoflinearequationshasnofreevariables, then

ithasauniquesolution.

e.

f.

If an augmented matrix Œ A b  is transformed into

Œ C d  byelementaryrowoperations, thentheequations Ax D b and C x D d haveexactlythesamesolutionsets.

Ifasystem Ax D b hasmorethanonesolution, thenso

doesthesystem Ax D 0.

g. If A isan m n matrixandtheequation Ax D b is

consistentforsome b, thenthecolumnsof A span Rm .

h. Ifanaugmentedmatrix Œ A b  canbetransformedby

elementaryrowoperationsintoreducedechelonform,

thentheequation Ax D b isconsistent.

o. If A isan m n matrix, iftheequation Ax D b hasat

leasttwodifferentsolutions, andiftheequation Ax D c

isconsistent, thentheequation Ax D c hasmanysolutions.

p. If A and B arerowequivalent m n matricesandifthe

columnsof A span Rm , thensodothecolumnsof B .

q. Ifnoneofthevectorsintheset S D fv1 ; v2 ; v3 g in R3 is

amultipleofoneoftheothervectors, then S islinearly

independent.

r.

If fu; v; wg islinearlyindependent, then u, v, and w are

notin R2 .

s.

Insomecases, itispossibleforfourvectorstospan R5 .

t.

If u and v arein Rm , then u isin Spanfu; vg.

u. If u, v, and w arenonzerovectorsin R2 , then w isalinear

combinationof u and v.

v.

If w isalinearcombinationof u and v in Rn , then u isa

linearcombinationof v and w.

w. Supposethat v1 , v2 , and v3 arein R5 , v2 isnotamultiple

of v1 , and v3 isnotalinearcombinationof v1 and v2 .

Then fv1 ; v2 ; v3 g islinearlyindependent.

x. A lineartransformationisafunction.

i.

Ifmatrices A and B arerowequivalent, theyhavethe

samereducedechelonform.

y.

j.

Theequation Ax D 0 hasthetrivialsolutionifandonly

iftherearenofreevariables.

If A isa 6 5 matrix, thelineartransformation x 7! Ax

cannotmap R5 onto R6 .

z.

If A isan m n matrixwith m pivotcolumns, thenthe

lineartransformation x 7! Ax isaone-to-onemapping.

k. If A isan m n matrixandtheequation Ax D b isconsistentforevery b in Rm , then A has m pivotcolumns.

l.

Ifan m n matrix A hasapivotpositionineveryrow,

thentheequation Ax D b hasauniquesolutionforeach

b in Rm .

m. Ifan n n matrix A has n pivotpositions, thenthe

reducedechelonformof A isthe n n identitymatrix.

n. If 3 3 matrices A and B eachhavethreepivotpositions, then A canbetransformedinto B byelementary

rowoperations.

2. Let a and b representrealnumbers. Describethepossible

solutionsetsofthe(linear)equation ax D b . [Hint: The

numberofsolutionsdependsupon a and b .]

3. Thesolutions .x; y; ´/ ofasinglelinearequation

ax C by C c´ D d

formaplanein R3 when a, b , and c arenotallzero. Construct

setsofthreelinearequationswhosegraphs(a)intersectin

asingleline, (b)intersectinasinglepoint, and(c)haveno

Chapter 1 Supplementary Exercises

pointsincommon. Typicalgraphsareillustratedintheﬁgure.

89

c. Deﬁneanappropriatelineartransformation T usingthe

matrixin(b), andrestatetheproblemintermsof T .

8. Describethepossibleechelonformsofthematrix A. Usethe

notationofExample1inSection 1.2.

a. A isa 2 3 matrixwhosecolumnsspan R2 .

b. A isa 3

Three planes intersecting

in a line

(a)

Three planes intersecting

in a point

(b)

3 matrixwhosecolumnsspan R3 .

Ä

5

9. Write the vector

as the sum of two vectors,

6

one on the line f.x; y/ W y D 2xg and one on the line

f.x; y/ W y D x=2g.

10. Let a1 ; a2 , and b bethevectorsin R2 shownintheﬁgure, and

let A D Œa1 a2 . Doestheequation Ax D b haveasolution?

Ifso, isthesolutionunique? Explain.

x2

b

a1

Three planes with no

intersection

(c)

Three planes with no

intersection

(c')

a2

4. Supposethecoefﬁcientmatrixofalinearsystemofthree

equations in three variables has a pivot position in each

column. Explainwhythesystemhasauniquesolution.

5. Determine h and k suchthatthesolutionsetofthesystem

(i)isempty, (ii)containsauniquesolution, and(iii)contains

inﬁnitelymanysolutions.

a.

b.

x1 C 3x2 D k

4x1 C hx2 D 8

2x1 C hx2 D

6x1 C kx2 D

1

2

6. Considertheproblemofdeterminingwhetherthefollowing

systemofequationsisconsistent:

4x1

8x1

2x2 C 7x3 D

3x2 C 10x3 D

5

3

a. Deﬁneappropriatevectors, andrestatetheproblemin

termsoflinearcombinations. Thensolvethatproblem.

b. Deﬁneanappropriatematrix, andrestatetheproblem

usingthephrase“columnsof A.”

c. Deﬁneanappropriatelineartransformation T usingthe

matrixin(b), andrestatetheproblemintermsof T .

7. Considertheproblemofdeterminingwhetherthefollowing

systemofequationsisconsistentforall b1 , b2 , b3 :

2x1

4x2

2x3 D b1

5x1 C x2 C x3 D b2

7x1

5x2

x1

3x3 D b3

a. Deﬁneappropriatevectors, andrestatetheproblemin

termsof Span fv1 ; v2 ; v3 g. Thensolvethatproblem.

b. Deﬁneanappropriatematrix, andrestatetheproblem

usingthephrase“columnsof A.”

11. Constructa 2 3 matrix A, notinechelonform, suchthat

thesolutionof Ax D 0 isalinein R3 .

12. Constructa 2 3 matrix A, notinechelonform, suchthat

thesolutionof Ax D 0 isaplanein R3 .

13. Writethe reduced echelonformofa 3 3 matrix A such

that

3 columns of A are pivot columns and

2 the3 ﬁrst2 two

0

3

A4 2 5 D 4 0 5 .

0

1

Ä

Ä

1

a

14. Determinethevalue(s)of a suchthat

;

is

a

aC2

linearlyindependent.

15. In(a)and(b), supposethevectorsarelinearlyindependent.

Whatcanyousayaboutthenumbers a; : : : ; f ? Justifyyour

2 3 2 3 2 3

2 3 2 3 2 3

a

b

d

d

b

a

617 6c7 6 e 7

6

7

6

7

6

5

4

5

4

5

4

0 , c , e

b. 4 5, 4 5, 4 7

a.

0

1

f 5

0

0

f

0

0

1

16. UseTheorem7inSection 1.7toexplainwhythecolumnsof

thematrix A arelinearlyindependent.

2

3

1

0

0

0

62

5

0

07

7

43

6

8

05

4

7

9 10

17. Explain why a set fv1 ; v2 ; v3 ; v4 g in R5 must be linearly

independentwhen fv1 ; v2 ; v3 g islinearlyindependentand v4

is not in Span fv1 ; v2 ; v3 g.

18. Suppose fv1 ; v2 g isalinearlyindependentsetin Rn . Show

that fv1 ; v1 C v2 g isalsolinearlyindependent.

90

CHAPTER 1

Linear Equations in Linear Algebra

19. Suppose v1 ; v2 ; v3 aredistinctpointsononelinein R3 . The

lineneednotpassthroughtheorigin. Showthat fv1 ; v2 ; v3 g

islinearlydependent.

20. Let T W Rn ! Rm bealineartransformation, andsuppose

T .u/ D v. Showthat T . u/ D v.

21. Let T W R ! R be the linear transformation that reﬂects each vector through the plane x2 D 0. That is,

T .x1 ; x2 ; x3 / D .x1 ; x2 ; x3 /. Findthestandardmatrixof T .

3

3

22. Let A bea 3 3 matrixwiththepropertythatthelinear

transformation x 7! Ax maps R3 onto R3 . Explainwhythe

transformationmustbeone-to-one.

23. A Givensrotation isalineartransformationfrom Rn to Rn

usedincomputerprogramstocreateazeroentryinavector

(usuallyacolumnofamatrix). Thestandardmatrixofa

Givensrotationin R2 hastheform

Ä

a

b

;

a2 C b 2 D 1

b

a

Ä

Ä

4

5

Find a and b suchthat

isrotatedinto

.

3

0

x2

(4, 3)

(5, 0)

A Givensrotationin R2 .

WEB

x1

24. ThefollowingequationdescribesaGivensrotationin R3 .

Find a and b .

2

a

40

b

0

1

0

32 3 2 p 3

b

2

2 5

0 54 3 5 D 4 3 5 ;

a

4

0

a2 C b 2 D 1

25. A large apartment building is to be built using modular

construction techniques. The arrangement of apartments

onanyparticularﬂooristobechosenfromoneofthree

basicﬂoorplans. PlanA has18apartmentsononeﬂoor,

including3three-bedroomunits, 7two-bedroomunits, and8

one-bedroomunits. EachﬂoorofplanB includes4threebedroomunits, 4two-bedroomunits, and8one-bedroom

units. EachﬂoorofplanC includes5three-bedroomunits,

3two-bedroomunits, and9one-bedroomunits. Supposethe

buildingcontainsatotalof x1 ﬂoorsofplanA, x2 ﬂoorsof

planB,and x3 ﬂoorsofplanC.

2 3

3

a. Whatinterpretationcanbegiventothevector x1 4 7 5?

8

b. Writeaformallinearcombination ofvectorsthat expresses the total numbers of three-, two-, and onebedroomapartmentscontainedinthebuilding.

c. [M] Isitpossibletodesignthebuildingwithexactly66

three-bedroomunits, 74two-bedroomunits, and136onebedroomunits? Ifso, istheremorethanonewaytodo

2

Matrix Algebra

INTRODUCTORY EXAMPLE

Computer Models in Aircraft Design

Todesignthenextgenerationofcommercialandmilitary

aircraft, engineersatBoeing’sPhantomWorksuse3D

modelingandcomputationalﬂuiddynamics(CFD).They

importantdesignquestionsbeforephysicalmodelsare

created. Thishasdrasticallyreduceddesigncycletimes

andcost—andlinearalgebraplaysacrucialroleinthe

process.

originalwire-framemodel. Boxesinthisgridlieeither

completelyinsideorcompletelyoutsidetheplane, orthey

intersectthesurfaceoftheplane. Thecomputerselects

theboxesthatintersectthesurfaceandsubdividesthem,

retainingonlythesmallerboxesthatstillintersectthe

surface. Thesubdividingprocessisrepeateduntilthegrid

isextremelyﬁne. A typicalgridcanincludeover400,000

boxes.

Thevirtualairplanebeginsasamathematical“wireframe”modelthatexistsonlyincomputermemoryand

ongraphicsdisplayterminals. (A modelofaBoeing

777isshown.) Thismathematicalmodelorganizesand

inﬂuenceseachstepofthedesignandmanufactureofthe

airplane—boththeexteriorandinterior. TheCFD analysis

concernstheexteriorsurface.

Theprocessforﬁndingtheairﬂowaroundtheplane

involvesrepeatedlysolvingasystemoflinearequations

Ax D b thatmayinvolveupto2millionequationsand

variables. Thevector b changeseachtime, basedondata

fromthegridandsolutionsofpreviousequations. Using

thefastestcomputersavailablecommercially, aPhantom

Worksteamcanspendfromafewhourstoseveraldays

settingupandsolvingasingleairﬂowproblem. Afterthe

teamanalyzesthesolution, theymaymakesmallchanges

totheairplanesurfaceandbeginthewholeprocessagain.

ThousandsofCFD runsmayberequired.

Althoughtheﬁnishedskinofaplanemayseem

smooth, thegeometryofthesurfaceiscomplicated. In

stabilizers, slats, ﬂaps, andailerons. Thewayairﬂows

aroundthesestructuresdetermineshowtheplanemoves

throughthesky. Equationsthatdescribetheairﬂoware

complicated, andtheymustaccountforengineintake,

engineexhaust, andthewakesleftbythewingsofthe

plane. Tostudytheairﬂow, engineersneedahighlyreﬁned

descriptionoftheplane’ssurface.

A computercreatesamodelofthesurfacebyﬁrst

superimposingathree-dimensionalgridof“boxes”onthe

Thischapterpresentstwoimportantconceptsthat

assistinthesolutionofsuchmassivesystemsofequations:

Partitioned matrices: A typical CFD system

ofequationshasa“sparse”coefﬁcientmatrix

withmostlyzeroentries. Groupingthevariables

zeroblocks. Section2.4introducessuchmatrices

anddescribessomeoftheirapplications.

91

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

10 Linear Models in Business, Science, and Engineering

Tải bản đầy đủ ngay(0 tr)

×