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10 Linear Models in Business, Science, and Engineering

10 Linear Models in Business, Science, and Engineering

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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

wouldthenaddsimilarproductsforsoyflourandwheyandsettheresultingsumequal

totheamountofproteinweneed. Analogouscalculationswouldhavetobemadefor

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.

Thedietconstructionproblemleadstothe linear equation(2)becausetheamount

ofnutrientssuppliedbyeachfoodstuffcanbewrittenasascalarmultipleofavector, as

in(1). Thatis, thenutrientssuppliedbyafoodstuffare proportional totheamountof



82



CHAPTER 1



Linear Equations in Linear Algebra



thefoodstuffaddedtothedietmixture. Also, eachnutrientinthemixtureisthe sum of

theamountsfromthevariousfoodstuffs.

Problemsofformulatingspecializeddietsforhumansandlivestockoccurfrequently. Usuallytheyaretreatedbylinearprogrammingtechniques. Ourmethodof

constructingvectorequationsoftensimplifiesthetaskofformulatingsuchproblems.



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

mathematicallyadynamicsystemthatchangesovertime. Severalfeaturesofthesystem

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

adifferenceequationmightarise.

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.

Supposedemographicstudiesshowthateachyearabout5%ofthecity’spopulation

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

isfoundby adding themovementofpeoplefromthedifferentareas.



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

additiontothenutrientslistedintheTable1forExample

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.

b. [M] Solvetheequationin(a)anddiscussyouranswer.

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

herfavoritelunchbyaddingbroccoliandcannedchicken.

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

about500cars, atthreelocations. A carrentedatonelocation

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



9. Inacertainregion, about7%ofacity’spopulationmoves

tothesurroundingsuburbseachyear, andabout5%ofthe

suburbanpopulationmovesintothecity. In2010, therewere

800,000residentsinthecityand500,000inthesuburbs.

Setupadifferenceequationthatdescribesthissituation,

where x0 istheinitialpopulationin2010. Thenestimate

the populations in the city and in the suburbs two years

later, in2012. (Ignoreotherfactorsthatmightinfluencethe

populationsizes.)

10. Inacertainregion, about6%ofacity’spopulationmoves

tothesurroundingsuburbseachyear, andabout4%ofthe

suburbanpopulationmovesintothecity. In2010, therewere

10,000,000residentsinthecityand800,000inthesuburbs.

Setupadifferenceequationthatdescribesthissituation,

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.

c. Finally, makeageneralconjectureaboutthecorrespondencefromthelistofeightboundarytemperaturestothe

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

1. MarkeachstatementTrueorFalse. Justifyeachanswer. (If

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

answers. [Hint: Useatheoremfor(b).]

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

AD6

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

it? Explainyouranswer.



2



Matrix Algebra



INTRODUCTORY EXAMPLE



Computer Models in Aircraft Design

Todesignthenextgenerationofcommercialandmilitary

aircraft, engineersatBoeing’sPhantomWorksuse3D

modelingandcomputationalfluiddynamics(CFD).They

studytheairflowaroundavirtualairplanetoanswer

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

additiontowingsandafuselage, anaircrafthasnacelles,

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

correctlyleadstoapartitionedmatrixwithmany

zeroblocks. Section2.4introducessuchmatrices

anddescribessomeoftheirapplications.

91



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