Math230 Pe Feb2010 Final (2)

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School of AdvancedTechnologies

and Mathematics

Practice Exam

MATH 230Linear Algebra

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MATH 230: Linear Algebra 109

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Practice Exam

Theformatofthispracticeexamissimilartothatofyourfinalexam.Foryourown

benefit,youshouldnotlookatthisexamuntilyouhavecompletedyourreviewofthe

course

and

have

submitted

and

received

feedback

on

all

assignments.

When

you

writethisexam,youshouldisolateyourselfforthreehours.Bysimulatingexam

conditionsinthisway,youwilllearnhowtomanageyourtimeonthefinal.Also

rememberthatthefinalexamisclosedbook.

ThePracticeExamAnswerKeyfoundonthepagesimmediatelyfollowingthis

examshouldprovidesufficientinformationtoindicatewhichconceptsyoumay

haveforgottenorneverreallylearned.Whenyouhavecompletedyourreviewofthe

exam,contactyourtutortodiscussanypointsofconcern.

Thenextpagescontaintheselfadministeredfinalexamandtheguideformarking

it.

DONOTturnthepageuntilyouarereadytotrytheexam!

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THOMPSON RIVERS UNIVERSITY,OPEN LEARNING

MATH 230 LINEAR ALGEBRA

PRACTICE EXAMINATION

Time Allowed: 3 Hours

Total Pages (Including This Page): 5

Total Marks: 100

Materials/Equipment Permitted:

ScientificCalculator

Materials Provided:

Twoexam

answer

booklets

Student: Please Complete This SectionPrint

Clearly.

Surname

_________________________________________

FirstName

_________________________________________

StudentNumber(ninedigits)

_________________________________________

TutorsName

_________________________________________

StudentsSignature(required)

_________________________________________

Date

_________________________________________

Tutor: Please Complete This Section

Print Clearly.

StudentsMark________%

TutorsName

_________________________________

TutorsI.D.No.

_________________________________

TutorsSignature

_________________________________

Date

_________________________________

Instructions

Placeyourphotoidentification(e.g.,driverslicence),examinationconfirmationletter,

ontheupperrighthandcornerofyourdeskforinspectionbytheinvigilator.

Writeyouranswersandcompletesolutionsintheexamanswerbookletsprovided.

Whenyouhavefinished,RETURNALLPAPERS,INCLUDINGTHEEXAMANDALLANSWERBOOKLETS(USEDANDUNUSED)INTHEENVELOPE.FailuretodosomayresultinaFAILgrade.

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112 Practice Exam

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ExamInstructions AttemptALLtenquestions.Eachquestionisworth10marks. Givecompleteanddetailedsolutionstoallproblems.Marksmaybededucted

forinsufficient

details.

Identifyallrow(orcolumn)operationsusedonmatrices.

Unitsmustbegiveninthefinalanswerofanapplication,whereapplicable.

1. UsetheGaussJordanEliminationMethodtofindthecurrents 1I , 2I and 3I forthe

electricalnetwork

shown

below,

where

1 2 1 2 33 volts, 4 volts, 4 , 3 , 1V V R ohms R ohms R ohm= = = = =

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2. a. IfAisanidempotentmatrix(thatis, )2 ,A A=

thenprovethat 0)Adet( = or

det( ) 1.A =

b.

Determine

if

the

matrix

B

below

is

diagonalizable.

1 0 0

0 1 1

0 1 1

B

=

3. Considerthefollowingsystemoflinearequations:

( )2

2 5

3 2 20

4 3 24 20

x y z

x y z

x y k k

=

+ =

+ = +

Forwhatvalue(s)ofkdoesthissystemhave:

a. Aninfinitenumberofsolutions?

b. Auniquesolution?

c. Nosolution?

4. IfAisaninvertiblematrix,provethat:

a. ( ) ( )1

1T

TA A

=

b. If AC AB, then C B.= =

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114 Practice Exam

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5. Giventhematrix

=

24

910A

a. Showthat

=4

6v

isaneigenvectorforAandfindthe

correspondingeigenvalue.

b.Calculate 3A u , where 12 .8

=

u

6. Let ( ) ( ){ }0, 2 , 2 , 1, 0, 2B = beabasisforasubspaceof 3R ,andlet( )2,4,1 =x beavectorinthesubspace.

a. Findthecoordinatesofx relativetoB.b. UsetheGramSchmidtOrthonormalizationprocessto

transformBintoanorthonormalset /B .7. Determinewhetherthefunction: ( ) ( )21212122 ,2,,: xxxxxxTRRT += is

alineartransformation.Ifitis,finditsstandardmatrixA.

8. ThelineartransformationTisgivenby ( ) vv AT = ,where1 2

1 0 .

1 1

A

=

Find:

a. AbasisforthekernelofTb. AbasisfortherangeofTc. TherankofTd. ThenullityofT

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9. Determinewhether { }20 1 2 0 1 2: 0S a a x a x a a a= + + + + = isasubspaceofP2.

10.Suppose , 2 31 1 2 2 3 3

u v u v u v u v= + + representsaninnerproducton 3R .For

( ) ( )2, 1,1 and 2,4,0u v= =

a. Findtheinnerproductofuandv

b. Findthe

distance

between

uand

v

c. VerifyTheTriangleInequalityforuandv: u v u v

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Math230 Pe Feb2010 Final (2)