extra_problems_for_math_2050_linear_algebra_i

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1、ExtraProblemsforMath2050LinearAlgebraI−−→1.FindthevectorABandillustratewithapictureifA=(−2,1)andB=(1,4).−−→−1−−→−12.FindB,givenA=(1,4)andAB=.A=(1,4)andAB=22−283.Ifpossible,expressx=7asascalarmultipleofu=−28,asascalarmultiple4−16207ofv=0,andasascalarmultipleofw=−1.Justifyyouransw

2、ers.0−47−4360−54.Givenu1=2,u2=−6,u3=−3,u4=0,u5=10,listallpairsof212−30−20parallelvectors.5.Findtheindicatedvectors.21023(a)4+2(b)31−20−4−1−313−52a136.Supposeu=,v=andw=.Findaandbif2u−v=w.−56−b17.Supposex,y,u,andvarevectors.Expressuandvintermsofxandygivenx−y=

3、uand2x+3y=v.−248.Givenu=andv=,findu−vandillustratewithapicturelikethatin−21Figure6.9.Shownattherightaretwononparallelvectorsuw1uandvandfourothervectorsw1,w2,w3,w4.Re-vproduceu,v,andw1inapicturebythemselvesw2andexhibitw1asthediagonalofaparallelogramwithsidesparalleltouandv.Guess(approxi-mate)valu

4、esofaandbsothatw1=au+bv.w4w372−110.Expressasalinearcombinationofand.7321−2311.Isitpossibletoexpressasalinearcombinationofand?Explain.2−23cMMVIIIEdgarG.Goodaire,Dept.ofMathematicsandStatistics,MemorialUniversityofNewfoundland2ExtraProblems14012.Letv1=2,v2=5,andv3=1.3601Is0

5、alinearcombinationofv1,v2,v3?01−213.Isitpossibletoexpressthezerovectorasalinearcombinationofandin4−8morethanoneway?Justifyyouranswer.14.Supposeuandvarevectors.Showthatanylinearcombinationof2uand−3visalinearcombinationofuandv.15.LetO(0,0),A(2,2),B(4,4),C(2,−1),D(4,1),E(6,3),F(2,−4),G(6,0),H(8,2

6、)be−→theindicatedpointsinthexy-plane.Letu=OAbethevectorpicturedbythearrow−−→fromOtoAandv=OCthevectorpicturedbythearrowfromOtoC.Finduandv.−−→ThenexpressthevectorODfrom0toDasacombinationofuandv.Forexample,−−→6OE==2u+v.3A16.SupposeABCisatriangle.LetDbetheDEmidpointofABandEbethemidpointofAC.Usevector

7、stoshowthatDEisparallelBCtoBCandhalfitslength.17.LetA=(1,2),B=(4,−3)andC=(−1,−2)bethreepointsintheEuclideanplane.FindafourthpointDsuchthattheA,B,CandDaretheverticesofaparallelogramandjustifyyouranswer.Isthe