3rd edition Linear Algebra and its Applications

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1、1.1SOLUTIONSNotes:Thekeyexercisesare7(or11or12),19–22,and25.Forbrevity,thesymbolsR1,R2,…,standforrow1(orequation1),row2(orequation2),andsoon.Additionalnotesareattheendofthesection.xx12+=571571.−−=275xx12−−−−275xx12+=57157ReplaceR2byR2+(2)R1ando

2、btain:39x2=039xx12+=57157ScaleR2by1/3:x2=3013x1=−8108−ReplaceR1byR1+(–5)R2:x2=3013Thesolutionis(x1,x2)=(–8,3),orsimply(–8,3).244xx12+=−244−2.571xx12+=15711xx12+=22−122−ScaleR1by1/2andobtain:571xx12+=15711xx12+=22−122−Repla

3、ceR2byR2+(–5)R1:−=32x21032−1xx12+=22−122−ScaleR2by–1/3:x2=−7017−x1=121012ReplaceR1byR1+(–2)R2:x2=−7017−Thesolutionis(x1,x2)=(12,–7),orsimply(12,–7).12CHAPTER1•LinearEquationsinLinearAlgebra3.Thepointofintersectionsatisfiesthesystemoftwo

4、linearequations:xx12+=57157xx12−=22−122−−xx12+=57157ReplaceR2byR2+(–1)R1andobtain:−=79x2−079−−xx12+=57157ScaleR2by–1/7:x2=9/7019/7x1=4/7104/7ReplaceR1byR1+(–5)R2:x2=9/7019/7Thepointofintersectionis(x1,x2)=(4/7,9/7).4.Thepointofi

5、ntersectionsatisfiesthesystemoftwolinearequations:xx12−=51151−375xx12−=375−xx12−=51151−ReplaceR2byR2+(–3)R1andobtain:82x2=082xx12−=51151−ScaleR2by1/8:x2=1/4011/4x1=9/4109/4ReplaceR1byR1+(5)R2:x2=1/4011/4Thepointofintersectionis(

6、x1,x2)=(9/4,1/4).5.Thesystemisalreadyin“triangular”form.Thefourthequationisx4=–5,andtheotherequationsdonotcontainthevariablex4.Thenexttwostepsshouldbetousethevariablex3inthethirdequationtoeliminatethatvariablefromthefirsttwoequations.Inmatrixnotation,t

7、hatmeanstoreplaceR2byitssumwith3timesR3,andthenreplaceR1byitssumwith–5timesR3.6.Onemorestepwillputthesystemintriangularform.ReplaceR4byitssumwith–3timesR3,which16401−−02704−produces.Afterthat,thenextstepistoscalethefourthrowby–1/5.00123−000

8、51−57.Ordinarily,thenextstepwouldbetointerchangeR3andR4,toputa1inthethirdrowandthirdcolumn.Butinthiscase,thethirdrowoftheaugmentedmatrixcorrespondstotheequation0x1+0x2+0x3=1,orsimply,0=1.Asystemcontainingthisconditionhasnosolution.Furth