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1、CHAPTER7EuclideanStructureInthischapterweabstracttheconceptofEuclideandistance.Wegainnogreatergenerality;wegainsimplicity,transparencyandflexibility.WereviewthebasicstructureofEuclideanspaces.Wechooseapoint0asorigininrealn-dimensionalEuclideanspace;thelengthofa
2、nyvectorxinspace,denotedas11x11,isdefinedasitsdistancetotheorigin.LetusintroduceaCartesiancoordinatesystemanddenotetheCartesiancoordinatesofxasxi,...,x,,.ByrepeateduseofthePythagoreantheoremwecanexpressthelengthofxintermsofitsCartesiancoordinates.xx,++x.(1)Thes
3、calarpmductoftwovectorsxandy,denotedas(x,),),isdefinedby(x,y)=>xiyi(2)Clearly,thetwoconceptsarerelated;wecanexpressthelengthofavectorasIIx112=(X,X).(2)'Thescalarproductiscommutative:(x,Y)=(Y'X)(3)andbilinear:(x+U,Y)_(x,y)+(u,y),(3)'(x,y+v)_(x,y)+(x,v).LinearAlg
4、ebraandItsApplications.SecondEdition,byPeterD.LaxCopyright``.i2007JohnWiley&Sons,Inc.7778LINEARALGEBRAANDITSAPPLICATIONSUsingthesealgebraicpropertiesofscalarproductwecanderivetheidentity(x-y,x-y)=(x,x)-2(x,y)+()',Y)Using(2)',wecanrewritethisidentityasIIx-YII2=I
5、Ix1I2-2(x,y)+IIYII?(4)Thetermontheleftisthedistanceofxfromy,squared;thefirstandthirdtermsontherightarethedistancesofxandyfrom0,squared.Thesethreequantitieshavegeometricmeaning;thereforetheyhavethesamevalueinanyCartesiancoordinatesystem.Iffollowsthereforefrom(4)
6、thatalsothescalarproduct(2)hasthesamevalueinallCartesiancoordinatesystems.Bychoosingspecialcoordinateaxes,thefirstonethroughx,thesecondsothatyiscontainedintheplanespannedbythefirsttwoaxes,wecanuncoverthegeometricmeaningof(x,y).IxIxThecoordinatesofthevectorxandy
7、inthiscoordinatesystemarex=(IIx11,0...0)andy=(II)'IIcos0...).Therefore(x,Y)=IIx1IIIYIIcose,(5)0theanglebetweenxandy.Thethreepoints0,x,yformatrianglewhosesidesarea=IIxII,b=IIyII,c=IIx-y11,forminganangleOat0:Relations(4)and(5)canbewrittenasc'=a2+b2-2abcos0.(4)'EU
8、CLIDEANSTRUCTURE79Thisistheclassicallawofcosine;aspecialcaseofit,0=n/2,isthePythagoreantheorem.Mosttextsderiveformula(5)forthescalarproductfromthelawofcosine.Thisisapedagogi
