Linear-Algebra-彭国华-第7章课后答案

AnswerstoSomeExercisesinChapter7GuohuaPengemail:[e-mail]4onpage341Letf1;f2;¢¢¢;fmbenon-zerovectorsinV¤.Showthatthereexistsavector®2Vsuchthatfi(®)6=0;fori=1;2;¢¢¢;m:SolutionToshowthatthereexistsavector®2Vsuchthatfi(®)6=0fori=1;2;¢¢¢;misequivalenttoshowthatm[i=1kerfi6=V:Notethatfi6=0implieskerfi6=V.Itsu±cestoprovethatm[i=1Vi6=V;ifViVfori=1;2;¢¢¢;m.Wemakeinductiononm.Assumem=2.IfV1µV2orV2µV1,thereisnothingtoprove.IfV1V2andV2V1,thenthereexist®2V1nV2;¯2V2nV1.Consequently®+¯62V1[V2.Assumetheresultistrueform¡1subspaces.Thentherewillbe®2Vsuchthat®62[m¡1i=1Vi.Thatis®62Viforall1·i·m¡1.If®62Vm,thentheresultfollows,since®62[mi=1Vi.Soweassume®2Vm.NotethatVmV.Therewillbesome¯62Vm.Then,foreach1·i·m,thereareatmostonek2Fsuchthat®+k¯2Vi.(Otherwise,if®+k1¯;®+k2¯2Viwithk16=k2,then(k1¡k2)¯2Vi,yielding¯2Vi.Thisisimpossiblefori=m.If1·i·m¡1,¯2Viimpliesthat®2Vi,whichisalsoimpossible.)Thuswehavec2Fsuchthat®+c¯62Viforall1·i·m.Itfollowsthat®+c¯62m[i=1Vi:1