(1.2.7)--英3.3.1 Definition of The Largest线性代数线性代数.ppt

LinearAlgebra3.3.1DefinitionofTheLargestIndependentGroupandRankofVectorGroups

R:255G:255B:0R:255G:0B:255R:0G:255B:2552550RedGreenBlue002552550002550Thepracticalmeaningofn-dimensionalvectors

? Visiblecolorsinnaturecanbeobtainedbymixingprimarycolorswithcertainproportions,thatis,anycolorisregardedasalinearcombinationofred,greenandbluecolorvectors.?

ColorvectorgroupI:red,green,blue,yellow,purpleQuestion1:

CanyoufindaminimalcombinationI0ofcolorvectors

torepresentcolorvectorgroupI?

ColorvectorgroupI:red,green,blue,yellow,purpleYellowvectorcanberepresentedbyred,greenandbluevectorsColorvectorgroupI1:red,green,blue,purpleRemovepurplevectorColorvectorgroupI0:red,green,blueRemoveyellowvectorPurplevectorcanberepresentedbyred,greenandbluevectorsColorvectorgroupI:red,green,blue,yellow,purplecanberepresentedbythepartialgroupI0ofI:red,green,blueRed,greenandbluevectorscannotberepresentedlinearlywitheachother

ColorvectorgroupI：?1,?2,?3,?4,?5belinearlydependentQuestion2:

CanyoufindasubsetassmallaspossibletorepresentthevectorgroupI?

?Forexample,?4canberepresentedbytheremainingvectors?Remove?1?Remove?4Forexample,?1canberepresentedbytheremainingvectors??

??Definition??LargestindependentgroupPartialgroupwiththemostnumberoflinearindependentvectors.mostAddinganyothervectortothelargestindependentgroupbecomeslinearlydependent.

Forexample,in11,?3=,10?1=01,?2=?1,?2arelinearlyindependent,?3canbelinearlyrepresentedby?1,?2.Itcanbeseenthat?1,?2isalargestindependentgroupof?1,?2,?3.Therankofthevectorgroupis2.?1,?2islinearlyindependent,?1,?2isalargestindependentgroupof?1,?2,?3.?3canberepresentedlinearlyby?1,?2，?1,?2,?3islinearlydependent.(i)VectorgroupI0islinearlyindependent;

(ii)