(1.2.4)--英3.2.1 Definition of Linear Depe线性代数线性代数.ppt

LinearAlgebra3.2.1DefinitionofLinearDependentofVectorGroups

???Geometry:?Thereisauniquerealscalark,suchthat?=k??

Geometry:?????????

?Otherwise,itiscalledlinearlyindependent.?

Proof:Method1(Observation)：??1+2?2+?3=?ExampleProvethat?1=arelinearlydependent.222,?2=,?3=101020Therearescalars?1、2、1thatarenotallzero,suchthat??1+2?2+?3=?So?1,?2,?3arelinearlydependent.

?????=222101020000ExampleProvethat?1=arelinearlydependent.222,?2=,?3=101020

Method2:???=222101020000??ExampleProvethat?1=arelinearlydependent.222,?2=,?3=101020So?1,?2,?3arelinearlydependent.

????????Itcanbeconcludedthatthecorrelationbetweenvectorscanberepresentedbymeansofmatrixorhomogeneouslinearequations.????????

ExampleProve??-dimensionalbasicunitvectorgroup?100…,?010…,?001……,arelinearlyindependent.Solution:??=?…=000…?100…?010…001…?

??ExampleProve??-dimensionalbasicunitvectorgroup?100…,?010…,?001……,arelinearlyindependent.Solution:??

???????????Solution:

????Solution:

Note⑵Zerovectorislinearlydependenttoanyvectors.??Note⑴NOTEs??

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