3.4.1TheDefinitionofVectorSpaceLinearAlgebraDefinition(VectorSpace)AVectorsetiscalledaVectorspaceifithas𝑉thefollowingthreeproperties.Vectorsetisnotempty𝑉(closedunderVectoraddition)Ifandbelongto,belongstoV𝑉(closedunderVectoraddition)Ifbelongstoandisascalar,belongstoV𝑉2)ThedefinitionalsospecifiesstepstoverifywhetheraVectorsetisaVectorspace:①Visnotempty;②VisclosedunderVectoraddition;③...
LinearAlgebra3.2.2PropertiesofLinearCorrelationofVectorGroupsTheorem1VectorgroupA:islinearlydependentThereisatleastoneoftheVectorsisalinearcombinationofother−1Vectors.𝑚Proof(Sufficiency)GivenaVectorin(suchas)thatcanbelinearlyrepresentedbytheremainingVectors,so𝛼𝑚=𝜆1𝛼1+𝜆2𝛼2+⋯+𝜆𝑚−1𝛼𝑚−1⇒𝜆1𝛼1+𝜆2𝛼2+⋯+𝜆𝑚−1𝛼𝑚−1+(−1)𝛼𝑚=𝑂Obviouslyarenotallzero,soislinearlydependent.Thatis...
