ju1j1 p 301111AStep2Letu2v2Proju1v2Proju1v2101111 12121201111A2 301111Au201011A2 301111A01323131Ae2u2 ju2j3 p 601323131A Step3Letu3v3Proju ID: 389486
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Example.WewillapplytheGram-Schmidtalgorithmtoorthonormalizethesetofvectors~v1=0@1�111A;~v2=0@1011A;~v3=0@1121A:ToapplytheGram-Schmidt,werstneedtocheckthatthesetofvectorsarelinearlyindependent.111�101112=1(0�1)�1((�1)(2)�(1)(1))+1((�1)(1)�0)=16=0:Thereforethevectorsarelinearlyindependent.Gram-Schmidtalgorithm:Step1.Let~u1=~v1=0@1�111A~e1=~u1 j~u1j=1 p 30@1�111A:Step2.Let~u2=~v2�Proj~u1~v2Proj~u1~v2=(1;0;1):(1;�1;1) 12+(�1)2+120@1�111A=2 30@1�111A~u2=0@1011A�2 30@1�111A=0@1=32=31=31A~e2=~u2 j~u2j=3 p 60@1=32=31=31A: Step3.Let~u3=~v3�Proj~u1~v3�Proj~u2~v3Proj~u1~v3=(1;1;2):(1;�1;1) 12+(�1)2+110@1�111A=2 30@1�111AProj~u2~v3=(1;1;2):(1=3;2=3;1=3) (1=3)2+(2=3)2+(1=3)20@1=32=31=31A=5 20@1=32=31=31A~u3=0@1121A�2 30@1�111A�5 20@1=32=31=31A=0@�1=201=21A~e3=~u3 j~u3j=p 20@�1=201=21A:Example.Considerthevectorsf[3;0;4];[�1;0;7];[2;9;11]gCheckthatthevectorsarelinearlyindependentandusetheGram-Schmidtprocesstondorthogonalvectors.Ans.f[3;0;4];[�4;0;3];[0;9;0]gCheckthatthevectorsaremutuallyorthog-onal.