已知A是n阶方阵,a1,a2,a3为n维列向量,且a1≠0,Aa1=a1,Aa2=a1+a2,Aa3= a2+a3,求证
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已知A是n阶方阵,a1,a2,a3为n维列向量,且a1≠0,Aa1=a1,Aa2=a1+a2,Aa3= a2+a3,求证a1,a2,a3线性无关
a1≠0,{a1}线性无关.
①证明{a1,a2}线性无关:假如{a1,a2}线性相关.a2=ka1.
Aa2=Aka1=kAa1=ka1=a1+a2=(1+k)a1,a1≠0,k=1+k,不可.
∴{a1,a2}线性无关.
②证明{a1,a2,a3}线性无关:假如{a1,a2,a3}线性相关.
则a3=ka1+ha2[无关相关表示定理]
Aa3=a2+a3=A(ka1+ha2)=ka1+ha1+ha2.
a3=(k+h)a1+(h-1)a2=ka1+ha2,∵{a1,a2}线性无关.
∴k+h=k.h-1=h,得到-1=0.不可.∴{a1,a2,a3}线性无关.
①证明{a1,a2}线性无关:假如{a1,a2}线性相关.a2=ka1.
Aa2=Aka1=kAa1=ka1=a1+a2=(1+k)a1,a1≠0,k=1+k,不可.
∴{a1,a2}线性无关.
②证明{a1,a2,a3}线性无关:假如{a1,a2,a3}线性相关.
则a3=ka1+ha2[无关相关表示定理]
Aa3=a2+a3=A(ka1+ha2)=ka1+ha1+ha2.
a3=(k+h)a1+(h-1)a2=ka1+ha2,∵{a1,a2}线性无关.
∴k+h=k.h-1=h,得到-1=0.不可.∴{a1,a2,a3}线性无关.
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