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

Question

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

Answer 1

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.