Question

证明题:设向量组a1,a2,a3,线性无关,证明向量组a1+2a2,a2+2a3,a3+2a1线性无关？

Answer 1

设k1,k2,k3使得
k1(a1+2a2)+k2( a2+2a3)+k3(a3+2a1)=0
(k1+2k3)a1+(2k1+k2)a2+(2k2+k3)a3=0
a1,a2,a3线性无关
所以 k1+ 2k3=0
2k1+k2=0
2k2+k3=0
解得：k1=k2=k3=0
所以向量组a1+2a2,a2+2a3,a3+2a1线性无关,5,若a1+2a2,a2+2a3,a3+2a1线性相关，则存在非零数组(m,n,l)使得
m(a1+2a2)+n(a2+2a3)+l(a3+2a1)=0
==> (m+2l)a1+(n+2m)a2+(l+2n)a3=0
因为a1，a2，a3线性无关，则有
m+2l=0, n+2m=0, l+2n=0
解的m=n=l=0,矛盾。
故a1+2a2,a2+2a...,1,