设n维向量组a1,a2,a3线性无关,证明向量组a1+2a2, a2+2a3, a3+2a4线性无关.求详细的解题过程

设存在一组数,k1,k2,k3使得
k1(a1+2a2)+k2( a2+2a3)+k3( a3+2a1)=0
整理得：(k1+2k3)a1+(2k1+k2)a2+(2k2+k3)a3=0
因为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