设存在K1,K2,K3使K1（a1+2a2）+K2（a2+2a3）+K3(a3+2a1）=0
整理得(K1+2K3)a1+(2k1+k2)a2+(K3+2k2)a3=0
因为a1,a2,a3线性无关
所以
(K1+2K3)=0
(2k1+k2)=0
(K3+2k2)=0解得：K1=K2=K3=0
所以
a1+2a2,a2+2a3,a3+2a1线性无关.