已知向量组a1,a2,a3线性无关,证明向量组a1+2a2,2a2+3a3,3a3+a1线性无关.

证明：设：k1(a1+2a2)+k2(2a2+3a3)+k3(3a3+a1)=0整理得：(k1+k3)a1+(2k1+2k2)a2+(3k2+3k3)a3=0∵a1,a2,a3线性无关∴k1+k3=02k1+2k2=03k2+3k3=0解得：k1=k2=k3=0故：向量组a1+2a2,2a2+3a3,3a3+a1线性无关...