若向量组a1,a2,a3线性无关,证明向量组b=a1+2a2,b2=a2+2a3,b3=a3+2a1线性无关
问答/421℃/2025-01-20 05:52:47
优质解答：
设
k1(a1+2a2)+k2(a2+2a3)+k3(a3+2a1)=0,
即证k1=k2=k3=0
(k1+2k3)a1+(2k1+k2)a2+(2k2+k3)a3=0
因为向量组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b=a1+2a2,b2=a2+2a3,b3=a3+2a1线性无关