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.

证明:因为 (a1+2a2,a2+2a3,a3+2a1)=(a1,a2,a3)K
其中K=
1 0 2
2 1 0
0 2 1
因为a1,a2,a3线性无关,所以r(a1+2a2,a2+2a3,a3+2a1)=r(K).
因为 |K|= 9
所以 r(a1+2a2,a2+2a3,a3+2a1)=r(K)=3
所以 a1+2a2,a2+2a3,a3+2a1 线性无关.