Question

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.？

Answer 1

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.,2,若有三数l,m,n使
l(a1+2a2)+m(a2+2a3)+n(a3+2a1)=0，则(l+2n)a1+(m+2l)a2+(n+2m)a3=0,
因a1,a2,a3 线性无关，所以l+2n=0, m+2l=0, n+2m=0
l=-2n,m=4n代入n+2m=0,得9n=0,所以n=0,因此又有 l=m=0，
故a1+2a2,a2+2a3,a3+2a1 线性无关。,0,