证明:因为a1,a2,a3线性无关,所以 r(a1,a2,a3)=3
由已知 (3a1+2a2,a2-a3,4a3-5a1)=(a1,a2,a3)K
K=
3 0 -5
2 1 0
0 -1 4
= 22 ≠0
所以K是可逆矩阵
所以 r(3a1+2a2,a2-a3,4a3-5a1)=r((a1,a2,a3)K)=r(a1,a2,a3)=3
所以 3a1+2a2,a2-a3,4a3-5a1线性无关.