已知向量组a1,a2,a3线性无关,b1=2a1+a2,b2=3a2+a3,b3= a1+4a3,试证向量组b1,b2,b3线性无关.
k1b1+k2b2+k3b3=0
k1(2a1+a2)+k2(3a2+a3)+k3(a1+4a3)=0
(2k1+k3)a1+(k1+3k2)a2+(k2+4k3)a3=0
=>
2k1+k3 =0 (1) and
k1+3k2=0 (2) and
k2+4k3 =0 (3)
2(2) -(1)
6k2-k3=0 (4)
6(3)-(4)
k3=0
k2=k1=0
=> b1,b2,b3 linearly independent
Ans: A