k1b1+k2b2+k3b3=0
k1(a1+a2)+k2(2a1+2a2+a3)+k3(a1-a2+2a3)=0
(k1+2k2+k3)a1+(k1+2k2-k3)a2+(k2+2k3)a3=0
=>
k1+2k2+k3=0 (1) and
k1+2k2-k3=0 (2) and
k2+2k3=0 (3)
(1)-(2) =>k3=0 (4)
from (4) and (3) => k2=0 (5)
from (1) and (4) and (5) => k1=0
=>向量组b1=a1+a2 b2=2a1+2a2+a3 b3=a1-a2+2a3也线性无关