k1(a1+a3)+k2(a1-2a3)+k3(a2+a3)=0
=> (k1+k2)a1+k3a2+(k1-2k2+k3)a3=0
=> k1+k2=0 (1) and
k3=0 (2) and
k1-2k2+k3=0 (3)
from (3) and (2)
k1-2k2 = 0 (4)
(1)-(4)
3k2=0
=> k2 =0
from (1)
=> k3=0
=> a1+a3,a1-2a3,a2+a3线性无关