Question

设向量组 a1 a2 a3 a4 a5 线性无关,证,a1+a2,a2+a3,a3+a4,a4+a5也线性无关.

Answer 1

k1(a1+a2)+k2(a2+a3)+k3(a3+a4)+k4(a4+a5)=0

k1a1+(k1+k2)a2+(k2+k3)a3+(k3+k4)a4+k4a5=0

=>

k1=0 (1) and

k1+k2=0 (2) and

k2+k3=0 (3) and

k3+k4=0 (4) and

k4 =0

from (1) (2)

k2=0

from (3)

k3=0

ie k1=k2=k3=k4=0

=> a1+a2,a2+a3,a3+a4,a4+a5也线性无关