Question

如图 5.6题 在线等

Answer 1

(5)
α1,α2,α3,β 线性无关
β1=α1+β
β2=α2+2β
β3=α3+3β
k1.β1+k2.β2+k3.β3+k4.β=0
k1(α1+β) + k2(α2+2β)+k3.(α3+3β)+k4.β=0
k1.α1+k2.α2+k3.α3 + (k1+2k2+3k3+k4)β=0
=>
k1=0
k2=0
k3=0
k1+2k2+3k3+k4=0
=> k1=k2=k3=k4=0
=> β1,β2,β3,β 线性无关
(6)
α1,α2,α3,...αs 线性无关
β1=α1
β2=(α1+α2)
...
βs=(α1+α2+...+αs)
k1.β1+k2.β2+...+ks.βs =0
k1.α1+k2.(α1+α2)+...+ks.(α1+α2+...+αs) =0
(k1+k2+..+ks)α1+(k2+k3+..+ks).α2+...+ks.αs =0
=>
k1+k2+...+ks =0 (1)
k2+k3+...+ks=0 (2)
...
...
ks = 0 (s)
=> k1=k2=...=ks=0
=> β1,β2,β3,...,βs 线性无关