设向量a1,a2,.as(s>=2)线性无关,且β1=a1+a2,β2=a2+a3,...βs-1=a(s-1)+as,bs=as_数学解答

设向量a1,a2,.as(s>=2)线性无关,且β1=a1+a2,β2=a2+a3,...βs-1=a(s-1)+as,bs=as+a1,讨论向量组β1,β2,.βs的线性相关性,

人气：151 ℃　时间：2020-03-18 16:44:40

解答

证明1:设 k1β1+k2β2+...+k(s-1)β(s-1)+ksβs = 0整理得:(k1+ks)a1 + (k1+k2)a2 + ...+ (k(s-1)+ks)as = 0由 a1,a2,a3,...,as线性无关,得k1+ks = 0k1+k2 = 0k2+k3 = 0...k(s-1)+ks = 0由 k1+k2 = 0 得 k1 = -k2由 ...