线性代数的题:已知向量a1,a2,a3线性无关,证明a1+2a2,a2+2a3,a3+2a1线性无关.
问题描述:
线性代数的题:已知向量a1,a2,a3线性无关,证明a1+2a2,a2+2a3,a3+2a1线性无关.
答
设存在K1,K2,K3使K1(a1+2a2)+K2(a2+2a3)+K3(a3+2a1)=0
整理得(K1+2K3)a1+(2k1+k2)a2+(K3+2k2)a3=0
因为a1,a2,a3线性无关
所以
(K1+2K3)=0
(2k1+k2)=0
(K3+2k2)=0解得:K1=K2=K3=0
所以
a1+2a2,a2+2a3,a3+2a1线性无关.