已知向量组a1,a2,a3线性无关,证明向量组a1+a2,3a2+2a3,a1-2a2+a3线性...答:重新分组:a1(k1+k3) + a2(k1+3k2-2k3) + a3(2k2+k3)=0 因为a1,a2,a3线性无关,所以有方程组:k1+k3=0; k1+3k2-2k3=0; 2k2+k3=0 .行列式:1 0 1 1 3 -2 0 2 1 不等于0,所以方程只有零解,即k1,k2,k3都等于0,所以向量组a1+a2,3a2+2a3,a1-2a2+a3线性无关.,...
证明题:设向量组a1,a2,a3,线性无关,证明向量组a1+2a2,a2+2a3,a3+2a1...答:设k1,k2,k3使得 k1(a1+2a2)+k2( a2+2a3)+k3(a3+2a1)=0 (k1+2k3)a1+(2k1+k2)a2+(2k2+k3)a3=0 a1,a2,a3线性无关 所以 k1+ 2k3=0 2k1+k2=0 2k2+k3=0 解得:k1=k2=k3=0 所以向量组a1+2a2,a2+2a3,a3+2a1线性无关,5,若a1+2a2,a2+2a3,a3+2a1线性相关,则...
设向量组a1,a2,a3 线性无关,证明向量组a1+2a2,a2+2a3,a3+2a1 线性无...答:其中K= 1 0 2 2 1 0 0 2 1 因为a1,a2,a3线性无关,所以r(a1+2a2,a2+2a3,a3+2a1)=r(K).因为 |K|= 9 所以 r(a1+2a2,a2+2a3,a3+2a1)=r(K)=3 所以 a1+2a2,a2+2a3,a3+2a1 线性无关.,2,若有三数l,m,n使 l(a1+2a2)+m(a2+2a3)+n(a3+2a1)=0,则(l+2n)...
已知向量组a1,a2,a3线性无关,若向量组a1+a2,a2+a3,λa1+a3线性无关,则...答:(a1+a2,a2+a3,λa1+a3) = (a1,a2,a3)K K= 1 0 λ 1 1 0 0 1 1 |K|=1+λ 由已知 r(K) = r(a1+a2,a2+a3,λa1+a3) = 3 所以 λ≠ -1.
设向量组a1,a2,a3,线性无关.证明:向量组a1+a2+a3,a2+a3,a3也线性无关...答:假设a1+a2+a3,a2+a3,a3线性相关,则k1(a1+a2+a3)+k2(a2+a3)+k3a3=0其中k1、k2、k3不全为0.化简成k1a1+(k1+k2)a2+(k1+k2+k3)a3=0 由于向量组a1,a2,a3,线性无关.所以k1=0、k1+k2=0、k1+k2+k3=0 则k1=0、k2=0、k3=0 与上述k1、k2、k3不全为0矛盾.所以向量组...
向量组a1,a2,a3线性无关,β=k1a1+k2a2+k3a3,证明若k1不等于0,β,a2,a...答:(β,a2,a3) = (a1,a2,a3)K K= k1 0 0 k2 1 0 k3 0 1 因为 a1,a2,a3 线性无关 所以 r(β,a2,a3) = r(K)所以 β,a2,a3 线性无关 <=> r(K)=3 <=> |K|≠0 <=> k1≠0.