设向量组a1,a2,a3线性无关.证明向量组a1+a3,a1-2a3,a2+a3也与线性无关
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设向量组a1,a2,a3线性无关.证明向量组a1+a3,a1-2a3,a2+a3也与线性无关
![设向量组a1,a2,a3线性无关.证明向量组a1+a3,a1-2a3,a2+a3也与线性无关](/uploads/image/z/4171868-44-8.jpg?t=%E8%AE%BE%E5%90%91%E9%87%8F%E7%BB%84a1%2Ca2%2Ca3%E7%BA%BF%E6%80%A7%E6%97%A0%E5%85%B3.%E8%AF%81%E6%98%8E%E5%90%91%E9%87%8F%E7%BB%84a1%2Ba3%2Ca1-2a3%2Ca2%2Ba3%E4%B9%9F%E4%B8%8E%E7%BA%BF%E6%80%A7%E6%97%A0%E5%85%B3)
k1(a1+a3)+k2(a1-2a3)+k3(a2+a3)=0
=> (k1+k2)a1+k3a2+(k1-2k2+k3)a3=0
=> k1+k2=0 (1) and
k3=0 (2) and
k1-2k2+k3=0 (3)
from (3) and (2)
k1-2k2 = 0 (4)
(1)-(4)
3k2=0
=> k2 =0
from (1)
=> k3=0
=> a1+a3,a1-2a3,a2+a3线性无关
=> (k1+k2)a1+k3a2+(k1-2k2+k3)a3=0
=> k1+k2=0 (1) and
k3=0 (2) and
k1-2k2+k3=0 (3)
from (3) and (2)
k1-2k2 = 0 (4)
(1)-(4)
3k2=0
=> k2 =0
from (1)
=> k3=0
=> a1+a3,a1-2a3,a2+a3线性无关
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