若向量组a1,a2,a3可用向量组β1,β2线性表出,证明向量a1,a2,a3线性相关
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设a1=x1β1+y1β2,a2=x2β1+y2β2,a3=x3β1+y3β2,且x1/y1,x2/y2,x3/y3,互不相等
则对于任意a1a2a3,
有a3=x3·(y2·a1-y1·a2)/(x1·y2-x2·y1)+y3·(x2·a1-x1·a2)/(y1·x1-y2·x2)
=[x3·y1/(x1·y2-x2·y1)+y3·x2/(y1·x1-y2·x2)]·a1-[x3·y1/(x1·y2-x2·y1)+y3·x1/(y1·x1-y2·x2)]·a2
即a3可以用a2与a1表示
即向量a1,a2,a3线性相关
则对于任意a1a2a3,
有a3=x3·(y2·a1-y1·a2)/(x1·y2-x2·y1)+y3·(x2·a1-x1·a2)/(y1·x1-y2·x2)
=[x3·y1/(x1·y2-x2·y1)+y3·x2/(y1·x1-y2·x2)]·a1-[x3·y1/(x1·y2-x2·y1)+y3·x1/(y1·x1-y2·x2)]·a2
即a3可以用a2与a1表示
即向量a1,a2,a3线性相关
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