已知向量a1,a2,a3线性无关.试确定数X1,X2,X3使得a1+X2*a2,a2+X3*a3,a3+X1*a1也线性无关
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已知向量a1,a2,a3线性无关.试确定数X1,X2,X3使得a1+X2*a2,a2+X3*a3,a3+X1*a1也线性无关
已知向量a1,a2,a3线性无关.试确定数X1,X2,X3使得a1+X2*a2,a2+X3*a3,a3+X1*a1也线性无关
已知向量a1,a2,a3线性无关.试确定数X1,X2,X3使得a1+X2*a2,a2+X3*a3,a3+X1*a1也线性无关
(a1+x2a2,a2+x3a3,a3+x1a1)=(a1,a2,a3)K.
其中 K =
1 0 x1
x2 1 0
0 x3 1
由已知a1,a2,a3线性无关,
所以a1+x2a2,a2+x3a3,a3+x1a1线性无关的充分必要条件是 |K|≠0.
而 |K|=1+x1x2x3
所以 x1x2x3 ≠ -1.
k1(a1+x2a2)+k2(a2+x3a3)+k3(a3+x1a1)=0
(k1+k3x1)a1+(k1x2+k2)a2+(k2x3+k3)=0
=>
k1+k3x1=0 (1) and
k1x2+k2=0 (2) and
k2x3+k3=0 (3)
from (1)
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k1(a1+x2a2)+k2(a2+x3a3)+k3(a3+x1a1)=0
(k1+k3x1)a1+(k1x2+k2)a2+(k2x3+k3)=0
=>
k1+k3x1=0 (1) and
k1x2+k2=0 (2) and
k2x3+k3=0 (3)
from (1)
k1+k3x1=0
k1=k3=0 , => x1 can be any number
from (2)
k1x2+k2=0
k1=k2=0, =>x2 can be any number
from (3)
k2x3+k3=0
k2=k3=0 ,=> x3 can be any number
x1,x2,x3 can be any number
a1+x2a2,a2+x3a3,a3+x1a1也线性无关
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