线性代数线性无关证明题?Let v1,v2 and v3 be vectors in a vector space V .Decide whether or notw1 = v1 + v2,w2 = v2 + v3 and w3 = v1 − v3 are linearly independent,givingreasons for your answer.怎么证明?

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线性代数线性无关证明题?Letv1,v2andv3bevectorsinavectorspaceV.Decidewhetherornotw1=v1+v2,w2=v2+v3andw3=v1−

线性代数线性无关证明题?Let v1,v2 and v3 be vectors in a vector space V .Decide whether or notw1 = v1 + v2,w2 = v2 + v3 and w3 = v1 − v3 are linearly independent,givingreasons for your answer.怎么证明?
线性代数线性无关证明题?
Let v1,v2 and v3 be vectors in a vector space V .Decide whether or not
w1 = v1 + v2,w2 = v2 + v3 and w3 = v1 − v3 are linearly independent,giving
reasons for your answer.怎么证明?

线性代数线性无关证明题?Let v1,v2 and v3 be vectors in a vector space V .Decide whether or notw1 = v1 + v2,w2 = v2 + v3 and w3 = v1 − v3 are linearly independent,givingreasons for your answer.怎么证明?
w1,w2,w3线性相关,因为w1-w2-w3=0,即存在不全为0的实数k1,k2,k3使得k1w1+k2w2+k3w3=0
所以w1,w2,w3线性相关

一般方法是这样的:
设 k1w1+k2w2+k3w3 = 0
则 k1(v1+v2)+k2(v2+v3)+k3(v1-v3) = 0.
得 (k1+k3)v1+(k1+k2)v2+(k2-k3) = 0
由于 v1,v2,v3 线性无关, 所以有
k1+k3 = 0
k1+k2 = 0
k2-k3 = 0
(若这个齐次线性方程组有非零...

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一般方法是这样的:
设 k1w1+k2w2+k3w3 = 0
则 k1(v1+v2)+k2(v2+v3)+k3(v1-v3) = 0.
得 (k1+k3)v1+(k1+k2)v2+(k2-k3) = 0
由于 v1,v2,v3 线性无关, 所以有
k1+k3 = 0
k1+k2 = 0
k2-k3 = 0
(若这个齐次线性方程组有非零解, 则w组线性相关, 否则线性无关)
因为行列式
1 0 1
1 1 0
0 1 -1
= 0
所以齐次线性方程组有非零解, 故w1,w2,w3线性相关. 证毕#
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