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?

来源：学生作业帮助网编辑：六六作业网时间：2024/12/25 22:29:44

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...

全部展开

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#
有问题请消息我或追问
搞定了就采纳

收起

线性代数证明线性无关线性代数.线性无关证明线性代数,线性无关证明线性代数证明线性无关大学,线性代数,证明线性无关. 求解线性代数线性无关证明线性代数证明题,证明向量线性无关的题目线性代数,线性无关证明证明B,C,D线性无关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? 线性代数关于向量线性无关的证明线性代数:关于线性无关的证明, 线性代数证明线性无关的充分必要条件工程数学线性代数证明题证明可逆矩阵的列向量线性无关线性代数相关无关证明 abcd线性无关,abce线性无关,abcf线性无关,若def线性无关,abcdef线性无关不成立吧（线性代数证明）假设{V1,V2.Vk}是Rn里线性无关的一组向量求证假如A是n*n的非奇异矩阵,那么{A*V1,A*V2...A*Vk}也是线性无关的一组向量（线性代数证明）假设{V1,V2.Vk}是Rn里线性无关的一组向量求证假如A是n*n的非奇异矩阵,那么{A*V1,A*V2...A*Vk}也是线性无关的一组向量线性代数线性无关问题线性代数,线性相关,线性无关

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 notw1 = v1 + v2,w2 = v2 + v3 and w3 = v1 − v3 are linearly independent,givingreasons for your answer.怎么证明?