线性相关证明题If η is a solution of AX=b ,which and R(A)=r.Letζ1ζ2ζ
来源:学生作业帮 编辑:作业帮 分类:英语作业 时间:2024/07/09 03:49:05
线性相关证明题
If η is a solution of AX=b ,which and R(A)=r.
Letζ1ζ2ζ3ζ4ζ5 be a basis of the solution space of AX=0.
Show η ζ1ζ2ζ3ζ4ζ5 that are linearly independent.
![](http://img.wesiedu.com/upload/5/29/529103d8f330afab2c686f64a035bfe7.jpg)
If η is a solution of AX=b ,which and R(A)=r.
Letζ1ζ2ζ3ζ4ζ5 be a basis of the solution space of AX=0.
Show η ζ1ζ2ζ3ζ4ζ5 that are linearly independent.
![](http://img.wesiedu.com/upload/5/29/529103d8f330afab2c686f64a035bfe7.jpg)
Suppose η,ζ1,ζ2,ζ3,ζ4,ζ5 are linearly dependent.
Then there are k1,k2,...,k6 of which not all are 0 such that k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.
So A(k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5)=0.
Because Aη=b,Aζi=0.
So k1b=0,namely k1=0.
So k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.
Because ζ1,ζ2,ζ3,ζ4,ζ5 are a basis of the solution space of AX=0,
so they are linearly independent.
So k2=k3=...=k6=0
That is impossible.
So η,ζ1,ζ2,ζ3,ζ4,ζ5 are linearly independent
Then there are k1,k2,...,k6 of which not all are 0 such that k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.
So A(k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5)=0.
Because Aη=b,Aζi=0.
So k1b=0,namely k1=0.
So k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.
Because ζ1,ζ2,ζ3,ζ4,ζ5 are a basis of the solution space of AX=0,
so they are linearly independent.
So k2=k3=...=k6=0
That is impossible.
So η,ζ1,ζ2,ζ3,ζ4,ζ5 are linearly independent
线性相关证明题If η is a solution of AX=b ,which and R(A)=r.Letζ1ζ2ζ
线代 已知r(A)=r,A是n阶矩阵,证明AX=b有n—r+1个线性无关解.
线性相关的证明向量组a1,a2,……a(r)线性无关(r>=2)任取r-1个数k1,k2,……k(r-1)构造向量组b1
AX=B 如何证明非齐次线性方程组无解时r(a,b)=r(a)+1 (a,b)为增广矩阵
怎么证明R(AB)>=R(A)+R(B)-N
线性代数的题,设A是4阶非零矩阵,a1a2a3a4是非齐次线性方程组AX=b的不同的解 1)若a1a2a3线性相关,证明
设向量组a b r线性无关,证明向量组a,a+b,a+b+r也线性无关.
在4元非齐次线性方程组AX=b中,已知r(A)=2 n1 n2 n3为方程组三个线性无关的解 则AX=b通解?
刘老师你好,请问一下一个向量组A可以被另一个向量组B线性表示则r(A)=r(A B)如何证明呢?
假设s×n矩阵A的秩为r.证明Ax=θ的任意n-r个线性无关的解都是其基础解析.
已知向量组A能由向量组B线性表示,为什么r(B) = r(B,A)?请老师帮我证明下,我不是太理解,
设a1,a2,a3 是四元非齐次线性方程组Ax=B的三个线性无关的解向量,且r(A)=2 ,则Ax=0的通解为