作者tibicos (tibicos)
看板Math
标题Re: [线代] least squares solution
时间Tue Jun 7 04:23:51 2011
※ 引述《mqazz1 (无法显示)》之铭言：
: let A be an 8*5 matrix of rank 3, and let b be a nonzero vector in N(A^T)
: (1) show that the system Ax=b must be inconsistent
: (2) how many least squares solutions will the system Ax=b have? explain
: 请问有人会这两题的证明吗?
: 可以指点一下吗 谢谢!!
(1) b is in N(A^T)
-> A^T(b)=0,
<-> b^T(A)=0; <-> b^T(A)x=b^T(Ax)=b^T(b)=||b||^2 >0 (because b is nonzero)
so no x can be found so that b=Ax;
(2) If p=Ax is the projection of b onto C(A)
(column space of A, the range of A),
then (A^T)(b-Ax)=0 -> A^T(Ax)=A^T(b)=0;
so p=Ax is in N(A^T)
<-> A^T(Ax)=0; -> (x^T)(A^T)(Ax)=0; -> ||Ax||^2=0; -> p=0;
therefore the projection p=0, the least squares solution x is in N(A)
with a degree of freedom 5-3=2
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1F：推 mqazz1 :谢谢!! 06/07 14:44
※ 编辑: tibicos 来自: 111.252.14.248 (06/07 19:54)