Re: [線代] least squares solution

[tibicos (tibicos)]

※ 引述《mqazz1 (無法顯示)》之銘言：
: 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
: 請問有人會這兩題的證明嗎?
: 可以指點一下嗎 謝謝!!

(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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※ 編輯: tibicos         來自: 111.252.14.248       (06/07 19:54)