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Linear algebra systems ,Ax=b,Ax=0

  1. Aug 19, 2011 #1
    1. The problem statement, all variables and given/known data

    Let be Ax=b any consistent system of linear equations, and let be x1 a fixed solution. Show that every solution to the system
    can be written in the form x=x1 +x0, where x0 is a solution Ax=0 . Show also that every matrix of this form is a solution

    2. Relevant equations



    3. The attempt at a solution

    well A(x1+x0)= Ax1+Ax0=b+0=b
    but this only proves that any matrix x=x1+x0 is a solution but not that every solution can be written as such and i dont know how to prove that
     
  2. jcsd
  3. Aug 19, 2011 #2

    HallsofIvy

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    Suppose x is a solution to Ax= b. Show that x- x1 is a solution to Ax= 0.
     
  4. Aug 19, 2011 #3
    wait x-x0 or x-x1?
     
  5. Aug 19, 2011 #4
    I mean A(x-x0)=B-0=b isnt a solution but maybe A(x-x1)=b-b = 0 hmm so you are saying there will always exist x0=x-x1 such that x=x1+x0 is a solution to Ax=b
     
  6. Aug 19, 2011 #5
    I would appreciate a comment about whether i am right or wrong cause i dont want to go to the next problem until I am sure i understood this one
     
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