Linear algebra systems ,Ax=b,Ax=0

  • Thread starter madah12
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Homework Statement



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

Homework Equations





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
 

Answers and Replies

  • #2
HallsofIvy
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Suppose x is a solution to Ax= b. Show that x- x1 is a solution to Ax= 0.
 
  • #3
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wait x-x0 or x-x1?
 
  • #4
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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
 
  • #5
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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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