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Linear algebra proof with trivial solution

  1. Aug 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Problem:
    Prove Ax=b has a solution for each b in R^m if and only if the equation A^T x = 0 has only the trivial solution.

    Hint: For the forward direction use theorem 1.4.4 to prove that the dimension of the null space pf A^T is zero

    2. Relevant equations

    Theorem 1.4.4: Let A be an mxn matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false.
    a. For each b in R^m, the equation Ax=b has a solution
    b. Each b in R^m is a linear combination of the columns of A
    c. The columns of A span R^m
    d. A has a pivot position in every row


    3. The attempt at a solution

    Can someone please help me through this proof? I'm not even sure how to begin it. Thank you
     
  2. jcsd
  3. Aug 7, 2010 #2
    Isn't there a simpler way of going about this ?

    Correct me if I'm wrong but isn't it true that

    B is invertible iff BX = 0 has only the trivial solution ?

    Therefore if A^T x = 0 has only the trivial solution. A^T is invertible then so is A.

    Which means Ax=b always has a solution .

    EDIT

    If Ax= 0 had non trivial solutions then A is not invertible, in which case it does not reduce to RREF( reduced row echeol form) and can not have unique solutions.

    In which case a row of zero exist somewhere and by putting a 1 on that row in the column matrix of B we can show that there is no solution.
     
    Last edited: Aug 7, 2010
  4. Aug 7, 2010 #3
    That does make sense, I don't see why it wouldn't work that way. The information above is just what I was given to do this problem. I like your way, its simple and makes sense.
     
  5. Aug 7, 2010 #4
    My only concern is that, you may still need to know perhaps the more difficult way of doing things, for exams or what not.
     
  6. Aug 7, 2010 #5
    Yeah, I'm not sure. I have an exam coming up but we usually don't get asked things that are too indepth since the exam only lasts an hour and 20 minutes. Do you understand how to do it the other way? I think it would be good to know just in case.
     
  7. Aug 7, 2010 #6
    Sorry I can't help you there. It's been a while since I took linear algebra. Heck, I can't even remember what it means by A has a pivot position in every row.
     
  8. Aug 7, 2010 #7
    lol thats ok. Thanks for your help with the simple proof though
     
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