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Linear equation Ax=b

  1. Oct 19, 2007 #1
    let A be a mxn matrix.
    prove that the system of linear equations Ax=b is consistnet for all column vectors b if and only if the rank of A is m.

    I have no idea how to start, can anyone helo me out?
  2. jcsd
  3. Oct 19, 2007 #2
    what does it mean if the matrix equation is consistent for all vectors b?
  4. Oct 19, 2007 #3
    i guess my problem is that i dont quite understand when it says "consistent for all column vectors b."
  5. Oct 19, 2007 #4
    also it would mean that b is in the column space of A.
  6. Oct 19, 2007 #5
    yes but any b?
    Last edited: Oct 19, 2007
  7. Oct 20, 2007 #6


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    A system of linear equations is consistent if it has a solution. Of course, this solution need not be unique.
  8. Oct 25, 2007 #7


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    A matrix equation, Ax= b, is "consistent" if it has at least one solution. "Ax= b is consistent for all b" means the equation Ax= b is consistent no matter what vector b is.

    The OP said earlier, "also it would mean that b is in the column space of A." Okay. And if b is to be any member of A, what must the column space be?
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