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Linear algebra question

  1. Jun 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Quote from my textbook:

    "The linear system Ax=b is consistent if and only if the number of nonzero rows of the augmented matrix [U| c], equals the number of nonzero rows in U."

    [U,c] is the rref of [A,b]

    2. Relevant equations

    3. The attempt at a solution

    I understand "only if" but not "if". "Only if" is true since the elementary row operations do not affect the solutions to the system and it is clear that a linear combination of zeros cannot equal a nonzero number. Can someone help me with "if"?
  2. jcsd
  3. Jun 16, 2008 #2


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    Homework Helper

    number of nonzero rows in U is the rank of U,r(U)
    the no. of nonzero rows of [U|c] is the rank of that matrix,r(U|c)

    For a system of eq'ns to be constisent

    [tex]r(U)=r(U|c) \leq n[/tex]

    n= no . of rows
  4. Jun 18, 2008 #3
    Thanks for responding to my question!

    Unfortunately, I am still confused. It seems like that just restated my question using the word "rank".

    I think the quote I gave came from a proof of this statement actually...
  5. Jun 18, 2008 #4


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    well saying "if and only if", like restricting the statement so that the statement will be true when that condition is satisfied.
  6. Jun 18, 2008 #5
    I figured it out. Thanks for your help.
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