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Homework Help: Range/column space

  1. Feb 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove or disprove with counter example:
    If A is an (n x n) matrix then range(A) = range(A^T)

    2. Relevant equations

    Range = { y in R^N, y = Ax}

    3. The attempt at a solution

    I thought that the range(A) = the row space of (A^T) however whenever I tried examples to disprove this statement like:

    1 2 3
    4 5 6
    7 8 9

    And then transposed it and found the solution to y = Ax and y = A^Tx I got that both of the range was z - x - 2y = 0

  2. jcsd
  3. Feb 17, 2010 #2


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    Science Advisor
    Homework Helper

    You are probably just unlucky with your counterexample. Pick a simpler one. Try a 2x2 matrix. And you had better pick a singular matrix. Do you see why?
  4. Feb 17, 2010 #3
    Ohhh duuuh! Thank you so much!
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