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EQUALITY OF ROW AND COLUMN RANK (O'Neil's proof) Is there smt wrong?

  1. Oct 14, 2011 #1
    EQUALITY OF ROW AND COLUMN RANK (o'Neil's proof) Is there smt wrong?

    http://www.mediafire.com/imageview.php?quickkey=znorkrmk3k1otjd&thumb=6

    Theorem 7.9: EQUALITY OF ROW AND COLUMN RANK
    Proof: Page 210.

    It writes:....
    so the dimension of this column space is AT MOST r (equal to r if these columns are linearly independent, less than r if they are not)

    I THINK THIS IS WRONG. Look at the r vectors:
    1 0
    0 1
    : 0
    0 :
    BETAr+1,1 BETAr+1,2
    :
    BETAm1 BETAm2


    The first r columns of these r vectors are e1,e2,...er. Hence, they are DEFINITELY LINEARLY INDEPENDENT.
    There is no way to obtain 1 in the first coordinate of the first of the r vectors from the remaining r-1 vectors since the 1st coordinate of all of the remaining r-1 vectors are all 0.

    Hence, the correct one should be:

    so the dimension of this column space is EXACTLY r.

    Where am I wrong? or O'neil's is really wrong as I indicated.
     
  2. jcsd
  3. Oct 14, 2011 #2

    mathwonk

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    there is no contradiction between his statement and yours, and in fact both statements are true.
     
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