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Math Help - EQUALITY OF ROW AND COLUMN RANK (o'Neil's proof) Is there smt wrong?

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

    OneilROWRANKCOLUMNRANK.jpg
    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.
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    Re: EQUALITY OF ROW AND COLUMN RANK (o'Neil's proof) Is there smt wrong?

    The book is doing a double inclusion proof. Right now the book know that it is possible less than r. However, when you go the other direction, you will get it equals r.
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