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.
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.