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.

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.