Question

what is the basic for the column space of the matrix
(0 6 6 3)
(1 2 1 1)
(4 1-34)
(1 3 2 0)

and what is the "rank"?

Answer 1

The matrix has row-reduced form of:

(1 | 0 | -1 | 0
0 | 1 | 1 | 0
0 | 0 | 0 | 1
0 | 0 | 0 | 0)

So you basically can swap the 4th column to the 3rd one to get echelon form. From that, it's obvious that the rank is 3 and one of the basis for the column space is the I_3, the identity matrix for 3 dimensions.