elemntery row operations do not change the row space of a matrix

in reduced row eschelon from we get

so the basis of the row space is (0,1,0,1) and (0,0,1,0) so the dimention of the row space is 2.

The leading 1's in the matrix above occur in column 2 and 3. The leading 1's let us know what columns are the basis for the column space. So we go back to the original matrix and choose columns 2 and 3.

so the basis of the column space is (1,0,1) and (1,0,0) so it has dimention 2.

using the rank nullity theorem?? I'm not sure it that is what it is called

Rank(A)+Nullity(A)=number of columns of A

2+Nullity(A)=4 nullity=2

The columspace of A is the row space of A transpose so

2+nullity(A trans)=3 nullity=1.

I hope this helps.