I was wondering if the following statements about ERO's (elementary row operations) are mathematically correct. Any help and insight is appreciated. Any additional facts to the table is also welcomed.
ERO's performed on a matrix $A$ will preserve
1) The row space of the matrix A
2) The null space of the matrix A
3) Any linear dependence or independence of the columns of $A$
ERO's performed on a matrix $A$ will NOT preserve
1) The column space of the matrix $A$
2) Any linear dependence or independence of the rows of $A$
note that the preservation of linear dependence / independence refers to preserving not just if the set of vectors is linearly dependent / independent - but also the exact relationships between the vectors themselves (e.g. vector #1 is exactly 2x that of vector #3)