Elementary row operations preserve the column space of the matrix a.

1. Do elementary row operations change column space?
2. Do column operations preserve column space?
3. What do row operations preserve?
4. Do elementary row operations alter the row space of a matrix?

Do elementary row operations change column space?

Elementary row operations affect the column space. So, generally, a matrix and its echelon form have different column spaces. However, since the row operations preserve the linear relations between columns, the columns of an echelon form and the original columns obey the same relations.

Do column operations preserve column space?

Just like elementary row operations preserve the row space of a matrix, elementary column operations preserve its column space.

What do row operations preserve?

Elementary row operations do not change the row space of a matrix but they do change the column space; only the dimension of the column space is preserved.

Do elementary row operations alter the row space of a matrix?

Elementary row operations do not affect the nullspace or the row space of the matrix. Hence, given a matrix A, first transform it to a matrix R in reduced row-echelon form using elementary row operations.