If A and B have the same reduced echelon form then the column space of A equals the column space of B.

Question

If A and B have the same reduced echelon form then the column space of A equals the column space of B.

I think this is true because if we do Gaussian reduction on the transpose of A we will still get the same reduced echelon form. This means that the column space of A and B will be the same.

Is this true?

Answer 1

If we reduce by columns it is trivially true but if we reduce by rows in general it is not true since row operations do not preserve the column space and we can only conclude that the dimension of the two column space is equal.