If A and B are row equivalent matrices, then the row vectors of A are linearly independent iff the row vectors of B are linearly independent. How exactly do you prove this?

This is my thought so far: First we assume the row vectors of A are linearly independent, hence they form a basis for the rowspace of A. Now since elementary row operations do not change the rowspace of a matrix, then the row vectors of B span the same space. So the row vectors of A also form a basis for the rowspace of B.