When doing matrix elimination (Gauss method) the row space and the column spaces

When doing matrix elimination (Gauss method) the row space and the column spaces of the initial matrix are changed. How do they change? How is the “column picture” changed after each step of elimination? How is the “row picture” changed after elimination? (The terms “column picture” and “row picture” can be found in page 34 of Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley-Cambridge Press, 2009. ISBN: 9780980232714)