Row operations do not change the row space

• From [2], row spaces are basically a set of linear combinations of the rows of a matrix, i.e. it’s the $\text{Span}$ of the rows.

Theorem 28.5 in references says that:

• If you have $A$ and you create $B$ by doing some row operations on $A$.
• Then the row space of each of those matrices is gonna be the same.
• This makes sense because $\text{Span}$ doesn’t change when you have more linearly dependent vectors in your spanning set.
• And row operations basically enable you to represent your rows as linear combinations of other rows.