Question about row operations changing linear independence of columns

Question

If I have some matrix that I put into rref how do I know that the row operations won't change the linear indepence? I have looked up this question but none of the answers I am really able to understand.. It's obvious why we keep linear independence for row swaps and scalar multiplication but when I add some row to another row it's not clear to me why the linear indepence between the columns stays the same could anyone explain? Like some type of proof and some intuition maybe would be nice

Answer 1

Maybe the easiest explanation as to why it works is because it's a reversible operation, which means it can't be changing the amount of "information" in the matrix.

If you have rows $R_1, R_2, \ldots, R_n$ and you replace $R_1$ with $R_1' := R_1 + R_2$, then the substitution $R_1'' := R_1' - R_2 = R_1 + R_2 - R_2 = R_1$ gets you back where you started.

If you're familiar with vector spaces and the fact that the rows of a matrix can be treated as vectors, then this is the same as saying that $\textrm{span}\{R_1, R_2, \ldots, R_n\} = \textrm{span}\{R_1 + R_2, R_2, \ldots, R_n \}$, which you can prove by noting that you can easily transform a linear combination of one of those sets of vectors into a linear combination of the other.