Professor Gilbert Strang writes in his notes ( does elimination on matrix $ U$ to reduce matrix into rref, denoted by matrix $R$ )
$$ ... U=\left[\begin{array}{llll} 1 & 2 & 2 & 2 \\ 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right] \rightarrow\left[\begin{array}{lllr} 1 & 2 & 0 & -2 \\ 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right] \rightarrow\left[\begin{array}{lllr} 1 & 2 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]=R $$ By exchanging some columns, $R$ can be rewritten with a copy of the identity matrix in the upper left corner, possibly followed by some free columns on the right. If some rows of $A$ are linearly dependent, the lower rows of the matrix $R$ will be filled with zeros: $$ R=\left[\begin{array}{ll} I & F \\ 0 & 0 \end{array}\right] $$
How did the professor swap columns and still call the result $R$ ?