[Math] Transpose of the reduced row echelon form

Question

I am reading the proof of a theorem and there is the following statement:

Let $U$ be the reduced row echelon form of $A$ and let $V$ be the
reduced row echelon form of $U^T$. Then $V^T$ is also in reduced row
echelon form

Why is $V^T$ in reduced row echelon form?

Answer 1

Because V is actually in a simple form [I 0; 0 0]

reason: Since U is in rref form, there exists a column exchange to swap the pivot columns to the first columns, let this exchange matrix be P, then

U = [I B; 0 0] * P

U' = P' [I 0; B' 0] = P'[I 0; -B' I][I 0; 0 0]

since rref form of U' is unique, we know that V = [I 0; 0 0]