I am trying to prove if R and R' are two reduced row echelon matrices, and have the same nullspaces, they are identical. I have observed this when testing them, but I have trouble finding a formal proof.
I tried reading this question and extracting a general solution, but this question is focused on a particular 2 by 3 matrix whereas I am trying to find a formal proof for an m by n matrix.
I believe I should start by writing two general reduced row echelon matrices, writing Rx=R'x=0 and finding a general answer for each row, then proving they are equal, but I have trouble proving it generally and without presumptions.
Thank you in advance.
Best Answer
Here is an attempt at a direct proof (which is unfortunately more complicated than I would have hoped).
It suffices to show that the RREF can be recovered from the nullspace. Let $R$ denote an RREF matrix with $n$ columns; we proceed inductively over $n$. For the case of $n=1$, there are two possible RREF matrices corresponding to the two possible nullspaces.
For $n > 1$, let $U$ denote the nullspace of $R$. Let $R'$ denote the matrix obtained by deleting the last column of $R$. We note that the nullspace of $R'$ has the same dimension as (and is "essentially equal" to) $$ U' = U \cap \operatorname{span}\{e_1,\dots,e_{n-1}\}, $$ where $e_1,\dots,e_n$ denote the standard basis of $\Bbb R^n$. There are two possibilities: in the case that $U' = U$, the final column of $R$ must be linearly independent of the rest, which means that the final column must be a pivot column. For the case that $U' \subsetneq U$, let $p_1<\cdots<p_r$ denote the indexes of the pivot columns of $r$, so that $Re_{p_i} = e_i$ for each $i = 1,\dots,k$. It follows that the final column $Re_n$ of $R$ satisfies $$ Re_n = \sum_{i=1}^k r_{in} e_i = \sum_{i=1}^k r_{in} R e_{p_i} \implies\\ R\left[e_n - \sum_{i=1}^k r_{in} e_{p_i}\right] = 0. $$ That is, if $v = e_n - \sum_{i=1}^k r_{in} e_{p_i}$, then $U = U' + \operatorname{span}(v)$.
I claim that for any $w = e_n - \sum_{i = 1}^k s_{in} e_{p_i}$, we can only have $U' + \operatorname{span}(v) = U' + \operatorname{span}(w)$ if $r_{in} = s_{in}$ for all $i$. If this is proved, then the final column of $R$ is uniquely determined by $U$, which would allow us to finish the proof.