Question-81
Two matrices \(\displaystyle A\) and \(\displaystyle B\) have the same reduced row echelon form. Which of the following subspaces are the same for both \(\displaystyle A\) and \(\displaystyle B\)? \(\displaystyle \mathcal{N}( \cdot )\) and \(\displaystyle \mathcal{C}( \cdot )\) refers to the nullspace and column space respectively.
The null spaces of the two matrices are the same.
To see this, first we note that if \(\displaystyle A\) and \(\displaystyle B\) have the same RREF, there exist invertible matrices \(\displaystyle E_{A}\) and \(\displaystyle E_{B}\) such that \(\displaystyle E_{A} A=E_{B} B\). This is because, each row operation corresponds to pre-multiplication by an elementary matrix and elementary matrices are invertible. With this information, we proceed:
\[ \begin{equation*} \begin{array}{ l c } & x\in \mathcal{N}( A)\\ \Longleftrightarrow & Ax=0\\ \Longleftrightarrow & E_{A} Ax=0\\ \Longleftrightarrow & E_{B} Bx=0\\ \Longleftrightarrow & Bx=0\\ \Longleftrightarrow & x\in \mathcal{N}( B) \end{array} \end{equation*} \]
Since row operations are reversible, \(\displaystyle A\) and \(\displaystyle B\) have the same row space. This implies that \(\displaystyle \mathcal{C}\left( A^{T}\right) =\mathcal{C}\left( B^{T}\right)\).
The other two spaces are not necessarily the same. As a counterexample, consider:
\[ \begin{equation*} A=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix} ,\ B=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 1 & 1 & 0 \end{bmatrix} \end{equation*} \]
Both \(\displaystyle A\) and \(\displaystyle B\) have the same RREF. We have:
\[ \begin{equation*} \begin{aligned} \mathcal{C}( A) & =\text{span}\{( 1,0,0) ,( 0,1,0)\}\\ \mathcal{C}( B) & =\text{span}\{( 1,0,1) ,( 0,1,1)\} \end{aligned} \end{equation*} \]
Now, consider \(\displaystyle A^{T}\) and \(\displaystyle B^{T}\):
\[ \begin{equation*} A^{T} =\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix} ,\ B^{T} =\begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 1\\ 0 & 0 & 0 \end{bmatrix} \end{equation*} \]
We have:
\[ \begin{equation*} \begin{aligned} \mathcal{N}( A) & =\text{span}\{( 0,0,1)\}\\ \mathcal{N}( B) & =\text{span}\{( 1,1,-1)\} \end{aligned} \end{equation*} \]