I will need to give some background to ask my question. Working on some linear algebra problems I came across a way to find a basis for the range of a matrix that I do not find discussed in the textbooks, but is all based on elementary facts.
If $A$ is an $n\times m$ matrix over $\mathbb R$, we regard it as a linear map $A:{\mathbb R}^m\rightarrow {\mathbb R}^n$. Since the range of this linear map is the subspace spanned by the columns of $A$, and doing row operations does not change the subspace spanned by the rows, a computationally efficient way to find a basis for the range of $A$ is to perform row operations on the transpose of $A$ until it is in row echelon form, and then the non-zero rows will be a basis for the range of $A$.
But there is another way to find a basis for the range of $A$. If $A$ has rank $r\leq \min(n,m)$, by doing row operations we can put it in Row Echelon Form $R=\begin{pmatrix}R_1\\0\end{pmatrix}$, where $R_1$ is $r\times n$ of rank $r$. The sequence of row operation is equivalent to left multiplication of $A$ by an invertible $n\times n$ matrix $E$, that we can write $E=\begin{pmatrix}E_1\\E_2\end{pmatrix}$ where $E_1$ is $r\times n$ and $E_2$ is $(n-r)\times n$. So $EA=\begin{pmatrix}E_1A\\E_2A\end{pmatrix}=\begin{pmatrix}R_1\\0\end{pmatrix}$, and $E_2A=0$. This means that a vector $w\in {\mathbb R}^n$ will be in the range of $A$ if and only if $E_2w=0$. In other words, the range of $A$ is the null space of $E_2$. So we can solve the homogenous system $E_2w=0$, whose solution will have $r$ free variables $t_1,t_2,\ldots, t_r$, and then setting $t_i=1$, $t_j=0$ for $j\neq i$ we find $r$ independent vectors that will form a basis for the range of $A$.
My question is about the matrix $E_2$. How does that fit in the general theory of linear operators on vector spaces? I feel the above outline is probably an elementary version of some general facts that involve important concepts for linear operators, but I do not know enough in that area to sort out where it all fits. A second question is about the computational efficiency of the second method to find a basis of the range of $A$, compared to the first one.