For matrices $A, B, C$, Does $\DeclareMathOperator{\Ran}{Ran}\Ran(A) = \Ran(B)$ imply $\Ran(AC) = \Ran(BC)$? The question is in the title. If we have matrices $A, B \in \mathbb{R}^{n \times m}$ such that $\Ran(A) = \Ran(B)$ (here, $\Ran(A)$ is the range/column space of $A$), does that imply $\Ran(AC) = \Ran(BC)$ for an arbitrary $C \in \mathbb{R}^{m \times p}$? We can assume that $AC \neq 0$ and $BC \neq 0$.
 A: No, that does not hold. An easy counterexample is given by
$$A= \begin{pmatrix}1&1\\0&1\end{pmatrix}
\qquad B=\begin{pmatrix}1&0\\0&1 \end{pmatrix} 
\qquad C=\begin{pmatrix}0&0\\0&1\end{pmatrix} 
\,.$$
Then $\mathrm{Im}(A)= \mathrm{Im}(B)= \mathbb R^2$, but
$$AC= \begin{pmatrix}0&1\\0&1 \end{pmatrix}
\qquad BC=\begin{pmatrix}0&0\\0&1\end{pmatrix} 
$$
have different image.
A: I would like to add some general remarks (too long for comment) to Desperado's excellent counterexample so this is not the independent answer but just the supplement to the Desperado's answer.
The answer concerns the problem:

*

*How to construct different images from linear transformations (which
have the same range) via multiplication?

Denote two matrices $A,B$ in the column form $A=[ a_1 \ \ a_2 \ \dots \ \ a_m]$, $B=[ b_1 \ \ b_2 \ \dots \ \ b_m]$,  -   their column vectors can be very different and in a different order, however  assume they have the same range.
Matrix $C$ can be constructed as   $C=[ e_1 \ \dots \ e_i \  \dots \ \ 0_j \dots e_p]$, where columns of $C$ are vectors of standard basis $e_i$ or zero vectors $0_j$
Such construction enables to choose columns of $A$ or $B$ because $Ae_i$ is   an operation with this effect, and $A0_j=0$ is equal to omitting  $j$ column of $A$ in defining range of product $AC$.
Products are equal to
$AC=   [ Ae_1 \ \dots \ Ae_i \  \dots \ \ A0_j \dots Ae_p]$,
and
$BC=   [ Be_1 \ \dots \ Be_i \  \dots \ \ B0_j \dots Be_p]$.
Range depends on columns of matrix and hence evidently the ranges of $AC$ and $BC$ can differ.
In fact appropriately constructed matrix $C$ can give the effect of different ranges for products even in the case when $A$ and $B$ have the same vectors as columns but permuted.
