Do equivalent matrices have the same image

Question

Let say we have R-module homomorphisms $T_1 ,T_2 : R^m \rightarrow R^n $ with matrices $A$ and $B$ respectively. If $B$ can be obtained from $A$ by just performing elementary column and row operations, must we have $\text{Im}(A)= \text{Im}(B)$ where $\text{Im}(A)= \{ Ax : x \in R^m \}.$

Answer 1

$\mathrm{Im}(A)$ in your definition is essentially the subspace generated by the columns of $A.$ So if $B$ is obtained from $A$ by row operations, nothing can be guaranteed that $\mathrm{Im}(B) = \mathrm{Im}(A).$

For example, $A = \begin{bmatrix}1& 0\\ 1&0\end{bmatrix}$ and $B = \begin{bmatrix}0 & 0 \\ 1&0\end{bmatrix}.$