@marathon Let's try it
Let $r_1,\dots,r_m$ be the row vectors of $A$. If $c$ is in the row space of $A$, then there exists coefficients $\alpha_1,\dots,\alpha_m$
such that $c=\sum_{i=1}^m\alpha_i r_i$. Let $x$ be a feasible solution. Then
$$\begin{align*}
c^\top x &= \sum_{i=1}^m\alpha_i r_i^\top x\\
&=\begin{bmatrix}\alpha_1 & \dots & \alpha_m\end{bmatrix}Ax\\
&=\begin{bmatrix}\alpha_1 & \dots & \alpha_m\end{bmatrix}b
\end{align*}
$$
Since all feasible solutions have the same value, they are all optimal