Let $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $x=\begin{bmatrix}2\\-1\end{bmatrix}$. Row products give $Ax=\begin{bmatrix}2-2\\6-4\end{bmatrix}=\begin{bmatrix}0\\2\end{bmatrix}$. By columns, the same result is $2(1,3)^T-(2,4)^T=(0,2)^T$. These are two views of the same operation.
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