$Ax = y$, possible to isolate matrix $A$? I have the matrix equation of real coefficients
$Ax = y$
where $y$ and $x$ are of different size. Is it possible to do a simple operation which will turn $A$ into a matrix function of both $y$ and $x$? I guess I can multiply on the right side with $x^{\top}$ and divide by the norm to yield the identity-matrix, but is there a smoother way to proceed?

Of course. Plug 
\begin{equation}
x=\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0\end{bmatrix},
\end{equation}
where $1$ is in the $j$-th position, into the equation. Now $y$ gives you the $j$-th column of $A$. You will need to do this for as many times as the columns of $A$, of course. 
