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T([1,2,-3]) = [1,0,4,2]

T([3,5,2]) = [-8,3,0,1]

T([-2,-3,-4]) = [0,2,-1,0]

And of course I know that:

T(x) = Ax

and I want to find the matrix A.

So, from the individual equations, I construct:

A[1, 2, -3] = [1, 0, 4, 2] (please forgive, these are actually col. vectors)

I do something similar for the other two, and come up with the equation below. I can solve this equation to obtain (the correct) matrix A, but I can't seem to find an explanation for why it is possible to throw it all together into "combined matrices." Could anyone help? Thanks!

[tex]

A\begin{pmatrix}

1 & 3 & -2\\

2 & 5 & -3\\

-3 & 2 & -4\end{pmatrix} =

\begin{pmatrix}

1 & -8 & 0\\

0 & 3 & 2\\

4 & 0 & -1\\

2 & 1 & 0\end{pmatrix}

[/tex]