Linear Transformations - formula

[notmuch]

Hello. I am given the following:

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!

$$A\begin{pmatrix} 1 & 3 & -2\\ 2 & 5 & -3\\ -3 & 2 & -4\end{pmatrix} = \begin{pmatrix} 1 & -8 & 0\\ 0 & 3 & 2\\ 4 & 0 & -1\\ 2 & 1 & 0\end{pmatrix}$$

 

[Galileo]

Because the last matrix is equivalent to the three individual equations. This just follows from matrix multiplication. Schematically, if the v_n's are column vectors, you can write:

$$A\left( \begin{array}{cccc} \vert & \vert & \vert & \vdots \\ v_1 & v_2 & v_3 & \vdots \\ \vert & \vert & \vert & \vdots \\ \end{array}\right)=\left( \begin{array}{cccc} \vert & \vert & \vert & \vdots \\ Av_1 & Av_2 & Av_3 & \vdots \\ \vert & \vert & \vert & \vdots \\ \end{array}\right)$$

 

[notmuch]

Got it. Thanks a lot!