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!

<br /> A\begin{pmatrix}<br /> 1 &amp; 3 &amp; -2\\<br /> 2 &amp; 5 &amp; -3\\<br /> -3 &amp; 2 &amp; -4\end{pmatrix} = <br /> \begin{pmatrix}<br /> 1 &amp; -8 &amp; 0\\<br /> 0 &amp; 3 &amp; 2\\<br /> 4 &amp; 0 &amp; -1\\<br /> 2 &amp; 1 &amp; 0\end{pmatrix} <br />

 

[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(<br /> \begin{array}{cccc}<br /> \vert &amp; \vert &amp; \vert &amp; \vdots \\<br /> v_1 &amp; v_2 &amp; v_3 &amp; \vdots \\<br /> \vert &amp; \vert &amp; \vert &amp; \vdots \\<br /> \end{array}\right)=\left(<br /> \begin{array}{cccc}<br /> \vert &amp; \vert &amp; \vert &amp; \vdots \\<br /> Av_1 &amp; Av_2 &amp; Av_3 &amp; \vdots \\<br /> \vert &amp; \vert &amp; \vert &amp; \vdots \\<br /> \end{array}\right)

 

[notmuch]

Got it. Thanks a lot!