Find a Transformation Matrix So I have two bases in R3. To find the transformation matrix, I have to solve a system of linear equations.
I have these

Gaussian Elimination for v:

Gaussian Elimination for w:

So v and w are bases. To calculate the transformation matrix, solve

with

And this is where it gets too much for me. How do I get these numbers

so you have:  
$\begin{bmatrix}w_{11}&w_{12}&w_{13}\\w_{21}&w_{22}&w_{23}\\w_{31}&w_{32}&w_{33}\end{bmatrix}\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}$=$\begin{bmatrix}v_{11}&v_{12}&v_{13}\\v_{21}&v_{22}&v_{23}\\v_{31}&v_{32}&v_{33}\end{bmatrix}$  
so now you need $a_{ij}$ Matrix  
You can augment Matrix V to Matrix W. And find Reduced Row echoleon form 
from there, you can easly get A matrix ,
Suppose u get RREF as:  
$\begin{bmatrix}1&0&0&v'_{11}&v'_{12}&v'_{13}\\0&1&0&v'_{21}&v'_{22}&v'_{23}\\0&0&1&v'_{31}&v'_{32}&v'_{33}\end{bmatrix}$
now $a_{11}$=$v'_{11}$, similarly $a_{21}$=$v'_{21}$... so on
