Interesting Linear Transformation question

***$\displaystyle U \text{ and } W \in R_4[x]$

and below I simply transform from $\displaystyle R_4[x] \rightarrow R^4$

$\displaystyle W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)$

Does there exist a linear transformation $\displaystyle T:R_4[x] \rightarrow R_4[x]$ so that $\displaystyle T(U)=W$ and $\displaystyle T(W) = U$

So I got to this...

$\displaystyle W = Sp{(1,0,0,0),(0,1,0,-1)}$

$\displaystyle U = Sp{(4,0,-1,0),(0,4,0,-1)} $

And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

I know in general that if you have a basis you can create any transformation you want but I don't know how to.

Thanks!