So I'm not sure how to phrase this in the first place, but here goes. There's this problem that gives two ordered bases, $B_1$=($e_1$, $e_2$, $e_3$, $e_4$) and $B_2$=($e_3$, $e_4$, $e_1$, $e_2$), and I have to find the matrix representations for $[L]_{B_1}$, $[L]_{B_2}$, $[L]_{B_1}^{B_2}$, $[L]_{B_2}^{B_1}$, where L is the linear transformation that shifts each entry down one spot. I found the first two just fine, which gave $$[L]_{B_1} = \begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0&1&0\\0&0&0&1\\1&0&0&0\end{matrix}$$ and $$[L]_{B_2} = \begin{matrix} 0 & 0 & 0 & 1 \\ 1 & 0&0&0\\0&1&0&0\\0&0&1&0\end{matrix}$$ But I'm running into a problem with the representations for $[L]_{B_1}^{B_2}$ and $[L]_{B_2}^{B_1}$, because I'm getting the same exact answer as the two above ($[L]_{B_1}$=$[L]_{B_1}^{B_2}$, $[L]_{B_2}$=$[L]_{B_2}^{B_1}$). I'm getting these results by computing the transformation then writing the result in terms of the basis I'm translating to (and transposing of course) - but if these two results are the same, what's the point of being an ordered basis?
Not sure if I'm even asking the right questions, maybe I'm missing something. Or maybe this is just a really simple example and my results are right. Any help is appreciated.