I have a doubt about performing changes of basis in linear algebra. I know that for each pair of bases, say $B$ and $C$, I can find a certain "translation" matrix $P$ that sends my vectors in base $B$ to the corresponding vectors in base $C$. Furthermore, the inverse $P^{-1}$ performs the same transformation, just the opposite way: sending the vectors in base $C$ to their correspondents in base $B$.
Now, let $A$ be a certain matrix that describes a transformation over the space in relation to vectors expressed in the base $B$. My teacher gave me the definition of the matrix $A'$, describing the same transformation of vectors in base $C$, as:
$$ A' = P^{-1}AP $$
which makes sense, as I am taking a vector in base $C$, throwing it into base $B$ by using $P$, performing my transformation $A$ and then throwing it back into base $C$ using $P^{-1}$. Problem is, I have also found the formula:
$$ A' = PAP^{-1} $$
which seems to do the exact same thing, although I don't really understand how: it would seem that I'd be taking my vector in base $C$ and then using the transformation from base $B$ to base $C$ on it, which I can't in anyway get to make sense. Hope someone can help clarify this for me.