Finding the basis with given transition matrix \begin{equation}
P = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 1
\end{bmatrix}
\end{equation}
a) P is the transition matrix from what basis B to the standard
basis S = {e1, e2, e3} for R3?
b) P is the transition matrix from the standard basis
S = {e1, e2, e3} to what basis B for R3?
My attempt:
For a), if PB=S (is this even right?), can we just multiply inverse of P both sides to get B?

Your idea looks correct. Since S is the standard basis, i.e. S is the identity matrix, in a) your basis B are the columns of $P^{-1}S = P^{-1}$.
In b) B equals P: $B = PS = P$.
