Consider $\mathbb{R}^3$ with two orthonormal bases: the canonical basis $e = (e_1,e_2,e_3)$ and the basis $f = (f_1, f_2, f_3)$, where
$f_1 = \sqrt{3}(1,1,1), f_2 = \sqrt{6}(1,−2,1), f_3 = \sqrt{2}(1,0,−1)$.
Find the matrix, $S$, of the change of basis transformation such that
$[v]_f =S[v]_e,\ \forall v \in\mathbb{R}^3$, where $[v]_b$ denotes the column vector of $v$ with respect to the basis $b$.
On
Note that the same vector $v$ is represented in two different forms in the two different bases.
So, you can write, $$v=A_f [v]_f=A_e [v]_e$$ where $A_f,A_e$ are the matrices formed with $f,e$ as their columns respectively.
Since both $A_f,A_e$ are invertible, you can now write, $$[v]_f=S [v]_e$$ where $S=A_f^{-1}A_e$.
Your matrix $S$ is $$(f_1\;f_2\;f_3)^{-1}$$ so choose your preferable method to invert a matrix and you are done.