Let's say I have a linear transformation $T: V\rightarrow V$ ($V$ finite-dimensional) and I am asked to find $[T]_\beta$, the matrix representation of $T$ with respect to the basis $\beta=\{v_1,\dots,v_n\}$.
As I understand it, I compute $T(v_j)$ for all $v_j\in\beta$. Then, I set that equal to $\sum\limits_{i=1}^{n}a_{ij}v_i$ and I get a system of linear equations from which I can solve for $a_{ij}$ which are the entries in $[T]_\beta$.
Now, if I want to instead find $[T]_{\beta'}$ (the matrix representation with respect to a new basis), is it true that $[T]_{\beta'}={([id_V]_{\beta'}^{\beta})}^{-1}[T]_\beta [id_V]_{\beta'}^{\beta}$? If not, how to I compute $[T]_{\beta'}$ using my knowledge of $[T_\beta]$?