I was reading this question: Change of basis for linear transformation - Linear algebra and the answer gave me a good idea of the effect of change of basis on representations of vectors and linear transformations. This got me wondering on how the representation of a metric matrix is affected by a change of basis (or even a general linear transformation). Let's say the metric is $M$ in the canonical basis, and we know its elements explicitly. This means that we know $\langle e_i,e_j\rangle$ explicitly for all $i,j$, where $\{e_i\}$ is the canonical basis (denote it by $\mathcal{A}$).
If we have another basis $\mathcal{B}=\{u_i\}$, the metric representation in the new basis will be $M'$ such that $M'_{ij}=\langle u_i,u_j\rangle$. If we have a change of basis transformation $T$ from $\mathcal{A}$ to $\mathcal{B}$, then I'm not sure if we have enough information to find out $\langle u_i,u_j\rangle$. Using $T$ we can only find the representation of $e_i$'s in the $\mathcal{B}$ basis. So I'm a bit confused here on how to deduce $\langle u_i,u_j \rangle$ - (per my understanding, $u_i,u_j$ here are the new basis vectors and not any specific representations of those).
Would appreciate any help! Thanks