This is a lemma for the Theorem of the Inverse of a Transition Matrix on p.211 of Larson's Elementary Linear Algebra 8e.
Let $B=(v_1,...,v_n), B' =(u_1,...,u_n)$ be bases of a vector space $V$.
If $v_1=c_{11}u_1+c_{21}u_2+...+c_{n1}u_n, ...,v_n=c_{1n}u_1+...+c_{nn}u_n$
then the transition matrix $Q$ from $B$ to $B'$ is all the $c_{ij}$ in columns. What would the inverse $Q^-1$ be to go from $B'$ to $B$? I tried augmenting $[QI]\rightarrow[IQ^-1]$ and think it is going to be the same because we are talking about basis vectors but unsure.
If you are looking for a expression for $Q^{-1}$ in terms of all the $c_{ij}$'s, the only generic way you can do so is either
1) Compute $Q^{-1}$ directly or
2) Express each $u_i$'s as a linear combination of all the $v_i$'s and take it from there.
However if both $B$ and $B'$ are both orthonormal bases, one can show that the transition matrix from one basis to another is in fact orthogonal, whose inverse can be simply obtained by taking transpose.