Change of basis matrix for inner product space

Question

If $B = (e_1,e_2,\ldots, e_n)$ is a basis for an inner product space $V$ and $B' = (f_1,f_2,\ldots,f_n)$ is an orthonormal basis of $V$.

Is the change of basis matrix $P$ necessarily orthogonal?

Answer 1

The change basis matrix $P$ is $$P=\left(f_1 f_2\cdots f_n\right)$$ where $f_i$ is a column vector and since the basis $B'$ is orthonormal we have $$f_i \ . f_j=\delta_{ij}$$ where $\delta$ is the Kroneker symbol and hence $P$ is orthogonal

Answer 2

Take any non-orthogonal invertible matrix $Q$. We wish to find a basis $e_1,\dots,e_n$ such that $Q$ is the change of basis from $(e_1,\dots,e_n)$ to $(f_1,\dots,f_n)$. But for this we may just set $e_i = Q_{1i} f_1 + \dots + Q_{ni} f_n$.