Let $V_1$ and $V_2$ be $n$-dimensional inner product spaces with inner products $\langle\,,\,\rangle_1$ and $\langle\,,\,\rangle_2$ respectively. Let $T:V_1\rightarrow V_2$ be a linear isomorphism.
Question: Is it possible to choose $n$ vectors $e_1,\ldots,e_n$ that are orthogonal in $V_1$ such that $Te_1,\ldots,Te_n$ are orthogonal in $V_2$?
Comment: it seems to me this is possible, from looking at some examples in $\mathbb{R}^2$, but I can't quite see how to show this in general. Perhaps there is a different phrasing that makes this obvious.