According to my reasoning it must, because of the following argument:
Let $f:V\to V$ be an invertible inner product preserving linear operator, and let $g=f^{-1}$. Hence, $$ <g(u)|g(w)>\ =\ <f(g(u))|f(g(w))>\ =\ <u|w> $$ for any $u,w\in V$.
For some reason, in any source I found, the claim in the title was preceded by the assumption that $V$ is finite dimensional, so I wanted to make sure that there isn't any flaw in the very simple argument above.