I'm having trouble with the following problem:

Let $\displaystyle V$ be a real vector space equipped with an inner product $\displaystyle \langle\cdot,\cdot\rangle$ (and thus with a norm induced by that inner product).

Let $\displaystyle T:V\to V$ be a bijective linear transformation such that if $\displaystyle \langle u,v\rangle=0$, then $\displaystyle \langle T(u),T(v)\rangle=0$.

Given an orthonormal basis $\displaystyle \{b_1,...,b_n\}$ is $\displaystyle V$, it clearly is the case that $\displaystyle \{T(b_1),...,T(b_n)\}$ is an orthogonal basis of $\displaystyle V$ and hence $\displaystyle \left\{\frac{T(b_1)}{\|T(b_1)\|},...,\frac{T(b_n)} {\|T(b_n)\|}\right\}$ is an orthonormal basis.

I now have to show that $\displaystyle \|T(b_1)\|=\cdots=\|T(b_n)\|$, but everything I've tried so far has lead me to a dead end.

Any hint or tip from someone who knows how to prove this would be greatly appreciated (I'm not looking for someone to post a complete solution).