First of all I would like to thank you for taking a closer look at the following problem, which unfortunately I have been struggling to solve for days:
Let $W$ be a n-dimensional Euclidean vector space and let $f \in \text{End}(W)$ such that $(\det(f))^2 = 1$ holds.
Also assume $||f(w)|| \leq 1$ holds for all $ w \in W $ with $||w|| \leq 1$.Prove that $f$ is a linear congruent transformation.
Since a linear congruent transformation is distance-preserving, I know that one needs to show: $\quad ||f(w)|| = ||w|| \quad (\forall w \in W)$
I've tried many ways to construct an orthonormal basis for $W$ such that I get a mapping matrix of $f$ with respect to that basis, but unfortunately I've been unsuccessful.
I'm deeply grateful for any kind of help.