Are isometries always linear?

linear-algebra normed-spaces

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0.

Must $f$ be linear in this case ?

Note : I am NOT assuming that the norm of $E$ comes from a quadratic form (otherwise I know the answer is yes, as per Should isometries be linear?). I expect the answer to my question should be no, but I don't have any counter example.


By the Mazur-Ulam theorem every bijective isometry between normed spaces, in particular between identical normed spaces, is affine. The linked paper also contains an example of injective non-affine isometry. See also Should isometries be linear?

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