# A function $\mathbb{R}^n\to\mathbb{R}^n$ that preserves distances must be a linear map followed by a translation [duplicate]

Consider the set of all functions $\varphi : \mathbb{R}^n \rightarrow \mathbb{R}^n$ that preserve distance.

That is $\forall a,b : \| a-b\| = \|\varphi(a) - \varphi(b)\|$

Is the following statement true or false?

$\forall \varphi: \exists v \in \mathbb{R}^n, M \in \mathbb{R}^{n\times n}: \varphi(x) = Mx + v$

Why or why not? (ie rough sketch of proof)

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## marked as duplicate by Arturo Magidin, Gerry Myerson, Benjamin Lim, copper.hat, t.b.May 16 '12 at 7:27

This is a repeat of this question –  Arturo Magidin May 16 '12 at 5:55
And indeed, $M$ must be orthogonal. Presumably the distance is the $2$-norm. –  copper.hat May 16 '12 at 5:59
See also this question –  t.b. May 16 '12 at 7:28