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I am looking for a thorough proof of the following theorem

Theorem: If an isometry of $\mathbb{R^n}$ fixes a non-empty set of points F, then it fixes the smallest affine subspace of $\mathbb{R^n}$ which contain F.

aribaldi
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    Use the fact that this space is just the space of (barycentric) combinations $x_1X_1+...+x_nX_n$ with $X_i\in F$ and $\sum _1^n x_1=1$. If $g$ fixes the $X_i$, it fixes their barycenter. – Thomas Jun 27 '16 at 14:57
  • And, of course, use the fact that any isometry of $\Bbb R^n$ is of the form $f(x)=Ax+b$ for an orthogonal matrix $A$ (in particular, is an affine linear map). – Ted Shifrin Jun 27 '16 at 16:38

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