In this question , $S\in SO(n)$ , the main idea is to decompose $S$ to reflection. But I really don't know why $S(v_n)=v_n$ and $SR_n\cdot\cdot\cdot R_1=Id$ ? Who can explain it ? Thanks .
2026-04-06 23:32:58.1775518378
Some details in proof of SO(n) is product of reflection
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Hints: Step 1. Prove the claim in the case of orthogonal transformations in the plane. Step 2: Use the normal form for orthogonal transformations in the general dimension (existence of invariant decomposition of the space as the direct sum of pairwise orthogonal subspaces of dimension at most 2).