Just learned about quotient spaces using the definition $V/W:=\{x+W|x\in V\}$ where $W$ is a subspace of $V$. Came up with some examples to see how it was working, but couldn't figure out what $\mathrm{O}(n)/\mathrm{SO}(n)$ was, even for $n = 2$. Any tips would be appreciated!
2026-03-28 12:13:25.1774700005
Figuring out what $\mathrm{O}(n)/\mathrm{SO}(n)$ is.
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Hint/Solution:
Note that $det:O(n) \to \frac {\mathbb Z}{2 \mathbb Z}(=\{-1,1\}$) is a surjective group homomorphism.
What is the kernel of this map?