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I am a little stumped on the following question. Not sure how to begin.

If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ then is $f$ defined by $f(x) = -x$ orientation preserving?

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  • $\begingroup$ How are you defining an orientation of $\mathbb{R}^n$? $\endgroup$ – Zev Chonoles Mar 31 '13 at 5:35
  • $\begingroup$ Not exactly sure. I know what orientable is, just not really what orientation preserving is. $\endgroup$ – Susan Mar 31 '13 at 5:40
  • $\begingroup$ Does a reflection preserve orientation? $\endgroup$ – Will Jagy Mar 31 '13 at 5:52
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The orientation is preserved if the determinant of the transformation's associated matrix is $+1$. What is the determinant of the matrix associated with $x\mapsto -x$?

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  • $\begingroup$ So it would be no if n is odd, and yes if n is even? $\endgroup$ – Susan Mar 31 '13 at 17:34
  • $\begingroup$ That's how it seems to me. $\endgroup$ – Ian Coley Mar 31 '13 at 17:36

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