how can I Show that the mapping $\phi: (a_1,a_2,\dots,a_n)\to (-a_1,-a_2,\dots,-a_n)$ is an automorphism of the group $\mathbb{R}^n$ under componentwise addition. AND How can I Describe the action of $\phi$ geometrically. Actually, I am not sure what does componentwise addition mean in this context. how to find a map of it?

  • $\begingroup$ Componentwise addition means you add the respective components: $(a,b)+(c,d)=(a+c,b+d)$. To get the first component of the sum, you sum the first components; to get the second component of the sum, you sum the second components, and so on. $\endgroup$ – symplectomorphic Mar 17 '15 at 3:00
  • $\begingroup$ And to see what the map does geometrically, draw a picture for a simple case, like $\mathbb{R}^2$. If the point $(a,b)$ gets sent to $(-a,-b)$, what's happening geometrically? $\endgroup$ – symplectomorphic Mar 17 '15 at 3:03
  • $\begingroup$ Have you tried to draw what it does geometrically? It might be easier to draw $(x,y) \mapsto (-x,y)$ and then draw $(-x,y) \mapsto (-x,-y)$ (that is break it up into easier to visualize pieces), I also recommend drawing the $x,y$ axis differently (like different colors or $x$ axis dotted, etc) so the mapping is more obvious. $\endgroup$ – Paul Plummer Mar 17 '15 at 4:31

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