- #1
Thales Castro
- 11
- 0
Homework Statement
The n-dimentional Euclidean group ## E^{n} ## is made of an n-dimentional translation ## a: x \mapsto x+a ## (##x,a \in \mathbb{R}^{n}## ) and a ## O(n) ## rotation ## R: x \mapsto Rx ##, ##R \in O(n) ##. A general element ## (R,a) ## of ## E^{n} ## acts on ## x ## by ## (R,a): x \mapsto Rx + a ##. The product is defined by ## (R_{2},a_{2}) \times (R_{1},a_{1}): x \mapsto R_{2}(R_{1}x + a_{1}) + a_{2} ##, that is, ## (R_{2},a_{2}) \circ (R_{1},a_{1}) = (R_{2}R_{1},R_{2}a_{1}+a_{2}) ##. Show that the maps ## a ##, ## R ##, and ## (R,a) ## are bijections. Find their inverse maps.
The Attempt at a Solution
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It is not very hard to prove that they're all injective. For instance, if ## x,x' \in \mathbb{R}^{n} ## with ## x \neq x' ##, we can do a proof by contradiction. If the map is not injective, then it is possible that ## x + a = x' + a ##, which implies that ## x = x' ##. So, we can conclude that the translation map is injective. The proofs for ## R ## and ## (R,a) ## are analogous.
I don't know how to prove that they're surjective. It makes intuitive sense to me (if we think of a plane, for example, it's easy to see that every point on the plane after a rotation/translation must come from a point on the "standard" plane), but I'm not sure how to formally write this.
Another question is: do I really need to prove that they are injective and then prove that they are surjective? Would it be enough to just find an inverse map and prove that the composition is the identity?
Thanks in advance!