I am taking a class on geometry, and am a bit confused by this paragraph in our textbook:
Each isometry is one~to-one-because points at nonzero distance cannot have images at zero distance-but it is not clear that each isometry is onto
I imagine one-to-one just means injective, and onto means surjective.
Firstly, we are saying every isometry is injective, meaning every point in the plane is sent to one and only one point in the transformed plane. I imaigine this comes from the property of metric preservance, but how would one go about proving this?
Secondly, to be surjective in this situation would mean every point in the transformed plane is mapped to by some point in the untransformed plane. Again, how would one show this? I imagine the span of the matrix of the isometry may be useful, but I am unsure.
One-to-one is somewhat bad terminology since every function maps one point to only one point (this is the vertical line test). What it means to be one-to-one is that every point in the image has only one point which gets mapped to it. The only way for this to fail is if some point in the image is hit by at least two points in the domain. An isometry preserves distances, so since distinct points are at a positive distance from one another, they are mapped to distinct points in the image. (In particular, an isometry never maps distinct points onto the same point, which is zero distance from itself.)
As for onto, it should be clear that things like rotations, translations, and reflections are all isometries and that they are all onto (if it isn’t clear, try to prove it using the definitions). It’s a theorem (not an obvious one) that these are the building blocks of all isometries of the plane, in some sense.