(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Verify that the following mappings are isometries onR^2

Reflection Through the Origin

Translation

Rotation

2. Relevant equations

Qualities of a metric:

d(x,y) = d(y,x)

d(x,x) = 0

d(x,y) = 0 <=> x = y

d(x,y) =< d(x,z) +d(z,y)

3. The attempt at a solution

As a metric hasn't been specified, I have been trying to prove this for a general metric using just the intrinsic qualities. I haven't had much luck, though.

I know that all three are straightforward to prove in Euclidean Space, which gives a metric. But is there a simple proof for a general metric?

I may have misunderstood the meaning of Verify, but would nevertheless like a proof if there is one.

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# Prove that the following mappings are Isometries.

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