Let f : R n → R n be a map where ‖ f ( a ) − f ( b ) ‖ = ‖ a − b ‖ and f(0)=0, prove a · b = f ( a ) · f ( b )Obviously this is a linear transformation, but how do we go about proving this statement?

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Let   f  :            R        n    →            R        n   be a map where   ‖  f  (  a  )  −  f  (  b  )  ‖  =  ‖  a  −  b  ‖ and f(0)=0, prove   a  ·  b  =  f  (  a  )  ·  f  (  b  )Obviously this is a linear transformation, but how do we go about proving this statement?

Marisa Colon · Accepted Answer

First, it is obvious that   ‖  f  (  x  )  ‖  =  ‖  x  ‖ for all   x  ∈            R        n   since f(0)=0.Now, for all a and   b  ∈            R        n  ,∥f(a)−f(b)∥2=∥a−b∥2.It follows that  ‖  f  (  a  )      ‖    2    +  ‖  f  (  b  )      ‖    2    −  2  f  (  a  )  ⋅  f  (  b  )  =  ‖  a      ‖    2    +  ‖  b      ‖    2    −  2  a  ⋅  b  .Since   ‖  f  (  a  )  ‖  =  ‖  a  ‖ and   ‖  f  (  b  )  ‖  =  ‖  b  ‖, we can deduce that  a  ⋅  b  =  f  (  a  )  ⋅  f  (  b  )  .

Let f:RR^n to RR^n be a map where norm(f(a)−f(b))=norm(a−b) and f(0)=0, prove ab=f(a)f(b).

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