 “Unique up to positive affine transformation” means that any utility function \(u'\) that also represents the preference relation can be transformed into \(u\) by multiplying by a constant and adding a constant (to borrow an example from a different domain: temperature scales are unique up to positive affine transformation, because although temperature can be represented by Celsius or Fahrenheit, Celsius can be transformed into Fahrenheit by multiplying by 9/5 and adding 32).