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perturbation comments on Terminology suggestion: Say "degrees utility" instead of "utils" to prompt affine thinking - Less Wrong
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(Moving a bit more to the concrete side for the sake of those who fall closer to the engineer perspective on the mathematician-engineer continuum):
Affine transformation == Linear transformation + translation.
It preserves ratios of distances, but not (necessarily) angles or distances themselves. Utility functions are only defined up to an affine transformation, which means that preferences are preserved, but that "doubling the utility assigned to an action" will not necessarily give the same results even if two people have the same utility function. (Just like doubling 15 degrees C == 30 degrees C, but the corresponding 59 degrees F and 86 degrees F are not doubles of each other.)