A few weeks ago, I was having a chat with Todd and some others in the office and it was in the conversational mix that cardinal utility had the property of preserving “intervals.” It was occasionally also mentioned that such utility representations were closed under “linear transformations.” I was confused by the discussion and at first I didn’t know why. On my walk home that day, I remembered I had heard those sorts of claims before. I typically think of a *linear transformations* as any mapping from a vector space to a vector space , both over a field , with the following properties of *linearity*:

- if is in the field then for , ;
- if , then .

For example, the equation is a linear transformation from the vector space of the set of reals back into itself. So, . When we speak of the algebra on in one dimension, is the underlying set for the vector space as well as the field.

Note that has the first property; suppose for example that and . Then

It also has the second property; for example, let and ; then

But clearly, this linear transformation does not preserve intervals:

I didn’t think I could be wrong about my understanding of the conventional use of the term “linear.” I thought maybe what people mean instead of “linear” in this context is that cardinal utility was closed under the class *affine transformations*. That is, the class containing all transformations of the following form (where is a scalar value in the field of and ):

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