In principle, thermometers made of different material ( e . g ., coloured alcohol thermometers ) might be expected to give different intermediate readings due to different expansion properties; in practice the substances used are chosen to have reasonably linear expansion characteristics as a function of true thermodynamic temperature, and so give similar results.
22.
If you're worried about the change of basis, just note that if PAP ^ {-1 } is the change-of-basis-ed diagonal matrix above, then Pe _ ke _ k ^ TP ^ {-1 } = v _ kv _ K ^ T, so the linear expansion you wrote down is preserved by changes of basis.
