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I'm going through a stats intro and got puzzled by the concept of interval and ratio data. Celsius temps is an example of interval data that can not be multiplied and divided, because there is no true 0. Yet, a simple linear transformation changes Celsius into Kelvin, where 0 is defined. Kelvin temps are divided and multiplied all over physics, so clearly a ratio data.

To me it seems counterintuitive that a simple linear transformation allows for definition of a whole new mathematical operation. Is it just me and there is actually nothing strange about it, or am I missing something?

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Interesting question!

Firstly Celsius to Kelvin is not a linear transform in its mathematical sense, it's an affine transform.

To answer the question of why affine transform allows more operations like addition and multiplication, we can take a similar example from linear algebra.

Think of a plane (linear space) $H$ that does not go through the origin e.g. $\{x\in \mathbb R^n|x^Tc=b\neq0\}$,$c$ is a constant vector. Then obviously you cannot multiply / add vectors from $H$ -- thee result will be out of $H$, this is the same case as Celcius.

But doing an affine transform to each point $x'=x+\frac{bc}{\|c\|^2}$, $\{x'\in \mathbb R^n|x'^Tc=0\}$ this space become a linear subspace, and now you can multiply / add vectors from it.

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    $\begingroup$ To paraphrase what you are saying, I confused linear and affine. No linear transform would result in inclusion of 0. However, this particular affine transform has this property that it shifts temps scale to include 0. There is a bunch of other affine transforms that would not have the same effect, but this particular one does. Is that right? $\endgroup$ May 22 at 12:01
  • $\begingroup$ Yeah that is my understanding :D $\endgroup$ May 22 at 12:06

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