# Is it possible that linear transform changes interval data to ratio?

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?

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.