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Suppose y=ax+b, where a and b are constants. This is to say that y is equal to x up to an affine transformation. Is there any notation to express the relationship between y and x that is similar to $\propto$ (the notation for proportional to)? Thanks!

ExcitedSnail
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    This function actually isn't linear in the sense that transformations are linear: consider that for any linear transformtion $T(x), 0T(x) = 0 \to T(0) = 0,$ but here $y(0) = b.$ – Stephen Donovan Dec 10 '21 at 02:16
  • @StephenDonovan Thanks! You are right, the one I asked about is called affine transformation, right? – ExcitedSnail Dec 10 '21 at 02:20
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    Affine transformations are often used is dimension $>1$. – markvs Dec 10 '21 at 02:42
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    You can transform any given point to any other given point by an affine transformation. So all points are equal up to affine transformation, so that's not an interesting enough notion to warrant a unique symbol. – Vercassivelaunos Dec 10 '21 at 05:48
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    @Vercassivelaunos The same applies to “proportional” if you apply it to individual points (except for $0$). But $y \propto x$ does not mean that the point $y$ is proportional to $x$ but rather that $x$ and $y$ are functions and $y(t) = \alpha x(t)$ for all $t$, where $\alpha$ is independent of $t$. The same concept can be applied to “up to an affine transformation” and it is not trivial in that case. Alas, I know of no special notation for this. – Eike Schulte Dec 10 '21 at 09:05
  • @markvs Thanks! Agree. I'm actually considering the relationship between two random variables, i.e., two measurable functions. – ExcitedSnail Dec 11 '21 at 14:14
  • @Vercassivelaunos Thanks! It's very helpful. Sorry for my unclear question, the y and x are actually random variables in my problem. So each of them is a function. – ExcitedSnail Dec 11 '21 at 14:16
  • @EikeSchulte Thank you! It's very helpful! – ExcitedSnail Dec 11 '21 at 14:17

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