Change of Variables and Their Corresponding Transformation

Question

If I have the change of variables $t = x + c$ and $s = y + b$, where $x$, $y$, $t$, and $s$ are variables, and $c$ and $b$ are constants, is this the transformation $(s, t) \to (x, y)$ (as opposed to $(x, y) \to (s, t))$? If so, lease explain why this is.

Answer 1

They way you written it is a transformation which given $x,y$ gives you $t,s$, so you can say $(x,y)\to (t,s)$, but it does not really say much about how the $t,s$ look like. More formally you can view $t$ as a function of $x$, i.e. $t:\mathbb{R}\to\mathbb{R}$ and $t(x)=x-c$ (assuming you work in reals), similar for $s(y)$. Even better, you can view it as a function $f:\mathbb{R^2}\to\mathbb{R^2}$ with $f(x,y)=(x+c,y+b)$ (which happens to be linear). This is also sometimes noted as $(x,y)\mapsto(x+c,y+b)$.

In this case you can also go the other way around since this transformation is invertible, so in terms of function above, we have $f^{-1}:\mathbb{R^2}\to\mathbb{R^2}$ where $f^{-1}(t,s)=(t-c,s-b)$, or $(t,s)\mapsto(t-c,s-b)$.

I don't know if this helps, since this mostly just about notation, but hopefully you can take something of it.