We have a transformation of projective 2-space, which my lecture notes assert map the point at infinity is mapped to $0$. Whilst I intuitively understand why this is the case, I am not sure how to formalize this.
My difficulty stems from the fact that the transformation is given in terms of affine coordinate. Specifically, we are working over $\mathbb{P}^2(\mathbb{R})$, and the birational transformation (in affine coordinates) is given by
$$ x' = -\frac{x}y, \quad y' = -\frac1y $$
My lecture notes asserts that this maps the projective point at infinity, $(0;1;0)$ to the affine origin. Similarly, we would expect the affine point $(a, 0)$ to be mapped to the point at infinity, $(a; 1; 0)$. Whilst I can intuitively see why this is the case (at the level of "$1/\infty = 0$"), I want to formalize this. However, I'm not sure how to convert the above transformation into one involving projective coordinates so that I can show this algebraically.
What I have tried so far: (Disclaimer, I am studying this as part of a course on elliptic curves, but have not studied any projective geometry on its own prior to this.)
It seems reasonable to replace $x$ with $X/Z$ and $x'$ with $X'/Z'$ in the equations for the transformations above (and similarly for $y$ and $y'$). Doing this and then rearranging gives
$$ X'Y = -XZ', \quad YY' = - ZZ' $$
This seems to show that $(0;1;0)$ maps to $(0;0;1)$, but doesn't seem to give a well defined map if $Y=0$ (e.g. for the point $(a;0;1)$, all we can deduce is that $Z' = 0$; on the other hand, intuitively, this point should map to $(a; 1; 0)$).
Is this is the right projective transformation to extend the affine transformation given above? Is this even the right way to go about it?