I'm self-studying Artin's Algebra and I'm getting a bit thrown through a loop on isometries (in Chapter 6, in case you're curious). Specifically, a section about change of coordinates:
Let $P$ denote an $n$-dimensional space. The formula $t_a\phi$ for an isometry depends on our choice of coordinates [...]. To analyze the effect of such a change, we begin with an isometry $f$, a point $p$ of $P$, and its image $q=f(p)$, without reference to coordinates. When we introduce our coordinates, the space $P$ becomes identified with $\mathbb{R}^n$ and the points $p$ and $q$ have coordinates say [...].
I don't understand how we can have an isometry $f$ before we've chosen coordinates and identified the space with $\mathbb{R}^n$, we don't even have a distance to preserve under $f$. I completely understand how the formula for an isometry changes if we have the formula for a set of coordinates and want to express it in another, but I don't get how we can have an isometry existing without respect to any coordinates as the quote mentions (in this case - I get that we could have something more general like a metric space, but Artin's not talking about that here).