In "Finite dimensional vector spaces" by Halmos (page 23) I read at some point:

The concept of dual space was defined without any reference to coordinate systems.

Also in an older thread here at math.stackexchange I read that:

Now remember that when a vector space is not given to you as $\mathbb{R}^n$, it doesn't normally have a canonical basis, so we don't have a canonical way to write its elements down via coordinates, and so we can't describe hyperplanes by explicit equations like above. (Or better, we can, but only after choosing coordinates, and this is not canonical.)

What I don't understand is what is meant by coordinates? Given the linear functional $l$ with the mapping $\ell:(x_1,\ldots,x_n) \mapsto a_1 x_1 + \cdots a_n x_n$, what are the coordinates here - the $x_{i}$ or the $a_{i}$? What exactly is meant by the phrase "coordinates" and "canonical" in this context? Could someone help me out with this?

Thanks!