That’s active vs passive transformation and AFAIK mathematicians often think in passive ones but physicists would use both. And the one you quoted is passive.

> It's weird to say applying a function doesn't affect a change.

It does affect a change--from one representation into another representation. The underlying object it is representing stays the same, however, which I think is reasonable to emphasize.

> effect*

effect*

> > effect*

> affect*

I said I don't like the wording.

> Given a vector x expressed in the standard basis, we change coordinates by applying

The set of lines in Euclidean 2-space is a vector space. There is an isomorphism into R^3: choose an origin, choose x,y axes, and then ax + by + c = 0 gives tuples in R^3.

The line is x, the tuple in R^3 is t but the wording uses x when it should use t (which is never introduced). The wording conflates vectors with coordinate vectors.

I am sorry, but what you are saying is not correct at all. The set of lines is not a vector space. There is also no isomorphism onto R^3 in any sense, but rather onto RP^2 minus a point, as a manifold. I'm also not sure what this has to do with changing coordinates.

I say this not to argue, but to hopefully correct some misunderstanding.