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.
> 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 say this not to argue, but to hopefully correct some misunderstanding.
effect*
> affect*