If a vector is simply "any quantity having magnitude and

direction", then how can a vector's components NOT transform according to the rule of transformation for the components of a vector when we apply a transformation of the coordinate system? Or alternatively, given three numbers (for the 3D case), how can we say that they are NOT the components of a vector in a given coordinate system, if we admit that they transform according to the rule of transformation for vector components when we change the coordinate system? I'm not sure anymore if a vector is simply "any quantity having magnitude and direction", I mean, in a given x-y coordinate system (2D case), I can define a vector, say (3,5) - but it's clear that (3,5) are NOT the components of my vector in another coordinate system. Does that mean that my vector is not a vector? I think it just means that (3,5) are not the components of my vector. But then, how do I specify my vector independently of the coordinate system, for isn't a vector an entity that exists regardless of any coordinate system?

I know my question seems confusing. Please read my following analysis before answering:

Vector or not vector?