inertial frames - Are vectors truly independent of coordinate systems?

I have been told to think of vectors as existing independent of a coordinate system. This means that the magnitude of a vector should be independent of any coordinate system we choose. Galilean transformations of the form

$$ x' = x - vt $$

do not preserve the magnitude of the velocity vectors however. How is it possible to have a vector that has a different magnitude in a different coordinate system?

Answer

I have been told to think of vectors as existing independent of a coordinate system.

Yes. Because vectors represent physical facts. That thing is halfway between those two things. This other object is moving directly toward Toledo. And so on.

But not things like 'it's x-coordinate is +7 meters' which isn't just a fact about the thing but one that entangles the choice of origin and orientation of the axes into the description. There is no reason to expect a fact that depends on the origin to be independent of the origin. Nor a priori any reason to expect a fact that depends on the orientation of the axes to be independent of that orientations.

Which leads us to:

This means that the magnitude of a vector should be independent of any coordinate system we choose.

No. The physical fact represented by the vector remains the same, but the numeric values used top represent that fact depend on how you chose to measure them (and 'an agreement on how to measure positions' is a reasonable definition of a coordinate system).

Now there are numeric facts about quantities that are independent of certain transformation of coordinate systems. The magnitude of Cartesian vectors are invariant on rotations of the coordinate system. The magnitude of Lorentz vectors are independent of those rotations and of boosts. And so on. But the statement quoted here over generalizes that.

How is it possible to have a vector that has a different magnitude in a different coordinate system?

When positions are treated as vectors positions (which is often done in introductory courses) they are displacements from the origin. But that makes it explicit that changing the origin will change the vector that you use. A Galilean transform is one that represents a continuously changing origin, so positions and their derivatives may not be invariant on such transformations.