So I understand that vectors are entities that exist in a vector space independently of of any basis we choose, therefore the magnitude of a vector will be a scalar. When studying non-inertial frames however this appears to break down. If there was a vector $ \mathbf{V} $ that pointed normal to the surface of the Earth which is rotating with an angular velocity $ \mathbf{\Omega} $, an observer stationary in the Earth frame would say
$$ \frac{d \mathbf{V}}{dt} = 0, $$
however a stationary observer external to Earth would say
$$ \frac{d \mathbf{V}}{dt} = \mathbf{\Omega} \times \mathbf{V}. $$
The magnitudes of these velocity vectors $\frac{d \mathbf{V}}{dt} $ disagree. If they are describing the same object in the same vector space how can this be?
However if the observers frames were rotated an angle $\theta$ w.r.t. each other, or translated w.r.t. each other, both observers would agree on $\frac{d \mathbf{V}}{dt} $. It seems that some frame transformations preserve scalars and others don't.
What is the difference between a frame of reference and a vector space, and why do they disagree sometimes and other times not?