Pray tell, which dimension do we lose when we normalize, say a 2D vector?
Before normalization, the vector lies in R^n, which is an n-dimensional manifold.
After normalization, the vector lies in the unit sphere in R^n, which is an (n-1)-dimensional manifold.
>>> Magnitude is not a dimension [...] To prove this normalize any vector and then try to de-normalize it again.
Say you have the vector (18, -5) in a normal Euclidean x, y plane.
Now project that vector onto the y-axis.
Now try to un-project it again.
What do you think you just proved?
Pray tell, which dimension do we lose when we normalize, say a 2D vector?