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Magnitude is not a dimension, it’s information about each value that is lost when you normalize it. To prove this normalize any vector and then try to de-normal
by extasia 2y ago
Magnitude is not a dimension, it’s information about each value that is lost when you normalize it. To prove this normalize any vector and then try to de-normalize it again.
- immibis 2y agoImpossible because... you lost a dimension.
- extasia 2y agoThat’s not mathematically accurate though, is it? We haven’t reduced the dimension of the vector by one. Pray tell, which dimension do we lose when we normalize, say a 2D vector?
- deleted 2y ago[deleted]
- thaumasiotes 2y agoMagnitude, obviously. >>> 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?
- creata 2y agoMathematically, it's fine to say that you've lost the magnitude dimension. 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.
- langcss 2y agoA circle circumference is a line, is 1D?
- kccqzy 2y agoMagnitude is a dimension. Any 2-dimensional vector can be explicitly transformed into the polar (r, theta) coordinate system where one of the dimensions is magnitude. Any 3-dimensional vector can be transformed into the spherical (r, theta, phi) coordinate where one of the dimensions is magnitude. This is high school mathematics. (Okay I concede that maybe the spherical coordinate system isn't exactly high school material, then just think about longitude, latitude, and distance from the center.)
- ekianjo 2y agoyou dont lose anything when you normalize things. not sure what you are tallking about.