Undergrad Coordinate transformation of a vector of magnitude zero

[Battlemage!]

Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude?

Since the formula for the magnitude of a vector is √(x₁²+x₂²+...x_n²), I can see no way for it to have magnitude zero unless every component is zero. Therefore it has to be the zero vector. Furthermore, since vectors are independent of the coordinate system they are in in Euclidean geometry, even if a coordinate transformations change coordinates, it seems to me a zero vector must contain the same coordinates of 0 all the way through.

But if that is true, is there some weird geometry where it doesn't hold? In which a vector of zero magnitude transforms to a vector whose magnitude is not zero?

 

[S.G. Janssens]

Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is zero. Therefore it has to be the zero vector.

Yes.

But if that is true, is there some weird geometry where it doesn't hold? In which a vector of zero magnitude transforms to a vector whose magnitude is not zero?

Every linear transformation maps the origin to itself, but what about translation (which is a non-linear but rather an affine transformation)?

 

[Battlemage!]

Yes.

Every linear transformation maps the origin to itself, but what about translation (which is a non-linear but rather an affine transformation)?

Ah! That should have been obvious. For context this question came about because I had thought I heard a math lab person say a zero vector must remain zero due to a linear transformation, but that it didn't hold in general. It is clear from what you just said that non-linear transformations do not.

Could this feature of an affine transformation be used to define it as such? I.e. an affine transformation is a transformation such that the origin does not necessarily map to itself?

 

[S.G. Janssens]

Could this feature of an affine transformation be used to define it as such? I.e. an affine transformation is a transformation such that the origin does not necessarily map to itself?

No. $T : \mathbb{R} \to \mathbb{R}$ defined by $T(x) = x + \tfrac{1}{2}\cos{x}$ is bijective and $T(0) = \tfrac{1}{2}$ but I don't think you would want to regard $T$ as an affine transformation.