Coordinate transformation of a vector of magnitude zero

In summary, the conversation discusses the concept of a coordinate transformation of a vector of magnitude zero and whether it can result in a non-zero vector. It is concluded that in Euclidean geometry, a zero vector must always remain zero after a coordinate transformation, but in non-linear transformations, this may not hold. The example of an affine transformation is also brought up, but it is determined that it cannot be defined solely based on the feature of the origin not mapping to itself.
  • #1
Battlemage!
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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+...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. 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?
 
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  • #2
Battlemage! said:
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.
Battlemage! said:
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)?
 
  • #3
Krylov said:
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?
 
  • #4
Battlemage! said:
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.
 

What is a coordinate transformation?

A coordinate transformation is a mathematical process that converts the coordinates of a vector from one coordinate system to another. This is commonly done for convenience, accuracy, or to align with a specific reference frame.

What is a vector of magnitude zero?

A vector of magnitude zero, also known as a zero vector, is a vector with a length of zero. This means that it has no direction and no magnitude, and is essentially a point in space. It is typically denoted by the symbol 0.

Why would a coordinate transformation be necessary for a vector of magnitude zero?

A coordinate transformation may be necessary for a vector of magnitude zero in order to accurately describe its location or motion in a specific coordinate system. It may also be necessary to align the vector with other objects in the same reference frame.

How is a coordinate transformation of a vector of magnitude zero performed?

A coordinate transformation of a vector of magnitude zero is typically performed using mathematical equations or matrices that relate the coordinates in one coordinate system to the coordinates in another. These equations or matrices take into account the relative orientations and scales of the two coordinate systems.

What are some real-world applications of coordinate transformations for vectors of magnitude zero?

Coordinate transformations for vectors of magnitude zero are used in many different fields, such as navigation, robotics, and computer graphics. For example, in navigation, a coordinate transformation may be used to convert the location of an object from GPS coordinates to a local coordinate system. In computer graphics, a coordinate transformation may be used to align an object with a specific point of view in a virtual scene.

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