Vector Transformation Law and Vector Spaces: Is it Abuse?

In summary: This is because in general relativity, vectors in the tangent space transform inversely to the way that the coordinates change, while covectors transform the same way. In summary, there is a difference in terminology between abstract vector spaces and tangent and cotangent spaces in differential geometry and general relativity. In these contexts, "vector" refers specifically to elements of tangent spaces, while "covector" refers to elements of cotangent spaces. Alternatively, these objects can be described as type-(1,0) and type-(0,1) tensors, or contravariant and covariant vectors.
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Rburto
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Typically an element of a vector space is called a vector, but Carroll's GR book repeatedly refers to elements of tangent spaces as "transforming as a vector" when they change coordinates as Vμ = ∂xμ/∂xν Vν. However, dual vectors are members of vector spaces (cotangent space) but obey ωμ = ∂xv/∂xμ ωv. Is this abuse of terminology? If so, what is a more exact way of describing objects in vector spaces obeying the vector transformation law?
 
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It's unfortunate that some words have different meanings in different branches of maths and physics.

Generally in linear algebra "vector" means an element of any kind of abstract vector space, but in differential geometry and its applications such as GR, "vector" has a more specialised meaning as a member of a tangent space. "Covector" is used to refer to a member of a cotangent space. If you want to avoid the word vector, you can describe tangent vectors as "type-(1,0) tensors" and cotangent covectors as "type-(0,1) tensors". Any type of tensor is technically a vector in the general linear algebra sense (is a member of a vector space), but in tensor theory it's (usually) only the type-(1,0) tensors that are actually called "vectors".

The other terminology you may see is "contravariant vectors" in the tangent space and "covariant vectors" in the cotangent (or dual) space.
 
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1. What is the vector transformation law?

The vector transformation law is a mathematical principle that states how vectors are affected by linear transformations. It states that a linear transformation of a vector can be represented by a matrix multiplication.

2. How is the vector transformation law used in vector spaces?

The vector transformation law is used to transform vectors in vector spaces. It allows us to perform operations on vectors, such as rotation, scaling, and reflection, which are essential in many areas of science, including physics, engineering, and computer graphics.

3. What is the difference between a vector transformation and a vector space transformation?

A vector transformation refers to the transformation of a vector itself, while a vector space transformation refers to the transformation of the entire vector space. In other words, a vector transformation is applied to individual vectors, while a vector space transformation is applied to all vectors within a space.

4. How do we know if the vector transformation law is being abused?

The vector transformation law can be abused if it is used in a way that does not follow the principles of linear transformations. This can happen if the matrix representing the transformation is not invertible, if the transformation is not linear, or if the transformation results in a vector that is not in the same vector space.

5. Can the vector transformation law be applied to non-numeric vectors?

Yes, the vector transformation law can be applied to non-numeric vectors as long as the vectors can still be represented as matrices. This includes geometric vectors, such as points in space, and abstract vectors, such as polynomials or functions.

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