Vector Transformation Law and Vector Spaces: Is it Abuse?

[Rburto]

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 V^μ = ∂x^μ/∂x^ν V^ν. However, dual vectors are members of vector spaces (cotangent space) but obey ω_μ = ∂x^v/∂x^μ ω_v. Is this abuse of terminology? If so, what is a more exact way of describing objects in vector spaces obeying the vector transformation law?

 

[DrGreg]

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.