# Vector Transformation Law and Vector Spaces

• I
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?

DrGreg