# Affine tensor

An element of the tensor product of copies of an -dimensional vector space and copies of the dual vector space . Such a tensor is said to be of type , the number defining the valency, or degree, of the tensor. Having chosen a basis in , one defines an affine tensor of type with the aid of components which transform as a result of a change of basis according to the formula

where . It is usually said that the tensor components undergo a contravariant transformation with respect to the upper indices, and a covariant transformation with respect to the lower.