from to , or also by . Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while and can both be expressed as n-by-n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together. The total number of indices () required to identify each component uniquely is equal to the dimension or the number of ways of an array, which is why an array is sometimes referred to as an -dimensional array or an -way array. The total number of indices is also called the order, degree, rank or modes of a tensor, although the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors), where the new basis vectors are expressed in terms of the old basis vectors as, Here R ji are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components vi of a column vector v transform with the inverse of the matrix R, where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, wi, of a covector (or row vector), w, transform with the matrix R itself, This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation