matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript). As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array that transforms under a change of basis matrix by . For the individual matrix entries, this transformation law has the form so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1). Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above: , where is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like can immediately be seen to be geometrically identical in all coordinate systems. Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components are given by . These components transform contravariantly, since The transformation law for an order tensor with p contravariant indices and q covariant indices is thus given as, Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type . The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or