versa for the inverse function: for all i, j. A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix L, the bases transform according to the matrix inverse L−1, and conversely if the coordinates transform according to inverse L−1, the bases transform according to the matrix L. The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are linearly transformed according to the following rules: {| class="wikitable" |- ! Vector elements ! Contravariant transformation law ! Covariant transformation law |- ! Coordinates | | |- ! Basis | | |- ! Any vector | | |- |} where Lij represents the entries of the transformation matrix (row number is i and column number is j) and (L−1)ik denotes the entries of the inverse matrix of the matrix Lik. If L is an orthogonal transformation (orthogonal matrix), the objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in which the norm of the vector x is preserved (and distances are preserved). The determinant of L is det(L) = ±1, which corresponds to two types of orthogonal transformation: (+1) for rotations and (−1) for improper rotations (including reflections). There are considerable algebraic simplifications, the matrix transpose is the inverse from the definition of an orthogonal transformation: From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indices, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponents. All indices will be lowered in the remainder of this article. One can determine the actual raised and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules. Exactly the same transformation rules apply to any vector a, not only the position vector. If its components ai do not transform according to the rules, a is not a vector. Despite the similarity between the expressions above, for the change of coordinates such as , and the action of a tensor on a vector like , L is not a tensor, but T is. In the change of coordinates, L is a matrix, used to relate two