In a basis of vector fields , any smooth tangent vector field can be written in the form

for some uniquely determined smooth functions . Upon changing the basis by a nonsingular matrix , the coefficients change in such a way that equation () remains true. That is,

Consequently, . In other words, the components of a vector transform "contravariantly" (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix . The contravariance of the components of is notationally designated by placing the indices of in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields define the dual basis to be the linear functionals such that

That is, , the Kronecker delta. Let

Under a change of basis for a nonsingular matrix , transforms via

Any linear functional on tangent vectors can be expanded in terms of the dual basis 

where denotes the row vector . The components transform when the basis is replaced by in such a way that equation () continues to hold. That is,

whence, because , it follows that . That is, the components transform "covariantly" (by the matrix rather than its inverse). The covariance of the components of is notationally designated by placing the indices of in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding fixed, the function

of tangent vector defines a linear functional on the tangent space at . This operation takes a vector at a point and produces a covector . In a basis of vector fields , if a vector field has components , then the components of the covector field in the dual basis are given by the entries of the row vector

Under a change of basis , the right-hand side of this equation transforms via

so that : transforms covariantly. The operation of associating to the (contravariant) components of a vector field the (covariant) components of the covector field , where

is called lowering the index.

To "raise the index", one applies the same construction but with the inverse metric instead of the metric. If are the components of a covector in the dual basis , then the column vector

has components which transform contravariantly:

Consequently, the quantity does not depend on the choice of basis in an essential way, and thus defines a vector field on . The operation () associating to the (covariant) components of a covector the (contravariant) components of a vector given is called raising the index. In components, () is

Induced metric
Raising and lowering indices
দৈর্ঘ এবং কোণ
Signature of a metric