Editing Einstein notation (section)

=== Superscripts and subscripts versus only subscripts ===

In terms of [[covariance and contravariance of vectors]],
* upper indices represent components of [[Covariance and contravariance of vectors|contravariant vectors]] ([[coordinate vector|vector]]s),
* lower indices represent components of [[covariant vector|covariant]] vectors ([[covector]]s).

They transform contravariantly or covariantly, respectively, with respect to [[change of basis]].

In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its ''components'', as in:
<math display="block">\begin{align}
v = v^i e_i = \begin{bmatrix} e_1 & e_2 & \cdots & e_n \end{bmatrix} \begin{bmatrix} v^1 \\ v^2 \\ \vdots \\ v^n \end{bmatrix} \\
w = w_i e^i = \begin{bmatrix} w_1 & w_2 & \cdots & w_n \end{bmatrix} \begin{bmatrix} e^1 \\ e^2 \\ \vdots \\ e^n \end{bmatrix}
\end{align}</math>

where <math> v </math> is the vector and <math> v^i </math> are its components (not the <math> i </math>th covector <math> v </math>), <math> w </math> is the covector and <math> w_i </math> are its components. The basis vector elements <math>e_i</math> are each column vectors, and the covector basis elements <math>e^i</math> are each row covectors. (See also {{slink|#Abstract description}}; [[dual basis|duality]], below and the [[Dual basis#Examples|examples]])

In the presence of a [[Degenerate bilinear form|non-degenerate form]] (an [[isomorphism]] {{math|''V'' → ''V''{{i sup|∗}}}}, for instance a [[Riemannian metric]] or [[Minkowski metric]]), one can [[raising and lowering indices|raise and lower indices]].

A basis gives such a form (via the [[dual basis]]), hence when working on {{math|'''R'''<sup>''n''</sup>}} with a [[Euclidean metric]] and a fixed [[orthonormal basis]], one has the option to work with only subscripts.

However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see [[Covariance and contravariance of vectors]].