# Covariant vector

An element of the vector space $E ^ {*}$ dual to an $n$- dimensional vector space $E$, that is, a linear functional (linear form) on $E$. In the ordered pair $( E, E ^ {*} )$, an element of $E$ is called a contravariant vector. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.
The coordinate notation for a covariant vector is particularly simple if one chooses in $E$ and $E ^ {*}$ so-called dual bases $e _ {1} \dots e _ {n}$ in $E$ and $e ^ {1} \dots e ^ {n}$ in $E ^ {*}$, that is, bases such that $( e ^ {i} e _ {j} ) = \delta _ {j} ^ {i}$( where $\delta _ {j} ^ {i}$ is the Kronecker symbol); an arbitrary covariant vector $\omega \in E ^ {*}$ is then expressible in the form $\omega = f _ {i} e ^ {i}$( summation over $i$ from 1 to $n$), where $f _ {i}$ is the value of the linear form $\omega$ at the vector $e _ {i}$. On passing from dual bases $( e _ {i} )$ and $( e ^ {j} )$ to dual bases $( \overline{e}\; _ {i ^ \prime } )$ and $( \overline{e}\; {} ^ {j ^ \prime } )$ according to the formulas
$$\overline{e}\; _ {i ^ \prime } = \ p _ {i ^ \prime } ^ {i} e _ {i} ,\ \ \overline{e}\; {} ^ {j ^ \prime } = \ q _ {i} ^ {j ^ \prime } e ^ {i} ,\ \ p _ {k ^ \prime } ^ {i} q _ {j} ^ {k ^ \prime } = \ \delta _ {j} ^ {i} ,$$
the coordinates $x ^ {i}$ of the contravariant vector $x = x ^ {i} e _ {i}$ change according to the contravariant law $\overline{x}\; {} ^ {i ^ \prime } = q _ {i} ^ {i ^ \prime } x ^ {i}$, while the coordinates $f _ {i}$ of the covariant vector $\omega$ change according to the covariant law $\overline{f}\; _ {i ^ \prime } = p _ {i ^ \prime } ^ {i} f _ {i}$( i.e. they change in the same way as the basis, whence the terminology "covariant vectorcovariant" ).