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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|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|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" ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,   "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,   "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,   "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1970–1975)  pp. 1–5</TD></TR></table>