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'' $  V $
over a field  $  k $''

An element  $  t $
of the vector space

$$ 
T  ^ {p,q} ( V)  = \ 
\left ( \otimes ^ { p }  V \right )
 \otimes \ 
\left ( \otimes ^ { q }  V  ^ {*} \right ) ,
$$

where  $  V  ^ {*} =  \mathop{\rm Hom}  ( V, k) $
is the dual space of  $  V $. 
The tensor  $  t $
is said to be  $  p $
times contravariant and  $  q $
times covariant, or to be of type  $  ( p, q) $. 
The number  $  p $
is called the contravariant valency, and  $  q $
the covariant valency, while the number  $  p + q $
is called the general valency of the tensor  $  t $. 
The space  $  T  ^ {0,0} ( V) $
is identified with  $  k $. 
Tensors of type  $  ( p, 0) $
are called contravariant, those of the type  $  ( 0, q) $
are called covariant, and the remaining ones are called mixed.

Examples of tensors.

1) A vector of the space  $  V $(
a tensor of type  $  ( 1, 0) $).

2) A covector of the space  $  V $(
a tensor of type  $  ( 0, 1) $).

3) Any covariant tensor

$$ 
t  = \ 
\sum _ {i = 1 } ^ { s } 
h _ {i1} \otimes \dots \otimes h _ {iq} ,
$$

where  $  h _ {ij} \in V  ^ {*} $, 
defines a  $  q $-
linear form  $  \widehat{t}  $
on  $  V $
by the formula

$$ 
\widehat{t}  ( x _ {1} \dots x _ {q} )  = \ 
\sum _ {i = 1 } ^ { s } 
h _ {i1} ( x _ {1} ) \dots
h _ {iq} ( x _ {q} );
$$

the mapping  $  t \mapsto \widehat{t}  $
from the space  $  T  ^ {0,q} $
into the space  $  L  ^ {q} ( V) $
of all  $  q $-
linear forms on  $  V $
is linear and injective; if  $   \mathop{\rm dim}  V < \infty $, 
then this mapping is an isomorphism, since any  $  q $-
linear form corresponds to some tensor of type  $  ( 0, q) $.

4) Similarly, a contravariant tensor in  $  T  ^ {p,0} ( V) $
defines a  $  p $-
linear form on  $  V  ^ {*} $, 
and if  $  V $
is finite dimensional, the converse is also true.

5) Every tensor

$$ 
t  = \ 
\sum _ {i = 1 } ^ { s } 
x _ {i} \otimes h _ {i} \ 
\in  T  ^ {1,1} ( V),
$$

where  $  x _ {i} \in V $
and  $  h _ {j} \in V  ^ {*} $, 
defines a linear transformation  $  \widehat{t}  $
of the space  $  V $
given by the formula

$$ 
\widehat{t}  ( y)  = \ 
\sum _ {i = 1 } ^ { s } 
h _ {i} ( y) x _ {i} ;
$$

if  $   \mathop{\rm dim}  V < \infty $, 
any linear transformation of the space  $  V $
is defined by a tensor of type  $  ( 1, 1) $.

6) Similarly, any tensor of type  $  ( 1, 2) $
defines in  $  V $
a bilinear operation, that is, the structure of a  $  k $-
algebra. Moreover, if  $   \mathop{\rm dim}  V < \infty $, 
then any  $  k $-
algebra structure in  $  V $
is defined by a tensor of type  $  ( 1, 2) $, 
called the structure tensor of the algebra.

Let  $  V $
be finite dimensional, let  $  v _ {1} \dots v _ {n} $
be a basis of it, and let  $  v  ^ {1} \dots v  ^ {n} $
be the dual basis of the space  $  V  ^ {*} $. 
Then the tensors

$$ 
v _ {i _ {1}  \dots i _ {p} } ^ {i _ {1} \dots i _ {q} }  = \ 
v _ {i _ {1}  } \otimes \dots \otimes v _ {i _ {p}  } \otimes
v ^ {j _ {1} } \otimes \dots \otimes v ^ {j _ {q} }
$$

form a basis of the space  $  T  ^ {p,q} ( V) $. 
The components  $  t _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $
of a tensor  $  t \in T  ^ {p,q} ( V) $
with respect to this basis are also called the components of the tensor  $  t $
with respect to the basis  $  v _ {1} \dots v _ {n} $
of the space  $  V $. 
For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases  $  ( v _ {i} ) $
and  $  ( v  ^ {j} ) $; 
the components of a tensor of type  $  ( 0, 2) $
coincide with the entries of the matrix corresponding to the bilinear form; the components of a tensor of type  $  ( 1, 1) $
coincide with the entries of the matrix of the corresponding linear transformation, and the components of the structure tensor of an algebra coincide with its structure constants. If  $  \widetilde{v}  _ {1} \dots \widetilde{v}  _ {n} $
is another basis of  $  V $, 
with  $  \widetilde{v}  _ {j} = a _ {j}  ^ {i} v _ {i} $, 
and  $  \| b _ {j}  ^ {i} \| = ( \| a _ {j}  ^ {i} \|  ^ {T} )  ^ {-} 1 $, 
then the components  $  t _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $
of the tensor  $  t $
in this basis are defined by the formula

$$ \tag{1 }
\widetilde{t}  {} _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots j _ {p} }  = \ 
b _ {k _ {1}  } ^ {i _ {1} } \dots
b _ {k _ {p}  } ^ {i _ {p} }
a _ {j _ {1}  } ^ {l _ {1} } \dots
a _ {j _ {q}  } ^ {l _ {q} }
t _ {l _ {1}  \dots l _ {q} } ^ {k _ {1} \dots k _ {p} } .
$$

Here, as often happens in tensor calculus, Einstein's [[summation convention]] is applied: with respect to any pair of equal indices of which one is an upper index and the other is a lower index, it is understood that summation from 1 to  $  n $
is carried out. Conversely, if a system of  $  n ^ {p + q } $
elements of a field  $  k $
depending on the basis of the space  $  V $
is altered in the transition from one basis to another basis according to the formulas (1), then this system is the set of components of some tensor of type  $  ( p, q) $.

In the vector space  $  T  ^ {p,q} ( V) $
the operations of addition of tensors and of multiplication of a tensor by a scalar from  $  k $
are defined. Under these operations the corresponding components are added, or multiplied by the scalar. The operation of multiplying tensors of different types is also defined; it is introduced as follows. There is a natural isomorphism of vector spaces

$$ 
T  ^ {p,q} ( V) \otimes
T  ^ {r,s} ( V)  \cong \ 
T ^ {p + r, q + s } ( V),
$$

mapping

$$ 
( x _ {1} \otimes \dots \otimes
x _ {p} \otimes h _ {1} \otimes
{} \dots \otimes h _ {q} ) \otimes
$$

$$ 
\otimes 
( x _ {1}  ^  \prime  \otimes \dots
\otimes x _ {r}  ^  \prime  \otimes
h _ {1}  ^  \prime  \otimes \dots \otimes h _ {s} )
$$

to

$$ 
x _ {1} \otimes \dots \otimes
x _ {p} \otimes
x _ {1}  ^  \prime  \otimes \dots
\otimes x _ {r}  ^  \prime  \otimes
$$

$$ 
\otimes 
h _ {1} \otimes \dots
\otimes h _ {q} \otimes
h _ {1}  ^  \prime  \otimes \dots
\otimes h _ {s}  ^  \prime  .
$$

Consequently, for any  $  t \in T  ^ {p,q} ( V) $
and  $  u \in T  ^ {r,s} ( V) $
the element  $  v = t \otimes u $
can be regarded as a tensor of type  $  ( p + r, q + s) $
and is called the tensor product of  $  t $
and  $  u $. 
The components of the product are computed according to the formula

$$ 
v _ {j _ {1}  \dots j _ {q + s }  } ^ {i _ {1} \dots i _ {p + r }  }  = \ 
t _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }
u _ {j _ {q + 1 }  \dots j _ {q + s }  } ^ {i _ {p + 1 }  \dots i _ {p + r }  } .
$$

Let  $  p > 0 $, 
$  q > 0 $, 
and let the numbers  $  \alpha $
and  $  \beta $
be fixed with  $  1 \leq  \alpha \leq  p $
and  $  1 \leq  \beta \leq  q $. 
Then there is a well-defined mapping  $  Y _  \beta   ^  \alpha  :  T  ^ {p,q} ( V) \rightarrow T ^ {p - 1, q - 1 } ( V) $
such that

$$ 
Y _  \beta   ^  \alpha 
( x _ {1} \otimes \dots \otimes x _ {p} \otimes
h _ {1} \otimes \dots \otimes h _ {q} ) =
$$

$$ 
= \ 
h _  \beta  ( x _  \alpha  ) x _ {1} \otimes \dots
\otimes x _ {\alpha - 1 } 
\otimes x _ {\alpha + 1 } 
\otimes \dots \otimes
x _ {p} \otimes
$$

$$ 
\otimes 
h _ {1} \otimes \dots \otimes
h _ {\beta - 1 } 
\otimes h _ {\beta + 1 } 
\otimes \dots \otimes h _ {q} .
$$

It is called contraction in the  $  \alpha $-
th contravariant and the  $  \beta $-
th covariant indices. In components, the contraction is written in the form

$$ 
( Y _  \beta   ^  \alpha  t) _ {j _ {1}  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p - 1 }  }  = \ 
t _ {j _ {1}  \dots j _ {\beta - 1 } 
ij _ {\beta + 1 }  \dots j _ {q} } ^ {i _ {1} \dots i _ {\alpha - 1 } 
ii _ {\alpha + 1 }  \dots i _ {p} } .
$$

For instance, the contraction  $  Y _ {1}  ^ {1} t $
of a tensor of type  $  ( 1, 1) $
is the trace of the corresponding linear transformation.

A tensor is similarly defined on an arbitrary unitary module  $  V $
over an associative commutative ring with a unit. The stated examples and properties of tensors are transferred, with corresponding changes, to this case, it being sometimes necessary to assume that  $  V $
is a free or a finitely-generated free module.

Let a non-degenerate bilinear form  $  g $
be fixed in a finite-dimensional vector space  $  V $
over a field  $  k $(
for example,  $  V $
is a Euclidean or pseudo-Euclidean space over  $  \mathbf R $); 
in this case the form  $  g $
is called a metric tensor. A metric tensor defines an isomorphism  $  \gamma :  V \rightarrow V  ^ {*} $
by the formula

$$ 
\gamma ( x) ( y)  =  g ( x, y),\ \ 
x, y \in V.
$$

Let  $  p > 0 $, 
and let the index  $  \alpha $, 
$  1 \leq  \alpha \leq  p $, 
be fixed. Then the formula

$$ 
x _ {1} \otimes \dots \otimes
x _ {p} \otimes
h _ {1} \otimes \dots \otimes
h _ {q\ } \mapsto
$$

$$ 
\mapsto \ 
x _ {1} \otimes \dots \otimes
x _ {\alpha - 1 }  \otimes x _ {\alpha + 1 }  \otimes \dots
\otimes x _ {p} \otimes
$$

$$ 
\otimes 
\gamma ( x _  \alpha  )
\otimes h _ {1} \otimes \dots
\otimes h _ {q}  $$

defines an isomorphism  $  \gamma  ^  \alpha  :  T  ^ {p,q} ( V) \rightarrow T ^ {p - 1, q + 1 } ( V) $, 
called lowering of the  $  \alpha $-
th contravariant index. In other terms,

$$ 
\gamma  ^  \alpha  ( t)  = \ 
Y _ {1}  ^  \alpha 
( g \otimes t).
$$

In components, lowering an index has the form

$$ 
\gamma  ^  \alpha  ( t) _ {j _ {1}  \dots j _ {q + 1 }  } ^ {i _ {1} \dots i _ {q - 1 }  }  = \ 
g _ { ij _ 1 } t _ {j _ {2}  \dots j _ {q + 1 }  } ^ {i _ {1} \dots i _ {\alpha - 1 } 
ii _ {\alpha + 1 }  \dots i _ {p - 1 }  } .
$$

Similarly one defines the isomorphism of raising the  $  \beta $-
th covariant index  $  ( 1 \leq  \beta \leq  q) $:

$$ 
\gamma _  \beta  : \ 
x _ {1} \otimes \dots \otimes
x _ {p} \otimes
h _ {1} \otimes \dots \otimes
h _ {q\ } \mapsto
$$

$$ 
\mapsto \ 
x _ {1} \otimes \dots \otimes
x _ {p} \otimes \gamma  ^ {-} 1 ( h _  \beta  ) \otimes
$$

$$ 
\otimes 
h _ {1} \otimes \dots \otimes
h _ {\beta - 1 }  \otimes h _ {\beta + 1 } 
\otimes \dots \otimes h _ {q} ,
$$

which maps  $  T  ^ {p,q} ( V) $
onto  $  T ^ {p + 1, q - 1 } ( V) $. 
In components, raising an index is written in the form

$$ 
\gamma _  \beta  ( t) _ {j _ {1}  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p + 1 }  }  = \ 
g ^ {ji _ {p + 1 }  }
t _ {j _ {1}  \dots j _ {\beta - 1 } 
ij _  \beta  \dots j _ {q - 1 }  } ^ {i _ {1} \dots i _ {p} } ,
$$

where  $  \| g  ^ {kl} \| = (\| g _ {ij} \|  ^ {T} )  ^ {-} 1 $. 
In particular, raising at first the first, and then also the remaining covariant index of the metric tensor  $  g $
leads to a tensor of type  $  ( 2, 0) $
with components  $  g  ^ {kl} $(
a contravariant metric tensor). Sometimes the lowered (raised) index is not moved to the first (last) place, but is written in the same place in the lower (upper) group of indices, a point being put in the empty place which arises. For instance, for  $  t \in T  ^ {2,0} ( V) $
the components of the tensor  $  \gamma  ^ {2} ( t) $
are written in the form  $  t _ {j} ^ {i. } = g _ {kj} t  ^ {ik} $.

Any linear mapping  $  f:  V \rightarrow W $
of vector spaces over  $  k $
defines in a natural way linear mappings

$$ 
T  ^ {p,0} ( f  )  = \ 
\otimes ^ { p }  f: \ 
T  ^ {p,0} ( V)  \rightarrow  T  ^ {p,0} ( W)
$$

and

$$ 
T  ^ {q,0} ( f ^ { * } )  = \ 
\otimes ^ { q }  f ^ { * } : \ 
T  ^ {0,q} ( W)  \rightarrow  T  ^ {0,q} ( V).
$$

If  $  f $
is an isomorphism, the linear mapping

$$ 
T  ^ {p,q} ( f  ): \ 
T  ^ {p,q} ( V)  \rightarrow  T  ^ {p,q} ( W)
$$

is also defined and  $  T  ^ {0,q} ( f  ) = T  ^ {q,0} ( f ^ { * } )  ^ {-} 1 $. 
The correspondence  $  f \mapsto T  ^ {p,q} ( f  ) $
has functorial properties. In particular, it defines a linear representation  $  a \mapsto T  ^ {p,q} ( a) $
of the group  $   \mathop{\rm GL} ( V) $
in the space  $  T  ^ {p,q} ( V) $(
the tensor representation).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley  (1974)  pp. Chapt.1;2  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   "Lectures on linear algebra" , Interscience  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,   Yu.I. Manin,   "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Postnikov,   "Linear algebra and differential geometry" , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>

{{SourceattributionEOM|Tensor on a vector space}}
[[Category:Mathematics]]