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A fundamental operation in the [[Exterior algebra|exterior algebra]] of tensors defined on an  $  n $-
dimensional vector space  $  V $
over a field  $  K $.

Let  $  e _ {1} \dots e _ {n} $
be a basis of  $  V $, 
and let  $  a $
and  $  b $
be  $  p $- 
and  $  q $-
forms:

$$ 
a  =  a ^ {i _ {1} \dots {i _ {p} } }
e _ {i _ {1}  }  \otimes \dots \otimes  e _ {i _ {p}  } ,
$$

$$ 
b  =  b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1}  }  \otimes \dots \otimes  e _ {j _ {q}  } .
$$

The exterior product of the forms  $  a $
and  $  b $
is the  $  ( p + q) $-
form  $  c $
obtained by [[Alternation|alternation]] of the tensor product  $  a \otimes b $. 
The form  $  c $
is denoted by  $  a \wedge b $; 
its coordinates are skew-symmetric:

$$ 
c ^ {k _ {1} \dots k _ {p+ q }  }  = \ 

\frac{1}{p! q! }

\delta _ {i _ {1}  \dots i _ {p} j _ {1} \dots j _ {q} } ^ {k _ {1} \dots \dots \dots k _ {p+ q }  }
a ^ {i _ {1} \dots i _ {p} } b ^ {j _ {1} \dots j _ {q} } ,
$$

where  $  \delta _ {i _ {1}  \dots j _ {q} } ^ {k _ {1} \dots k _ {p+} q } $
are the components of the generalized [[Kronecker symbol|Kronecker symbol]]. The exterior product of covariant tensors is defined in a similar manner.

The basic properties of the exterior product are listed below:

1)  $  ( ka) \wedge b = a \wedge ( kb) = k( a \wedge b) $, 
$  k \in K $(
homogeneity);

2)  $  ( a+ b) \wedge c = a \wedge c + b \wedge c $(
distributivity);

3)  $  ( a \wedge b ) \wedge c = a \wedge ( b \wedge c) $(
associativity).

4)  $  a \wedge b = (- 1)  ^ {pq} b \wedge a $; 
if the characteristic of  $  K $
is distinct from two, the equation  $  a \wedge a = 0 $
is valid for any form  $  a $
of odd valency.

The exterior product of  $  s $
vectors is said to be a decomposable  $  s $-
vector. Any [[Poly-vector|poly-vector]] of dimension  $  s $
is a linear combination of decomposable  $  s $-
vectors. The components of this combination are the ( $  s \times s $)-
minors of the ( $  n \times s $)-
matrix  $  ( a _ {j}  ^ {i} ) $, 
$  1 \leq  i \leq  n $, 
$  1 \leq  j \leq  s $, 
of the coefficients of the vectors  $  a _ {1} \dots a _ {s} $. 
If  $  s = n $
their exterior product has the form

$$ 
\alpha _ {n}  =  a _ {1} \wedge \dots \wedge a _ {n}  = \ 
 \mathop{\rm det} ( a _ {j}  ^ {i} )  e _ {1} \wedge \dots \wedge e _ {n} .
$$

Over fields of characteristic distinct from two, the equation  $  a _ {1} \wedge \dots \wedge a _ {n} = 0 $
is necessary and sufficient for vectors  $  a _ {1} \dots a _ {n} $
to be linearly dependent. A non-zero decomposable  $  s $-
vector  $  \alpha _ {s} $
defines in  $  V $
an  $  s $-
dimensional oriented subspace  $  A $, 
parallel to the vectors  $  a _ {1} \dots a _ {s} $, 
and the [[Parallelotope|parallelotope]] in  $  A $
formed by the vectors  $  a _ {1} \dots a _ {s} $
issuing from one point, denoted by  $  [ a _ {1} \dots a _ {s} ] $. 
The conditions  $  a \in A $
and  $  \alpha _ {s} \wedge a = 0 $
are equivalent.

For references see [[Exterior algebra|Exterior algebra]].

====Comments====
Instead of exterior product the phrase  "outer product"  is sometimes used. The condition  $  a \wedge b = (- 1)  ^ {pq} b \wedge a $
for  $  a $
of degree  $  p $
and  $  b $
of degree  $  q $
is sometimes called graded commutativity.