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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735302.png" />-vector, over a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735303.png" />''
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An element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735304.png" />-th exterior degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735305.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735306.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735307.png" /> (see [[Exterior algebra|Exterior algebra]]). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735308.png" />-vector can be understood as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p0735309.png" />-times skew-symmetrized contravariant tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353010.png" />. Any linearly independent system of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353011.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353012.png" /> defines a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353013.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353014.png" />; such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353016.png" /> generate the same subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353017.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353019.png" />. For any non-zero poly-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353020.png" />, its annihilator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353021.png" /> is a subspace of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353022.png" />, and the poly-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353023.png" /> is pure if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353024.png" />. The pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353025.png" />-vectors of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353026.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353027.png" /> form an algebraic variety in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353028.png" />; the corresponding projective algebraic variety is a [[Grassmann manifold|Grassmann manifold]]. Any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353029.png" />-vector or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353030.png" />-vector in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353031.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353032.png" /> is pure, but a bivector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353033.png" /> is pure if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353034.png" />.
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{{TEX|done}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353035.png" /> is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353037.png" />, then the coordinates of the poly-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353038.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353039.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353040.png" /> are the minors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353042.png" />, of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353043.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353044.png" />,
+
'' $  p $-
 +
vector, over a vector space $  V $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353045.png" /></td> </tr></table>
+
An element of the  $  p $-
 +
th exterior degree  $  \wedge  ^ {p} V $
 +
of the space  $  V $
 +
over a field  $  K $(
 +
see [[Exterior algebra|Exterior algebra]]). A  $  p $-
 +
vector can be understood as a  $  p $-
 +
times skew-symmetrized contravariant tensor on  $  V $.
 +
Any linearly independent system of vectors  $  x _ {1} \dots x _ {p} $
 +
from  $  V $
 +
defines a non-zero  $  p $-
 +
vector  $  x _ {1} \wedge \dots \wedge x _ {p} $;
 +
such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems  $  x _ {1} \dots x _ {p} $
 +
and  $  y _ {1} \dots y _ {p} $
 +
generate the same subspace in  $  V $
 +
if and only if  $  y _ {1} \wedge \dots \wedge y _ {p} = cx _ {1} \wedge \dots \wedge x _ {p} $,
 +
where  $  c \in K $.
 +
For any non-zero poly-vector  $  t \in \wedge  ^ {p} V $,
 +
its annihilator  $  \mathop{\rm Ann}  t = \{ {v \in V } : {t \wedge v = 0 } \} $
 +
is a subspace of dimension  $  \leq  p $,
 +
and the poly-vector  $  t $
 +
is pure if and only if  $  \mathop{\rm dim}  \mathop{\rm Ann}  t = p $.
 +
The pure  $  p $-
 +
vectors of an  $  n $-
 +
dimensional space  $  V $
 +
form an algebraic variety in  $  \wedge  ^ {p} V $;  
 +
the corresponding projective algebraic variety is a [[Grassmann manifold|Grassmann manifold]]. Any non-zero  $  n $-
 +
vector or  $  ( n- 1) $-
 +
vector in an  $  n $-
 +
dimensional space  $  V $
 +
is pure, but a bivector  $  t $
 +
is pure if and only if  $  t \wedge t = 0 $.
  
If one specifies a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353046.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353047.png" />, a duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353048.png" />-vectors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353049.png" />-vectors is obtained, i.e. a natural isomorphism
+
If $  v _ {1} \dots v _ {n} $
 +
is a basis of  $  V $
 +
and  $  x _ {i} = \sum_{j=1} ^ {n} x _ {i}  ^ {j} v _ {j} $,
 +
then the coordinates of the poly-vector $  t = x _ {1} \wedge \dots \wedge x _ {p} $
 +
in the basis  $  \{ {v _ {i _ {1}  } \wedge \dots \wedge v _ {i _ {p}  } } : {i _ {1} < \dots < i _ {p} } \} $
 +
of the space  $  \wedge  ^ {p} V $
 +
are the minors  $  t ^ {i _ {1} \dots i _ {p} } =  \mathop{\rm det}  \| x _ {i} ^ {i _ {k} } \| $,  
 +
$  i _ {1} < \dots < i _ {p} $,  
 +
of the matrix  $  \| x _ {i}  ^ {j} \| $.  
 +
In particular, for  $  p = n $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353050.png" /></td> </tr></table>
+
$$
 +
x _ {1} \wedge \dots \wedge x _ {n}  = \
 +
\mathop{\rm det}  \| x _ {i}  ^ {j} \|  v _ {1} \wedge \dots \wedge v _ {n} .
 +
$$
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353051.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353053.png" />.
+
If one specifies a non-zero  $  n $-
 +
vector  $  \omega \in \wedge  ^ {n} V $,
 +
a duality between  $  p $-
 +
vectors and $  ( n- p) $-
 +
vectors is obtained, i.e. a natural isomorphism
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353054.png" /> and let an inner product be defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353055.png" />, then in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353056.png" /> an inner product is induced with the following property: For any orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353058.png" /> the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353060.png" /> is also orthonormal. The scalar square
+
$$
 +
\pi : \wedge  ^ {p} ( V)  \rightarrow \
 +
( \wedge  ^ {n-p} V)  ^ {*}  \cong  \wedge  ^ {n-p} ( V  ^ {*} )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353061.png" /></td> </tr></table>
+
such that  $  t \wedge u = \pi ( t)( u) \omega $
 +
for all  $  t \in \wedge  ^ {p} V $
 +
and  $  u \in \wedge  ^ {n-p} V $.
  
of a pure poly-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353062.png" /> coincides with the square of the volume of the parallelopipedon in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353063.png" /> constructed on the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353064.png" />. If one specifies an orientation in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353065.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353066.png" /> (which is equivalent to choosing an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353067.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353068.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353069.png" />), then the above duality leads to a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353070.png" />. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353071.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353072.png" /> corresponds to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353073.png" />, called the vector product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073530/p07353074.png" />.
+
Let  $  k = \mathbf R $
 +
and let an inner product be defined in  $  V $,
 +
then in  $  \wedge  ^ {p} V $
 +
an inner product is induced with the following property: For any orthonormal basis  $  v _ {1} \dots v _ {n} $
 +
in  $  V $
 +
the basis  $  \{ {v _ {i _ {1}  } \wedge \dots \wedge v _ {i _ {p}  } } : {i _ {1} < \dots < i _ {p} } \} $
 +
in  $  \wedge  ^ {p} V $
 +
is also orthonormal. The scalar square
 +
 
 +
$$
 +
( t, t)  =  \sum _ {i _ {1} < \dots < i _ {p} }
 +
( t ^ {i _ {1} \dots i _ {p} } )  ^ {2}
 +
$$
 +
 
 +
of a pure poly-vector $  t = x _ {1} \wedge \dots \wedge x _ {p} $
 +
coincides with the square of the volume of the parallelopipedon in $  V $
 +
constructed on the vectors $  x _ {1} \dots x _ {p} $.  
 +
If one specifies an orientation in the $  n $-
 +
dimensional Euclidean space $  V $(
 +
which is equivalent to choosing an $  n $-
 +
vector $  \omega $
 +
for which $  ( \omega , \omega ) = 1 $),  
 +
then the above duality leads to a natural isomorphism $  \gamma : \wedge  ^ {p} V \rightarrow \wedge  ^ {n-p} V $.  
 +
In particular, the $  ( n- 1) $-
 +
vector $  t = x _ {1} \wedge \dots \wedge x _ {n-1} $
 +
corresponds to a vector $  \gamma ( t) \in V $,  
 +
called the vector product of the vectors $  x _ {1} \dots x _ {n-1} $.
  
 
====References====
 
====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">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "Linear algebra and differential geometry" , Moscow  (1979)  (In Russian)</TD></TR></table>
+
<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">  A.I. Kostrikin,  Yu.I. Manin,  "Linear algebra and geometry" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,  "Linear algebra and differential geometry" , Moscow  (1979)  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 18:59, 9 January 2024


$ p $- vector, over a vector space $ V $

An element of the $ p $- th exterior degree $ \wedge ^ {p} V $ of the space $ V $ over a field $ K $( see Exterior algebra). A $ p $- vector can be understood as a $ p $- times skew-symmetrized contravariant tensor on $ V $. Any linearly independent system of vectors $ x _ {1} \dots x _ {p} $ from $ V $ defines a non-zero $ p $- vector $ x _ {1} \wedge \dots \wedge x _ {p} $; such a poly-vector is called factorable, decomposable, pure, or prime (often simply a poly-vector). Here two linearly independent systems $ x _ {1} \dots x _ {p} $ and $ y _ {1} \dots y _ {p} $ generate the same subspace in $ V $ if and only if $ y _ {1} \wedge \dots \wedge y _ {p} = cx _ {1} \wedge \dots \wedge x _ {p} $, where $ c \in K $. For any non-zero poly-vector $ t \in \wedge ^ {p} V $, its annihilator $ \mathop{\rm Ann} t = \{ {v \in V } : {t \wedge v = 0 } \} $ is a subspace of dimension $ \leq p $, and the poly-vector $ t $ is pure if and only if $ \mathop{\rm dim} \mathop{\rm Ann} t = p $. The pure $ p $- vectors of an $ n $- dimensional space $ V $ form an algebraic variety in $ \wedge ^ {p} V $; the corresponding projective algebraic variety is a Grassmann manifold. Any non-zero $ n $- vector or $ ( n- 1) $- vector in an $ n $- dimensional space $ V $ is pure, but a bivector $ t $ is pure if and only if $ t \wedge t = 0 $.

If $ v _ {1} \dots v _ {n} $ is a basis of $ V $ and $ x _ {i} = \sum_{j=1} ^ {n} x _ {i} ^ {j} v _ {j} $, then the coordinates of the poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ in the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ of the space $ \wedge ^ {p} V $ are the minors $ t ^ {i _ {1} \dots i _ {p} } = \mathop{\rm det} \| x _ {i} ^ {i _ {k} } \| $, $ i _ {1} < \dots < i _ {p} $, of the matrix $ \| x _ {i} ^ {j} \| $. In particular, for $ p = n $,

$$ x _ {1} \wedge \dots \wedge x _ {n} = \ \mathop{\rm det} \| x _ {i} ^ {j} \| v _ {1} \wedge \dots \wedge v _ {n} . $$

If one specifies a non-zero $ n $- vector $ \omega \in \wedge ^ {n} V $, a duality between $ p $- vectors and $ ( n- p) $- vectors is obtained, i.e. a natural isomorphism

$$ \pi : \wedge ^ {p} ( V) \rightarrow \ ( \wedge ^ {n-p} V) ^ {*} \cong \wedge ^ {n-p} ( V ^ {*} ) $$

such that $ t \wedge u = \pi ( t)( u) \omega $ for all $ t \in \wedge ^ {p} V $ and $ u \in \wedge ^ {n-p} V $.

Let $ k = \mathbf R $ and let an inner product be defined in $ V $, then in $ \wedge ^ {p} V $ an inner product is induced with the following property: For any orthonormal basis $ v _ {1} \dots v _ {n} $ in $ V $ the basis $ \{ {v _ {i _ {1} } \wedge \dots \wedge v _ {i _ {p} } } : {i _ {1} < \dots < i _ {p} } \} $ in $ \wedge ^ {p} V $ is also orthonormal. The scalar square

$$ ( t, t) = \sum _ {i _ {1} < \dots < i _ {p} } ( t ^ {i _ {1} \dots i _ {p} } ) ^ {2} $$

of a pure poly-vector $ t = x _ {1} \wedge \dots \wedge x _ {p} $ coincides with the square of the volume of the parallelopipedon in $ V $ constructed on the vectors $ x _ {1} \dots x _ {p} $. If one specifies an orientation in the $ n $- dimensional Euclidean space $ V $( which is equivalent to choosing an $ n $- vector $ \omega $ for which $ ( \omega , \omega ) = 1 $), then the above duality leads to a natural isomorphism $ \gamma : \wedge ^ {p} V \rightarrow \wedge ^ {n-p} V $. In particular, the $ ( n- 1) $- vector $ t = x _ {1} \wedge \dots \wedge x _ {n-1} $ corresponds to a vector $ \gamma ( t) \in V $, called the vector product of the vectors $ x _ {1} \dots x _ {n-1} $.

References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2] A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)
[3] M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian)
How to Cite This Entry:
Poly-vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-vector&oldid=17085
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article