Poly-vector

$ 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)