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)