Alternating multilinear forms

Main articles: Alternating multilinear map
An important class of multilinear forms are the **alternating multilinear forms**, which have the additional property that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://mathoid-facade/v1/":): {\displaystyle f(x\_{\sigma(1)},\ldots, x\_{\sigma(k)}) = \sgn(\sigma)f(x\_1,\ldots, x\_k), }**

where

σ
:

N

k

→

N

k

{\displaystyle \sigma:\mathbf{N}\_k\to\mathbf{N}\_k}

 is a permutation and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://mathoid-facade/v1/":): {\displaystyle \sgn(\sigma)}**
 denotes its sign (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e.,

σ
(
p
# )
q
,
σ
(
q
# )
p

{\displaystyle \sigma(p)=q,\sigma(q)=p }

 and

σ
(
i
# )
i
,
1
≤
i
≤
k
,
i
≠
p
,
q

{\displaystyle \sigma(i)=i, 1\le i\le k, i\neq p,q }

):

f
(

x
1

,
…
,

x
p

,
…
,

x
q

,
…
,

x
k

# )
−
f
(

x
1

,
…
,

x
q

,
…
,

x
p

,
…
,

x
k

)
.

{\displaystyle f(x\_1,\ldots, x\_p,\ldots, x\_q,\ldots, x\_k) = -f(x\_1,\ldots, x\_q,\ldots, x\_p,\ldots, x\_k). }

With the additional hypothesis that the characteristic of the field

K

{\displaystyle K}

 is not 2, setting

x
p

=

x
q

=
x

{\displaystyle x\_p=x\_q=x }

 implies as a corollary that

f
(

x
1

,
…
,
x
,
…
,
x
,
…
,

x
k

# )
0

{\displaystyle f(x\_1,\ldots, x,\ldots, x,\ldots, x\_k) = 0 }

; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors use this last condition as the defining property of alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when

char
⁡
(
K
)
≠
2

{\displaystyle \operatorname{char}(K) \ne 2}

.

An alternating multilinear *k*-form on

V

{\displaystyle V}

 over

R

{\displaystyle \mathbf{R}}

 is called a **multicovector of degree *k*** or ***k*-covector**, and the vector space of such alternating forms, a subspace of

T

k

(
V
)

{\displaystyle \mathcal{T}^k(V)}

, is generally denoted

A

k

(
V
)

{\displaystyle \mathcal{A}^k(V)}

, or, using the notation for the isomorphic *k*th exterior power of

V
∗

{\displaystyle V^\*}

(the dual space of

V

{\displaystyle V}

),

⋀
k

V
∗

{\textstyle \bigwedge^k V^\*}

. Note that linear functionals (multilinear 1-forms over

R

{\displaystyle \mathbf{R}}

) are trivially alternating, so that

A

1

(
V
# )

T

1

(
V
# )

V
∗

{\displaystyle \mathcal{A}^1(V)=\mathcal{T}^1(V)=V^\*}

, while, by convention, 0-forms are defined to be scalars:

A

0

(
V
# )

T

0

(
V
# )

R

{\displaystyle \mathcal{A}^0(V)=\mathcal{T}^0(V)=\mathbf{R}}

.

The determinant on

n
×
n

{\displaystyle n \times n}

 matrices, viewed as an

n

{\displaystyle n}

 argument function of the column vectors, is an important example of an alternating multilinear form.

## Wedge product

The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each term, the **wedge product** (

∧

{\displaystyle \wedge}

) of multicovectors can be defined, so that if

f
∈

A

k

(
V
)

{\displaystyle f\in\mathcal{A}^k(V)}

 and

g
∈

A

ℓ

(
V
)

{\displaystyle g\in\mathcal{A}^\ell(V)}

, then

f
∧
g
∈

A

k
ℓ

(
V
)

{\displaystyle f\wedge g\in\mathcal{A}^{k+\ell}(V)}

:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://mathoid-facade/v1/":): {\displaystyle (f\wedge g)(v\_1,\ldots, v\_{k+\ell})=\frac{1}{k!\ell!}\sum\_{\sigma\in S\_{k+\ell}} (\sgn(\sigma)) f(v\_{\sigma(1)}, \ldots, v\_{\sigma(k)})g(v\_{\sigma(k+1)} ,\ldots,v\_{\sigma(k+\ell)}),}**

where the sum is taken over the set of all permutations over

k
ℓ

{\displaystyle k+\ell}

 elements,

S

k
ℓ

{\displaystyle S\_{k+\ell}}

. The wedge product is bilinear, associative, and anticommutative: if

f
∈

A

k

(
V
)

{\displaystyle f\in\mathcal{A}^k(V)}

 and

g
∈

A

ℓ

(
V
)

{\displaystyle g\in\mathcal{A}^\ell(V)}

 then

f
∧
# g
(
−
1

)

k
ℓ

g
∧
f

{\displaystyle f\wedge g=(-1)^{k\ell}g\wedge f}

.

Given a basis

(

v
1

,
…
,

v
n

)

{\displaystyle (v\_1,\ldots, v\_n)}

 for

V

{\displaystyle V}

 and dual basis

(

ϕ
1

,
…
,

ϕ
n

)

{\displaystyle (\phi^1,\ldots,\phi^n)}

 for

V
∗

=

A

1

(
V
)

{\displaystyle V^\*=\mathcal{A}^1(V)}

, the wedge products

ϕ

i
1

∧
⋯
∧

ϕ

i
k

{\displaystyle \phi^{i\_1}\wedge\cdots\wedge\phi^{i\_k}}

, with

1
≤

i
1

<
⋯
<

i
k

≤
n

{\displaystyle 1\leq i\_1<\cdots<i\_k\leq n}

 form a basis for

A

k

(
V
)

{\displaystyle \mathcal{A}^k(V)}

. Hence, the dimensionality of

A

k

(
V
)

{\displaystyle \mathcal{A}^k(V)}

 for *n*-dimensional

V

{\displaystyle V}

 is

(

n
k

)

=

n
!

(
n
−
k
)
!

k
!

{\textstyle \tbinom{n}{k}=\frac{n!}{(n-k)!\,k!}}

.