In mathematics, a - multilinear form is a function that assigns a value to arguments from - vector spaces and is linear in every component. In the more general case, that the image space itself is a vector space, or image and target spaces are modules , one speaks of a multilinear mapping .
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![v_ {i} \ in V_ {i}, \; i \ in \ {1, \ ldots, p \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1be31966da930b1d3502dfbebc690c795a4973fe)
![V_ {1}, \ ldots, V_ {p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35946e2f9af1c0989447f99e90e8d5fc9866d545)
![\ omega (v_ {1}, \ ldots, v_ {p}) \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1356f6e8f6c42c3ba96db3508e91b1ad42b4437)
definition
An illustration
![{\ begin {aligned} \ omega: \ V_ {1} \ times \ cdots \ times V_ {p} & \ rightarrow K \\ (v_ {1}, \ ldots, v_ {p}) \ & \ mapsto \ omega \ left (v_ {1}, \ dots, v_ {p} \ right) \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/301bd1697ab69ca699085334586dbce14a0ef08c)
is called multilinear form if the following two conditions are met for all and all :
![v_ {j} \ in V_ {j}, j \ in \ {1, \ ldots, p \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d45d82018d75d183d77951e7b8238f35d1cfde0)
![i \ in \ {1, \ ldots, p \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a50a95991f624cd213baa24b3c4a157a9fcc5e)
For all true
![\ lambda \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d95fcd9b07168a8162820f7fab4d8ee43366e8)
![\ omega \ left (v_ {1}, \ ldots, \ lambda \; v_ {i}, \ ldots, v_ {p} \ right) = \ lambda \; \ omega \ left (v_ {1}, \ ldots, v_ {i}, \ ldots, v_ {p} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4694379fbba5336bd1db0905d3675c085ff9613e)
and for everyone
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.
The set of all multilinear maps forms a vector space. In the case you write .
![{\ mathcal {J}} ^ {p} (V_ {1}, \ ldots, V_ {p})](https://wikimedia.org/api/rest_v1/media/math/render/svg/976e52900b1bdc9ffba568571515ac6e7decd4f9)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![V_ {1} = \ cdots = V_ {p} =: V](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccdc8cc7bd604d4338e932dda9c82004a2d10f56)
![{\ mathcal {J}} ^ {p} (V): = {\ mathcal {J}} ^ {p} (V, \ ldots, V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/38bbf377a2bd41d21bdfddd413b24e7c3adc68aa)
Alternating multilinear forms
A multilinear form is called alternating if it results in zero when the same vector is used twice, i.e. H.
![\ omega \ in {\ mathcal {J}} ^ {p} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1928e4bdd40fde5c11e4e0c968cebb186b44a6ad)
![\ omega \ left (\ dots, v, \ dots, v, \ dots \ right) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/abba3b9e917d0b0999f7d548665ddd08c8ea07fa)
for everyone .
![v \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb)
In this case it also follows that the form is skew-symmetrical, that is, that it changes its sign if any two arguments are interchanged , i.e.
![\ omega \ left (v_ {1}, \ dots, v_ {i}, \ dots, v_ {j}, \ dots, v_ {p} \ right) = - \ omega \ left (v_ {1}, \ dots , v_ {j}, \ dots, v_ {i}, \ dots, v_ {p} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c8a6edd44501ecf2a2f2f7fce252a191f054b3)
for everyone and . The reverse implication - that all skew-symmetric multilinear forms alternately are - but applies only if the characteristic of not 2, so for example .
![v_ {k} \ in V, \; k \ in \ {1, \ ldots, p \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76530f6b783d21dd9040b86ba88c379fefe7ef08)
![i, j \ in \ {1, \ ldots, p \}, \; i \ neq j](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e221acb327cb84a3c9eaacab6abc513257215c)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K = \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e)
If, more generally, is any permutation of the indices, then applies
![\ pi \ in S_ {p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424a4a4237c6874745ebabed911b1a8e04b609ef)
-
,
where denotes the sign of the permutation.
![\ operatorname {sign} (\ pi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae4d9c3bc2ae8cddb7d99a0096788d21d1ba1a0)
The set of all alternating multilinear forms is a subspace of . The structure of an algebra can also be defined on this set . This algebra is called Graßmann algebra . The special case is important . Then is a one-dimensional subspace of , and its elements are called determinant functions .
![\ Omega ^ {p} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/65b2477574743f273cf13bb6897cc92d0c2a856c)
![{\ mathcal {J}} ^ {p} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6fa441f399badd1bd5444555de8f15bc1bbb1d)
![\ p = \ dim V](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c8d332f7b7d8fe19cac14b8971c20d0afcbbc7)
![\ Omega ^ {p} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/65b2477574743f273cf13bb6897cc92d0c2a856c)
![{\ mathcal {J}} ^ {p} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6fa441f399badd1bd5444555de8f15bc1bbb1d)
Examples
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Linear forms are exactly the 1-multilinear forms.
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Bilinear forms are exactly the 2-multilinear forms. Antisymmetric bilinear forms are alternating multilinear forms (if the characteristic of is not 2).
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
- If a square matrix is formed from vectors by combining them, the determinant of this matrix is an alternating, normalized multilinear form. In the three-dimensional case it is defined by an alternating 3-multilinear form. The vectors are as follows shown in coordinates: .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![\omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
![\ omega \ left (v_ {1}, v_ {2}, v_ {3} \ right): = \ det {\ begin {pmatrix} v _ {{1x}} & v _ {{2x}} & v _ {{3x}} \\ v _ {{1y}} & v _ {{2y}} & v _ {{3y}} \\ v _ {{1z}} & v _ {{2z}} & v _ {{3z}} \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256da31add8fbff950c31b196260d4ced4447f0f)
![v_ {1}, v_ {2}, v_ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12e5ac2b8e42965e3d97dbe06d1031361bc310e6)
![v_ {1} = {\ begin {pmatrix} v _ {{1x}} \\ v _ {{1y}} \\ v _ {{1z}} \ end {pmatrix}}, \ quad \ quad v_ {2} = { \ begin {pmatrix} v _ {{2x}} \\ v _ {{2y}} \\ v _ {{2z}} \ end {pmatrix}}, \ quad \ quad v_ {3} = {\ begin {pmatrix} v_ {{3x}} \\ v _ {{3y}} \\ v _ {{3z}} \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835124eecd5a5354a348848b19f6ae6b8698f21d)
- Covariant tensors are multilinear forms: In the case that all vector spaces are identical (i.e. ), the -multilinear form is also a covariant tensor -th level. In the same case, the alternating -Multilinearforms are also totally antisymmetric tensors -th order.
![V_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c)
![V_ {i} = V](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3edf760a9e75d5833cfe9992db1772e60ce19a8)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
- A differential form assigns an alternating multilinear form to a point of a manifold on the associated tangent space .
literature
Individual evidence
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↑ a b Arkady L'vovich Onishchik : multilinear mapping . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).