See multilinear form on Wiktionary
Download JSON data for multilinear form meaning in All languages combined (5.6kB)
{
"derived": [
{
"_dis1": "52 48",
"word": "alternating multilinear form"
},
{
"_dis1": "52 48",
"word": "multilinear k-form"
}
],
"forms": [
{
"form": "multilinear forms",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "multilinear form (plural multilinear forms)",
"name": "en-noun"
}
],
"hyponyms": [
{
"_dis1": "50 50",
"sense": "multiply linear mapping to a field of scalars",
"word": "bilinear form"
},
{
"_dis1": "50 50",
"sense": "multiply linear mapping to a field of scalars",
"word": "covector"
},
{
"_dis1": "50 50",
"sense": "multiply linear mapping to a field of scalars",
"word": "linear form"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"_dis1": "52 48",
"word": "covariant"
},
{
"_dis1": "52 48",
"word": "differential form"
},
{
"_dis1": "52 48",
"word": "linear"
},
{
"_dis1": "52 48",
"word": "linear form"
},
{
"_dis1": "52 48",
"word": "multilinear algebra"
},
{
"_dis1": "52 48",
"word": "multilinear map"
},
{
"_dis1": "52 48",
"word": "multivector"
},
{
"_dis1": "52 48",
"word": "tensor"
}
],
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Linear algebra",
"orig": "en:Linear algebra",
"parents": [
"Algebra",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "51 49",
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"name": "English entries with incorrect language header",
"parents": [
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"Entry maintenance"
],
"source": "w+disamb"
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],
"examples": [
{
"text": "1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,\nFinally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators."
},
{
"text": "1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,\nA multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate m-multilinear form can more concisely be written as M!f."
},
{
"ref": "2003, Maks A. Akivis, Vladislav V. Goldberg, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55",
"text": "Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).",
"type": "quotation"
}
],
"glosses": [
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.",
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;"
],
"id": "en-multilinear_form-en-noun-b50LDZC1",
"links": [
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"linear algebra",
"linear algebra"
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"vector space",
"vector space"
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"scalar",
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],
[
"linear",
"linear"
],
[
"module",
"module"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "multilinear algebra",
"raw_glosses": [
"(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n"
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
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"kind": "topical",
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"name": "Linear algebra",
"orig": "en:Linear algebra",
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"Formal sciences",
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"Fundamental"
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"source": "w"
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{
"_dis": "51 49",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
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"source": "w+disamb"
}
],
"glosses": [
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.",
"a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R."
],
"id": "en-multilinear_form-en-noun-ZlCNN~C4",
"links": [
[
"linear algebra",
"linear algebra"
],
[
"vector space",
"vector space"
],
[
"scalar",
"scalar"
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[
"mapping",
"mapping"
],
[
"linear",
"linear"
],
[
"module",
"module"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "multilinear algebra",
"raw_glosses": [
"(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n"
],
"tags": [
"broadly"
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"_dis1": "50 50",
"sense": "multiply linear mapping to a field of scalars",
"word": "multicovector"
}
],
"translations": [
{
"_dis1": "50 50",
"code": "fr",
"lang": "French",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "forme multilinéaire"
},
{
"_dis1": "50 50",
"code": "de",
"lang": "German",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "Multilinearform"
},
{
"_dis1": "50 50",
"code": "es",
"lang": "Spanish",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "forma multilineal"
},
{
"_dis1": "50 50",
"code": "es",
"lang": "Spanish",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "función multilineal"
}
],
"wikipedia": [
"multilinear form"
],
"word": "multilinear form"
}
{
"categories": [
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"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns"
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"derived": [
{
"word": "alternating multilinear form"
},
{
"word": "multilinear k-form"
}
],
"forms": [
{
"form": "multilinear forms",
"tags": [
"plural"
]
}
],
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"expansion": "multilinear form (plural multilinear forms)",
"name": "en-noun"
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{
"sense": "multiply linear mapping to a field of scalars",
"word": "bilinear form"
},
{
"sense": "multiply linear mapping to a field of scalars",
"word": "covector"
},
{
"sense": "multiply linear mapping to a field of scalars",
"word": "linear form"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "covariant"
},
{
"word": "differential form"
},
{
"word": "linear"
},
{
"word": "linear form"
},
{
"word": "multilinear algebra"
},
{
"word": "multilinear map"
},
{
"word": "multivector"
},
{
"word": "tensor"
}
],
"senses": [
{
"categories": [
"English terms with quotations",
"en:Linear algebra"
],
"examples": [
{
"text": "1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,\nFinally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators."
},
{
"text": "1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,\nA multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate m-multilinear form can more concisely be written as M!f."
},
{
"ref": "2003, Maks A. Akivis, Vladislav V. Goldberg, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55",
"text": "Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).",
"type": "quotation"
}
],
"glosses": [
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.",
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;"
],
"links": [
[
"linear algebra",
"linear algebra"
],
[
"vector space",
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],
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"scalar"
],
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"mapping"
],
[
"linear",
"linear"
],
[
"module",
"module"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "multilinear algebra",
"raw_glosses": [
"(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n"
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"English terms with quotations",
"en:Linear algebra"
],
"glosses": [
"Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.",
"a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R."
],
"links": [
[
"linear algebra",
"linear algebra"
],
[
"vector space",
"vector space"
],
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"scalar"
],
[
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"mapping"
],
[
"linear",
"linear"
],
[
"module",
"module"
],
[
"commutative ring",
"commutative ring"
]
],
"qualifier": "multilinear algebra",
"raw_glosses": [
"(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;\n"
],
"tags": [
"broadly"
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
]
}
],
"synonyms": [
{
"sense": "multiply linear mapping to a field of scalars",
"word": "multicovector"
}
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "forme multilinéaire"
},
{
"code": "de",
"lang": "German",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "Multilinearform"
},
{
"code": "es",
"lang": "Spanish",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "forma multilineal"
},
{
"code": "es",
"lang": "Spanish",
"sense": "multiply linear mapping to a field of scalars",
"tags": [
"feminine"
],
"word": "función multilineal"
}
],
"wikipedia": [
"multilinear form"
],
"word": "multilinear form"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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