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Quaternionic vector space — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Quaternionic_vector_space
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T17:20:05.335Z
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Quaternionic vector space — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Quaternionic_vector_space (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_e212ef80dbac (Metraipolis archive dc_e212ef80dbac, sha256 0f5fa2f9e32e559176552059164958827ada0294a7e3ed29738022b217d3b2a5)

BibTeX

@misc{dce212ef80dbac,
  title = {Quaternionic vector space — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Quaternionic_vector_space},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_e212ef80dbac; sha256 0f5fa2f9e32e559176552059164958827ada0294a7e3ed29738022b217d3b2a5},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Quaternionic_vector_space. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division ring, these modules are referred to as "vector spaces". However, the quaternion algebra is noncommutative so we must distinguish left and right vector spaces. In left vector spaces, linear compositions of vectors v {\displaystyle v} and w {\displaystyle w} have the form a v + b w {\displaystyle av+bw} where a {\displaystyle a} , b ∈ H {\displaystyle b\in H} . In right vector spaces, linear compositions of vectors v {\displaystyle v} and w {\displaystyle w} have the form v a + w b {\displaystyle va+wb} . Similar to vector spaces over a field, if a quaternionic vector space has finite dimension n {\displaystyle n} , then it is isomorphic to the direct sum H n {\displaystyle H^{n}} of n {\displaystyle n} copies of the quaternion algebra H {\displaystyle H} . In this case we can use a standard basis which has the form e 1 = ( 1 , 0 , … , 0 ) {\displaystyle e_{1}=(1,0,\ldots ,0)} … {\displaystyle \ldots } e n = ( 0 , … , 0 , 1 ) {\displaystyle e_{n}=(0,\ldots ,0,1)} In a left quaternionic vector space H n {\displaystyle H^{n}} we use componentwise sum of vectors and product of vectors over scalars ( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})} q ( r 1 , … , r n ) = ( q r 1 , … , q r n ) {\displaystyle q(r_{1},\ldots ,r_{n})=(qr_{1},\ldots ,qr_{n})} In a right quaternionic vector space H n {\displaystyle H^{n}} we also use componentwise sum of vectors and product of vectors over scalars ( p 1 , … , p n ) + ( r 1 , … , r n ) = ( p 1 + r 1 , … , p n + r n ) {\displaystyle (p_{1},\ldots ,p_{n})+(r_{1},\ldots ,r_{n})=(p_{1}+r_{1},\ldots ,p_{n}+r_{n})} ( r 1 , … , r n ) q = ( r 1 q , … , r n q ) {\displaystyle (r_{1},\ldots ,r_{n})q=(r_{1}q,\ldots ,r_{n}q)}

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Linear algebra»).
Fonte: Wikipedia in inglese, voce «Quaternionic vector space» — https://en.wikipedia.org/wiki/Quaternionic_vector_space
Wikidata Q7269568 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q7269568
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Quaternionic_vector_space?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1334063732, consultata il 2026-10-07T16:51:20.517Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 687d2326f88010566f62460ef93d2afc5a6d9d502bf330d53ab874d1b8cf3581