Metraipolis › archive › dc_4d7791237e9a

Standard basis — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Standard_basis
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T19:43:52.354Z
SHA-256 of the archived text: a8f5ef984ab3f6fedcdd5c822b203b8b4c1a9681239e06f34b70603e858b6c4b
Stable URL of this page: https://metraipolis.com/documento/dc_4d7791237e9a

How to cite

Standard basis — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Standard_basis (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_4d7791237e9a (Metraipolis archive dc_4d7791237e9a, sha256 a8f5ef984ab3f6fedcdd5c822b203b8b4c1a9681239e06f34b70603e858b6c4b)

BibTeX

@misc{dc4d7791237e9a,
  title = {Standard basis — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Standard_basis},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_4d7791237e9a; sha256 a8f5ef984ab3f6fedcdd5c822b203b8b4c1a9681239e06f34b70603e858b6c4b},
  year = {2026}
}

Archived text

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

In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors e x = ( 1 , 0 ) , e y = ( 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1).} Similarly, the standard basis for the three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} is formed by vectors e x = ( 1 , 0 , 0 ) , e y = ( 0 , 1 , 0 ) , e z = ( 0 , 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).} Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},} the scalars vx, vy, vz being the scalar components of the vector v. In the n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the standard basis consists of n distinct vectors { e i : 1 ≤ i ≤ n } , {\displaystyle \{\mathbf {e} _{i}:1\leq i\leq n\},} where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices Mm×n, the standard basis consists of the m×n–matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices e 11 = ( 1 0 0 0 ) , e 12 = ( 0 1 0 0 ) , e 21 = ( 0 0 1 0 ) , e 22 = ( 0 0 0 1 ) . {\displaystyle {\begin{aligned}\mathbf {e} _{11}&={\begin{pmatrix}1&0\\0&0\end{pmatrix}},&\mathbf {e} _{12}&={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\\\mathbf {e} _{21}&={\begin{pmatrix}0&0\\1&0\end{pmatrix}},&\mathbf {e} _{22}&={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.\end{aligned}}}

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Linear algebra»).
Fonte: Wikipedia in inglese, voce «Standard basis» — https://en.wikipedia.org/wiki/Standard_basis
Wikidata Q2000912 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q2000912
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Standard_basis?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 1307165744, consultata il 2026-10-07T19:04:52.684Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 255111e9b9face858c2e2d4032ad53511461e5c2706106acfdc19a25aa1dca94