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{{short description|Himpunan vektor yang digunakan untuk mendefinisikan koordinat}}
{{Short description|In linear algebra, generated subspace}}
[[Berkas:Basis_for_a_plane.svg|ka|jmpl|280x280px|The cross-hatched plane is the linear span of '''u''' and '''v''' in '''R'''<sup>3</sup>.]]
{{redirect|Vektor basis|vektor basis dalam konteks kristal|Struktur kristal|konsep yang lebih umum dalam fisika|Kerangka acuan}}[[Berkas:3d_two_bases_same_vector.svg|jmpl|265x265px|Vektor yang sama (panah berwarna biru tua) dapat dinyatakan dengan menggunakan dua basis yang berbeda (panah-panah berwarna ungu dan berwarna merah).]]Dalam [[matematika]], sebarang [[Himpunan (matematika)|himpunan]] vektor {{mvar|B}} dalam suatu [[ruang vektor]] {{math|''V''}} disebut '''basis''', jika setiap elemen di {{math|''V''}} dapat dituliskan sebagai [[kombinasi linear]] terhingga yang unik dari elemen-elemen di {{mvar|B}}. Koefisien-koefisien pada kombinasi linear tersebut disebut sebagai ''koordinat'' dari vektor terhadap {{mvar|B}}. Elemen-elemen dari basis disebut sebagai ''vektor basis''. Basis juga dapat didefinisikan sebagai himpunan {{mvar|B}} yang elemen-elemennya saling [[Kebebasan linear|bebas linear]] dan setiap elemen di {{math|''V''}} adalah kombinasi linear dari elemen-elemen di {{mvar|B}}.<ref>{{cite book|last=Halmos|first=Paul Richard|year=1987|url=https://books.google.com/books?id=mdWeEhA17scC&pg=PA10|title=Finite-Dimensional Vector Spaces|location=New York|publisher=Springer|isbn=978-0-387-90093-3|edition=4th|page=10|author-link=Paul Halmos}}</ref> Dengan kata lain, basis adalah [[Rentang linear|himpunan merentang]] (''spanning'') yang bebas linear.
In [[mathematics]], the '''linear span''' (also called the '''linear hull'''<ref>{{Harvard citation text|Encyclopedia of Mathematics|2020}}. Linear Hull.</ref> or just '''span''') of a [[Set (mathematics)|set]] {{mvar|S}} of [[Vector space|vectors]] (from a [[vector space]]), denoted {{math|span(''S'')}},<ref name=":0">{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref> is defined as the set of all [[linear combinations]] of the vectors in {{mvar|S}}.<ref>{{Harvard citation text|Axler|2015}} p. 29, § 2.7</ref> For example, two [[linearly independent]] [[Vector (geometry)|vectors]] span a [[Plane (geometry)|plane]]. The linear span can be characterized either as the [[Intersection (set theory)|intersection]] of all [[Linear subspace|linear subspaces]] that contain {{mvar|S}}, or as the smallest subspace containing {{mvar|S}}. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to [[Matroid|matroids]] and [[Module (mathematics)|modules]].


To express that a vector space {{mvar|V}} is a linear span of a subset {{mvar|S}}, one commonly uses the following phrases—either: {{mvar|S}} spans {{mvar|V}}, {{mvar|S}} is a '''spanning set''' of {{mvar|V}}, {{mvar|V}} is spanned/generated by {{mvar|S}}, or {{mvar|S}} is a [[Generator (mathematics)|generator]] or generator set of {{mvar|V}}.
Suatu ruang vektor dapat memiliki beberapa basis; namun semua basis tersebut akan memiliki jumlah elemen yang sama, yang disebut sebagai [[Dimensi (ruang vektor)|''dimensi'' dari ruang vektor]]. Artikel ini secara umum membahas ruang-ruang vektor berdimensi hingga. Akan tetapi, banyak prinsip yang disampaikan juga berlaku untuk ruang vektor dimensi tak-hingga.


== Definisi ==
== Definition ==
Given a [[vector space]] {{mvar|V}} over a [[Field (mathematics)|field]] {{mvar|K}}, the span of a [[Set (mathematics)|set]] {{mvar|S}} of vectors (not necessarily finite) is defined to be the intersection {{mvar|W}} of all [[Linear subspace|subspaces]] of {{mvar|V}} that contain {{mvar|S}}. {{mvar|W}} is referred to as the subspace ''spanned by'' {{mvar|S}}, or by the vectors in {{mvar|S}}. Conversely, {{mvar|S}} is called a ''spanning set'' of {{mvar|W}}, and we say that {{mvar|S}} ''spans'' {{mvar|W}}.
Basis untuk ruang vektor <math>V</math> (atas [[Medan (matematika)|medan]] <math>F</math>) adalah suatu himpunan bagian <math>B\subset V</math> yang memenuhi:
# Setiap <math>\mathbf{v}\in V</math> dapat dituliskan sebagai <math>\mathbf{v}=\sum _{i=1}^ka_i\mathbf{b}_i</math> dengan <math>k\in\mathbb{N}, a_1,\ldots,a_k\in F, \mathbf{b}_1,\ldots,\mathbf{b}_k\in B</math>.
# Jika <math>\mathbf{v}=\sum _{i=1}^{\tilde{k}}\tilde{a}_i\tilde{\mathbf{b}}_i</math> representasi lain, maka <math>k=\tilde{k}</math> dan ada suatu permutasi <math>\iota:\{1,\ldots,k\}\to\{1,\ldots,k\}</math> yang <math>a_i=\tilde{a}_{\iota (i)}</math> dan <math>\mathbf{b}_i=\tilde{\mathbf{b}}_{\iota (i)}</math>.


Alternatively, the span of {{mvar|S}} may be defined as the set of all finite [[linear combinations]] of elements (vectors) of {{mvar|S}}, which follows from the above definition.<ref>{{Harvard citation text|Hefferon|2020}} p. 100, ch. 2, Definition 2.13</ref><ref name=":02">{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref><ref>{{Harvard citation text|Roman|2005}} pp. 41-42</ref><ref>{{Harvard citation text|MathWorld|2021}} Vector Space Span.</ref><math display="block"> \operatorname{span}(S) = \left \{ {\left.\sum_{i=1}^k \lambda_i \mathbf v_i \;\right|\; k \in \N, \mathbf v_i \in S, \lambda _i \in K} \right \}.</math>In the case of infinite {{mvar|S}}, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a [[Banach space]]) are excluded by the definition; a [[Linear combination#Generalizations|generalization]] that allows these is not equivalent.
Sebarang basis <math>B</math> dari suatu [[ruang vektor]] <math>V</math> atas [[Lapangan (matematika)|lapangan]] <math>F</math> (seperti [[bilangan riil]] <math>\R</math> atau [[bilangan kompleks]] <math>\C</math>) adalah suatu [[Himpunan bagian|subset]] dari <math>V</math> yang saling [[Kebebasan linear|bebas linear]] dan [[Rentang linear|merentang]] <math>V</math>. Hal ini mengartikan suatu subset <math>B</math> dari <math>V</math> merupakan basis jika memenuhi dua syarat berikut:
; ''kebebasan linear''
: Untuk setiap subset [[Himpunan hingga|terhingga]] <math>\{\mathbf v_1, \dotsc, \mathbf v_m\}</math> dari <math>B</math>, jika <math>c_1 \mathbf v_1 + \cdots + c_m \mathbf v_m = \mathbf 0</math> untuk suatu <math>c_1,\dotsc,c_m</math> di {{math|''F''}}, maka {{nowrap|<math>c_1 = \cdots = c_m = 0</math>;}}
; ''merentang linear''
: Untuk setiap vektor <math>\mathbf v \in V</math>, terdapat <math>n</math> skalar <math>a_1,\dotsc,a_n</math> di {{math|''F''}} dan <math>n</math> vektor <math>\mathbf v_1, \dotsc, \mathbf v_n</math> di {{mvar|B}}, sehingga {{nowrap|<math>\mathbf v = a_1 \mathbf v_1 + \cdots + a_n \mathbf v_n</math>.}}


== Examples ==
[[Skalar (matematika)|Skalar-skalar]] <math>a_i</math> disebut ''koordinat'' dari vektor <math>\mathbf v</math> terhadap basis <math>B</math>, dan berdasarkan sifat pertama, nilai mereka unik (tunggal). Ruang vektor disebut ''[[Dimensi (ruang vektor)|berdimensi hingga]]'' jika ruang vektor tersebut memiliki basis dengan total elemen yang berhingga.
The [[Real number|real]] vector space <math>\mathbb R^3</math> has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a [[Basis (linear algebra)|basis]]. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the [[Standard basis|canonical basis]] of <math>\mathbb R^3</math>.


Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, {{frac|1|2}}, 3), (1, 1, 1)}, but this set is not a basis, because it is [[Linear dependency|linearly dependent]].
Vektor-vektor basis seringkali (dan terkadang harus) perlu memiliki [[urutan total]] untuk mempermudah pembahasan. Sebagai contoh, ketika basis digunakan untuk membahas [[Orientasi (ruang vektor)|orientasi]], atau untuk membahas koefisien-koefisien vektor terhadap suatu basis tanpa perlu merujuk secara eksplisit elemen-elemen dari basis. Istilah ''basis terurut'' terkadang digunakan untuk mempertegas bahwa suatu urutan telah dipilih; yang sebenarnya menyebabkan basis bukan lagi sebagai suatu [[Himpunan (matematika)|himpunan]] tak-terurut, melainkan sebagai suatu [[barisan]] (atau sejenisnya). Pembahasan lebih lanjut tersedia di bagian [[Pengguna:Kekavigi/bak pasir#koordinat|Koordinat]] di bawah.


The set {{math|{(1, 0, 0), (0, 1, 0), (1, 1, 0)}}} is not a spanning set of <math>\mathbb R^3</math>, since its span is the space of all vectors in <math>\mathbb R^3</math> whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not <math>\mathbb R^3.</math> It can be identified with <math>\mathbb R^2</math> by removing the third components equal to zero.
== Contoh ==
[[Berkas:Basis graph (no label).svg|thumb|Gambar ini mengilustrasikan [[basis standar]] di <math>\R^2,</math> yang elemennya adalah vektor biru dan oranye. Vektor hijau dapat dinyatakan sebagai kombinasi linear dari vektor-vektor basis, mengakibatkan vektor ini [[bergantung linear]] pada mereka.]]Himpunan <math>\R^2</math> dari [[pasangan terurut]] [[bilangan riil]] adalah suatu ruang vektor dibawah operasi penjumlahan komponen-demi-komponen<math display="block">(a, b) + (c, d) = (a + c, b+d)</math>dan perkalian<math display="block">\lambda (a,b) = (\lambda a, \lambda b),</math>dengan <math>\lambda</math> adalah sebarang bilangan riil. Contoh basis yang sederhana dari ruang vektor ini adalah himpunan yang berisi vektor <math>\mathbf{e}_1 = (1,\,0)</math> dan <math>\mathbf{e}_2 = (0,\,1) </math>. Kedua vektor ini membentuk sebuah basis (yang disebut ''basis standar'') karena sebarang vektor <math>\mathbf{v} = (a,\,b) </math> di <math>\R^2</math> dapat ditulis secara unik sebagai<math display="block">\mathbf v = a \mathbf e_1 + b \mathbf e_2.</math>Sebarang pasangan vektor yang saling bebas linear di <math>\R^2</math>, seperti <math>(1,\,1)</math> dan <math>(-1,\,2)</math>, juga membentuk sebuah basis untuk <math>\R^2</math>.


The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in <math>\mathbb R^3</math>, and {(0, 0, 0)} is the intersection of all of these vector spaces.
Secara umum, jika <math>F</math> berupa [[Lapangan (matematika)|lapangan]], maka himpunan <math>F^n</math> yang berisi [[Rangkap|rangkap-''n'']] elemen-elemen dari <math>F^n</math> adalah sebuah ruang vektor, dibawah operasi penjumlahan dan perkalian yang serupa dengan contoh pembuka tadi. Misalkan<math display="block">\mathbf e_i = (0,\,\ldots,\,0,\,1,\,0,\,\ldots,\,0)</math>adalah rangkap-''n'' dengan semua komponennya bernilai 0, kecuali komponen ke-''i'', yang bernilai 1. Himpunan <math>\mathbf e_1, \ldots, \mathbf e_n</math> membentuk suatu basis (terurut) untuk <math>F^n,</math> yang disebut dengan ''basis standar'' dari <math>F^n.</math>


The set of [[Monomial|monomials]] {{mvar|x<sup>n</sup>}}, where {{mvar|n}} is a non-negative integer, spans the space of [[Polynomial|polynomials]].
Contoh yang berbeda terlihat pada [[gelanggang polinomial]]. Jika <math>F</math> berupa [[Lapangan (matematika)|lapangan]], himpunan <math>F[x]</math> dari semua [[polinomial]] satu-variabel dengan koefisien-koefisiennya berada di <math>F</math>, merupakan suatu ruang vektor. Salah satu basis untuk ruang ini adalah [[basis monomial]] {{mvar|B}}, yang berisi semua [[monomial]]:<math display="block">B=\{1, x, x^2, \ldots\}.</math>Contoh lain dari basis untuk ruang vektor tersebut adalah [[polinomial Bernstein|polinomial basis Bernstein]] dan [[polinomial Chebyshev]].


== Sifat-sifat ==
== Theorems ==
Banyak sifat dari basis terhingga merupakan hasil dari [[lema pertukaran Steinitz]], yang menyatakan bahwa, untuk sebarang ruang vektor <math>V</math>, dan sebarang penetapan [[Rentang linear|himpunan merentang]] <math>S</math> dan himpunan [[Kebebasan linear|bebas linear]] <math>L</math> berisi <math>n</math> elemen dari <math>V</math>, <math>n</math> elemen dari <math>S</math> dapat dipilih sedemikian rupa untuk ditukar dengan elemen-elemen di <math>L</math> sehingga menghasilkan suatu himpunan merentang yang: mengandung <math>L</math>, elemen-elemen yang lainnya berada di <math>S</math>, dan memiliki jumlah elemen yang sama dengan <math>S</math>. Sebagian besar sifat yang dihasilkan dari lema tersebut masih berlaku ketika tidak ada himpunan merentang yang terhingga, namun pembuktian untuk keadaan ini memerlukan [[aksioma pemilihan]] atau suatu bentuk yang lebih lemahnya, seperti [[lema ultrafilter]].


=== Equivalence of definitions ===
Jika <math>V</math> adalah ruang vektor atas lapangan <math>F</math>, maka:
The set of all linear combinations of a subset {{mvar|S}} of {{mvar|V}}, a vector space over {{mvar|K}}, is the smallest linear subspace of {{mvar|V}} containing {{mvar|S}}.


: ''Proof.'' We first prove that {{math|span ''S''}} is a subspace of {{mvar|V}}. Since {{mvar|S}} is a subset of {{mvar|V}}, we only need to prove the existence of a zero vector {{math|'''0'''}} in {{math|span ''S''}}, that {{math|span ''S''}} is closed under addition, and that {{math|span ''S''}} is closed under scalar multiplication. Letting <math>S = \{ \mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n \}</math>, it is trivial that the zero vector of {{mvar|V}} exists in {{math|span ''S''}}, since <math>\mathbf 0 = 0 \mathbf v_1 + 0 \mathbf v_2 + \cdots + 0 \mathbf v_n</math>. Adding together two linear combinations of {{mvar|S}} also produces a linear combination of {{mvar|S}}: <math>(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) + (\mu_1 \mathbf v_1 + \cdots + \mu_n \mathbf v_n) = (\lambda_1 + \mu_1) \mathbf v_1 + \cdots + (\lambda_n + \mu_n) \mathbf v_n</math>, where all <math>\lambda_i, \mu_i \in K</math>, and multiplying a linear combination of {{mvar|S}} by a scalar <math>c \in K</math> will produce another linear combination of {{mvar|S}}: <math>c(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) = c\lambda_1 \mathbf v_1 + \cdots + c\lambda_n \mathbf v_n</math>. Thus {{math|span ''S''}} is a subspace of {{mvar|V}}.
* Untuk sebarang subset bebas linear <math>L</math> dari sebarang himpunan merentang <math>S\subseteq V</math>, terdapat suatu basis <math>B</math> sehingga <math display="block">L\subseteq B\subseteq S.</math>
* <math>V</math> memiliki basis (hal ini dapat dihasilkan dari sifat sebelumnya dengan memilih <math>L</math> sebagai himpunan kosong, dan <math>S=V</math>).
* Setiap basis dari <math>V</math> memiliki [[kardinalitas]] yang sama, yang disebut dengan dimensi dari <math>V</math>. Pernyataan ini dikenal sebagai teorema dimensi.
* Sebarang himpunan pembangkit <math>S</math> adalah basis dari <math>V</math> jika dan hanya jika itu bersifat minimal, artinya, <math>S</math> bukan subset wajar (''proper subset'') dari sebarang himpunan yang bebas linear.


: Suppose that {{mvar|W}} is a linear subspace of {{mvar|V}} containing {{mvar|S}}. It follows that <math>S \subseteq \operatorname{span} S</math>, since every {{math|'''v'''<sub>''i''</sub>}} is a linear combination of {{mvar|S}} (trivially). Since {{mvar|W}} is closed under addition and scalar multiplication, then every linear combination <math>\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n</math> must be contained in {{mvar|W}}. Thus, {{math|span ''S''}} is contained in every subspace of {{mvar|V}} containing {{mvar|S}}, and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of {{mvar|S}}.
Jika <math>V</math> adalah ruang vektor berdimensi <math>n</math>, suatu subset berisi <math>n</math> elemen dari <math>V</math> merupakan basis dari <math>V</math> jika dan hanya jika:


=== Size of spanning set is at least size of linearly independent set ===
* Subset tersebut bebas linear;
Every spanning set {{mvar|S}} of a vector space {{mvar|V}} must contain at least as many elements as any [[Linear independence|linearly independent]] set of vectors from {{mvar|V}}.
* Subset tersebut himpunan merentang dari <math>V</math>.


: ''Proof.'' Let <math>S = \{ \mathbf v_1, \ldots, \mathbf v_m \}</math> be a spanning set and <math>W = \{ \mathbf w_1, \ldots, \mathbf w_n \}</math> be a linearly independent set of vectors from {{mvar|V}}. We want to show that <math>m \geq n</math>.
== Koordinat ==
Misalkan <math>V</math> adalah ruang vektor berdimensi <math>n</math> (hingga) atas lapangan <math>F</math>, dan <math display="block">B = \{\mathbf b_1, \ldots, \mathbf b_n\}</math>adalah basis dari <math>V</math>. Berdasarkan definisi dari basis, setiap <math>\mathbf v</math> di <math>V</math> dapat ditulis secara unik sebagai<math display="block">\mathbf v = \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n,</math>dengan koefisien-koefisien <math>\lambda_1, \ldots, \lambda_n</math> adalah skalar (yaitu, elemen-elemen dari <math>F</math>), yang disebut sebagai ''koordinat'' dari <math>\mathbf v</math> atas <math>B</math>. Akan tetapi, pembahasan terkait ''himpunan'' koefisien akan menghilangkan hubungan antara koefisien-koefisien dari elemen-elemen basis, dan beberapa vektor berbeda dapat memiliki ''himpunan'' koefisien yang sama. Sebagai contoh, vektor <math>3 \mathbf b_1 + 2 \mathbf b_2</math> dan <math>2 \mathbf b_1 + 3 \mathbf b_2</math> yang berbeda memiliki himpunan koefisien <math>\{2,\,3\}</math> yang sama. Oleh karena itu, konsep ''basis terurut'' umum digunakan untuk mempermudah pembahasan. Hal ini dilakukan dengan [[Himpunan indeks|mengindeks]] elemen-elemen basis menggunakan bilangan asli. Koordinat dari vektor juga diindeks dengan cara yang sama, sehingga vektor dapat dikarakterisasi seutuhnya dari barisan koordinat mereka.


: Since {{mvar|S}} spans {{mvar|V}}, then <math>S \cup \{ \mathbf w_1 \}</math> must also span {{mvar|V}}, and <math>\mathbf w_1</math> must be a linear combination of {{mvar|S}}. Thus <math>S \cup \{ \mathbf w_1 \}</math> is linearly dependent, and we can remove one vector from {{mvar|S}} that is a linear combination of the other elements. This vector cannot be any of the {{math|'''w'''<sub>''i''</sub>}}, since {{mvar|W}} is linearly independent. The resulting set is <math>\{ \mathbf w_1, \mathbf v_1, \ldots, \mathbf v_{i-1}, \mathbf v_{i+1}, \ldots, \mathbf v_m \}</math>, which is a spanning set of {{mvar|V}}. We repeat this step {{mvar|n}} times, where the resulting set after the {{mvar|p}}th step is the union of <math>\{ \mathbf w_1, \ldots, \mathbf w_p \}</math> and {{mvar|m - p}} vectors of {{mvar|S}}.


: It is ensured until the {{mvar|n}}th step that there will always be some {{math|'''v'''<sub>''i''</sub>}} to remove out of {{mvar|S}} for every adjoint of {{math|'''v'''}}, and thus there are at least as many {{math|'''v'''<sub>''i''</sub>}}'s as there are {{math|'''w'''<sub>''i''</sub>}}'s—i.e. <math>m \geq n</math>. To verify this, we assume by way of contradiction that <math>m < n</math>. Then, at the {{mvar|m}}th step, we have the set <math>\{ \mathbf w_1, \ldots, \mathbf w_m \}</math> and we can adjoin another vector <math>\mathbf w_{m+1}</math>. But, since <math>\{ \mathbf w_1, \ldots, \mathbf w_m \}</math> is a spanning set of {{mvar|V}}, <math>\mathbf w_{m+1}</math> is a linear combination of <math>\{ \mathbf w_1, \ldots, \mathbf w_m \}</math>. This is a contradiction, since {{mvar|W}} is linearly independent.
Misalkan, seperti biasa, <math>F^n</math> adalah himpunan [[Rangkap|rangkap-''n'']] dari elemen-elemen di <math>F</math>). Himpunan ini adalah ruang vektor-<math>F</math>, dengan operasi penjumlahan dan perkalian skalar-nya dilakukan secara komponen-demi-komponen. Pemetaan <math display="block">\varphi: (\lambda_1, \ldots, \lambda_n) \mapsto \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n</math>adalah suatu [[Peta linear|isomorfisme linear]] dari ruang vektor <math>F^n</math> pada (''onto'') <math>V</math>. Dalam kata lain, <math>F^n</math> adalah [[Ruang vektor#Ruang koordinat|ruang koordinat]] dari <math>V</math>, dan rangkap-''n'' <math>\varphi^{-1}(\mathbf v)</math> adalah vektor koordinat dari <math>\mathbf v</math>. Secara khusus, invers bayangan dari <math>\mathbf b_i</math> oleh <math>\varphi</math> adalah vektor <math>\mathbf e_i</math>, yang setiap komponennya bernilai 0, kecuali komponen ke-''i'' yang bernilai 1. Himpunan <math>\mathbf e_i</math> membentuk suatu basis terurut bagi <math>F^n</math>, yang disebut dengan ''basis standar'' atau ''basis kanonik''.


== Perubahan basis ==
=== Spanning set can be reduced to a basis ===
Let {{mvar|V}} be a finite-dimensional vector space. Any set of vectors that spans {{mvar|V}} can be reduced to a [[Basis (linear algebra)|basis]] for {{mvar|V}}, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the [[axiom of choice]] holds, this is true without the assumption that {{mvar|V}} has finite dimension. This also indicates that a basis is a minimal spanning set when {{mvar|V}} is finite-dimensional.
{{main|Perubahan basis}}
Let {{math|''V''}} be a vector space of dimension {{mvar|n}} over a field {{math|''F''}}. Given two (ordered) bases <math>B_\text{old} = (\mathbf v_1, \ldots, \mathbf v_n)</math> and <math>B_\text{new} = (\mathbf w_1, \ldots, \mathbf w_n)</math> of {{math|''V''}}, it is often useful to express the coordinates of a vector {{mvar|x}} with respect to <math>B_\mathrm{old}</math> in terms of the coordinates with respect to <math>B_\mathrm{new}.</math> This can be done by the ''change-of-basis formula'', that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to <math>B_\mathrm{old}</math> and <math>B_\mathrm{new}</math> as the ''old basis'' and the ''new basis'', respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has [[Expression (mathematics)|expressions]] involving the old coordinates, and if one wants to obtain equivalent expressions in terms of the new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates.


== Generalizations ==
---
Generalizing the definition of the span of points in space, a subset {{mvar|X}} of the ground set of a [[matroid]] is called a spanning set if the rank of {{mvar|X}} equals the rank of the entire ground set{{Citation needed|date=May 2016}}.


The vector space definition can also be generalized to modules.<ref>{{Harvard citation text|Roman|2005}} p. 96, ch. 4</ref><ref>{{Harvard citation text|Lane|Birkhoff|1999}} p. 193, ch. 6</ref> Given an {{mvar|R}}-module {{mvar|A}} and a collection of elements {{math|''a''<sub>1</sub>}}, ..., {{math|''a<sub>n</sub>''}} of {{mvar|A}}, the [[submodule]] of {{mvar|A}} spanned by {{math|''a''<sub>1</sub>}}, ..., {{math|''a<sub>n</sub>''}} is the sum of [[Cyclic module|cyclic modules]]<math display="block">Ra_1 + \cdots + Ra_n = \left\{ \sum_{k=1}^n r_k a_k \bigg| r_k \in R \right\}</math>consisting of all ''R''-linear combinations of the elements {{math|''a<sub>i</sub>''}}. As with the case of vector spaces, the submodule of ''A'' spanned by any subset of ''A'' is the intersection of all submodules containing that subset.
Maka {{math|''V''}} jadilah ''terang dan pagi~'' ruang vektor berdimensi {{mvar|n}} di atas bidang {{math|''F''}}. Diberikan dua pangkalan (order) <math>B_\mathrm {old}=(v_1, \ldots, v_n)</math> dan <math>B_\mathrm {new}=(w_1, \ldots, w_n)</math> dari {{math|''V''}}, sering kali berguna untuk menyatakan koordinat vektor {{mvar | x}} sehubungan dengan <math>B_\mathrm {old}</math> dalam hal koordinat sehubungan dengan <math>B_\mathrm {new}.</math> Ini dapat dilakukan dengan '' rumus perubahan-basis '', yang dijelaskan di bawah ini. Subskrip "lama" dan "baru" telah dipilih karena biasa digunakan untuk merujuk <math>B_\mathrm {old}</math> dan <math>B_\mathrm {new}</math> sebagai '' dasar lama '' dan '' dasar baru ''. Ini berguna untuk menggambarkan koordinat lama dengan yang baru, karena, secara umum, seseorang memiliki [[ekspresi (matematika) | ekspresi]] yang melibatkan koordinat lama, dan jika seseorang ingin mendapatkan ekspresi yang setara dalam hal koordinat baru; ini diperoleh dengan mengganti koordinat lama dengan ekspresi mereka dalam bentuk koordinat baru.


== Closed linear span (functional analysis) ==
Biasanya, vektor basis baru diberikan oleh koordinatnya di atas basis lama, yaitu
In [[functional analysis]], a closed linear span of a [[Set (mathematics)|set]] of [[Vector space|vectors]] is the minimal closed set which contains the linear span of that set.
:<math>w_j=\sum_{i=1}^n a_{i,j}v_i.</math>
If <math>(x_1, \ldots, x_n)</math> and <math>(y_1, \ldots, y_n)</math> are the coordinates of a vector {{mvar|x}} over the old and the new basis respectively, the change-of-basis formula is
:<math>x_i = \sum_{j=1}^n a_{i,j}y_j,</math>
for {{math|1=''i'' = 1, ..., ''n''}}.


Suppose that {{mvar|X}} is a normed vector space and let {{mvar|E}} be any non-empty subset of {{mvar|X}}. The '''closed linear span''' of {{mvar|E}}, denoted by <math>\overline{\operatorname{Sp}}(E)</math> or <math>\overline{\operatorname{Span}}(E)</math>, is the intersection of all the closed linear subspaces of {{mvar|X}} which contain {{mvar|E}}.
Rumus ini dapat ditulis secara ringkas dalam notasi [[matriks (matematika) | matriks]]. Misalkan {{mvar|A}} adalah matriks dari <math>a_{i,j},</math> dan
:<math>X= \begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}\quad</math> dan <math>\quad Y= \begin{pmatrix}y_1\\\vdots\\y_n\end{pmatrix}</math>
jadilah [[vektor kolom]] dari koordinat {{mvar|v}} di basis lama dan basis baru, maka rumus untuk mengubah koordinat adalah
:<math>X=AY.</math>


One mathematical formulation of this is
Rumusnya dapat dibuktikan dengan mempertimbangkan dekomposisi vektor {{mvar|x}} pada dua basa: satu memiliki
:<math>x=\sum_{i=1}^n x_i v_i,</math>
dan
:<math>\begin{align}
x&=\sum_{j=1}^n y_j w_j \\
&=\sum_{j=1}^n y_j\sum_{i=1}^n a_{i,j}v_i\\
&=\sum_{i=1}^n \left(\sum_{j=1}^n a_{i,j}y_j\right)v_i.
\end{align}</math>


: <math>\overline{\operatorname{Sp}}(E) = \{u\in X | \forall\varepsilon > 0\,\exists x\in\operatorname{Sp}(E) : \|x - u\|<\varepsilon\}.</math>
Rumus perubahan basis kemudian dari keunikan dekomposisi vektor atas basis, di sini <math>B_\mathrm {old};</math> adalah
:<math>x_i = \sum_{j=1}^n a_{i,j}y_j,</math>
untuk {{math|1=''i'' = 1, ..., ''n''}}.


The closed linear span of the set of functions ''x<sup>n</sup>'' on the interval [0, 1], where ''n'' is a non-negative integer, depends on the norm used. If the [[Lp space#Lp spaces and Lebesgue integrals|''L''<sup>2</sup> norm]] is used, then the closed linear span is the [[Hilbert space]] of [[Square-integrable function|square-integrable functions]] on the interval. But if the [[maximum norm]] is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the [[cardinality]] of the set of functions in the closed linear span is the [[cardinality of the continuum]], which is the same cardinality as for the set of polynomials.
== Change of basis ==


=== Notes ===
The linear span of a set is dense in the closed linear span. Moreover, as stated in the lemma below, the closed linear span is indeed the [[Closure (mathematics)|closure]] of the linear span.


Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, see [[Riesz's lemma]]).
Typically, the new basis vectors are given by their coordinates over the old basis, that is,<math display="block">\mathbf w_j = \sum_{i=1}^n a_{i,j} \mathbf v_i.</math>If <math>(x_1, \ldots, x_n)</math> and <math>(y_1, \ldots, y_n)</math> are the coordinates of a vector {{math|'''x'''}} over the old and the new basis respectively, the change-of-basis formula is<math display="block">x_i = \sum_{j=1}^n a_{i,j}y_j,</math>for {{math|1=''i'' = 1, ..., ''n''}}.


=== A useful lemma ===
This formula may be concisely written in [[Matrix (mathematics)|matrix]] notation. Let {{mvar|A}} be the matrix of the {{nowrap|<math>a_{i,j}</math>,}} and<math display="block">X= \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} \quad \text{and} \quad Y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}</math>be the [[Column vector|column vectors]] of the coordinates of {{math|'''v'''}} in the old and the new basis respectively, then the formula for changing coordinates is<math display="block">X = A Y.</math>The formula can be proven by considering the decomposition of the vector {{math|'''x'''}} on the two bases: one has<math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf v_i,</math>and<math display="block">\mathbf x =\sum_{j=1}^n y_j \mathbf w_j
Let {{mvar|X}} be a normed space and let {{mvar|E}} be any non-empty subset of {{mvar|X}}. Then{{ordered list|<math>\overline{\operatorname{Sp}}(E)</math> is a closed linear subspace of ''X'' which contains ''E'',|<math>\overline{\operatorname{Sp}}(E) = \overline{\operatorname{Sp}(E)}</math>, viz. <math>\overline{\operatorname{Sp}}(E)</math> is the closure of <math>\operatorname{Sp}(E)</math>,|<math>E^\perp = (\operatorname{Sp}(E))^\perp = \left(\overline{\operatorname{Sp}(E)}\right)^\perp.</math>|list-style-type=lower-alpha}}(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)
= \sum_{j=1}^n y_j\sum_{i=1}^n a_{i,j}\mathbf v_i
= \sum_{i=1}^n \biggl(\sum_{j=1}^n a_{i,j}y_j\biggr)\mathbf v_i.</math>The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here {{nowrap|<math>B_\text{old}</math>;}} that is<math display="block">x_i = \sum_{j=1}^n a_{i,j} y_j,</math>for {{math|1=''i'' = 1, ..., ''n''}}.


== Related notions ==
== See also ==


* [[Affine hull]]
=== Free module ===
* [[Conical combination]]
{{main|Free module|Free abelian group}}
* [[Convex hull]]
If one replaces the field occurring in the definition of a vector space by a [[Ring (mathematics)|ring]], one gets the definition of a [[Module (mathematics)|module]]. For modules, [[linear independence]] and [[Spanning set|spanning sets]] are defined exactly as for vector spaces, although "[[Generating set of a module|generating set]]" is more commonly used than that of "spanning set".


== Citations ==
Like for vector spaces, a ''basis'' of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A module that has a basis is called a ''free module''. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through [[Free resolution|free resolutions]].
<references />


== Sources ==
A module over the integers is exactly the same thing as an [[abelian group]]. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if {{mvar|G}} is a subgroup of a finitely generated free abelian group {{mvar|H}} (that is an abelian group that has a finite basis), then there is a basis <math>\mathbf e_1, \ldots, \mathbf e_n</math> of {{mvar|H}} and an integer {{math|0 ≤ ''k'' ≤ ''n''}} such that <math>a_1 \mathbf e_1, \ldots, a_k \mathbf e_k</math> is a basis of {{mvar|G}}, for some nonzero integers {{nowrap|<math>a_1, \ldots, a_k</math>.}} For details, see {{slink|Free abelian group|Subgroups}}.


=== Analysis ===
=== Textbooks ===
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term '''{{visible anchor|Hamel basis}}''' (named after [[Georg Hamel]]<ref>{{Harvnb|Hamel|1905}}</ref>) or '''algebraic basis''' can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are [[Orthogonal basis|orthogonal bases]] on [[Hilbert space|Hilbert spaces]], [[Schauder basis|Schauder bases]], and [[Markushevich basis|Markushevich bases]] on [[Normed linear space|normed linear spaces]]. In the case of the real numbers '''R''' viewed as a vector space over the field '''Q''' of rational numbers, Hamel bases are uncountable, and have specifically the [[cardinality]] of the continuum, which is the [[cardinal number]] {{nowrap|<math>2^{\aleph_0}</math>,}} where <math>\aleph_0</math> ([[aleph-nought]]) is the smallest infinite cardinal, the cardinal of the integers.


* {{Cite book|last=Axler|first=Sheldon Jay|year=2015|title=Linear Algebra Done Right|publisher=[[Springer Science+Business Media | Springer]]|isbn=978-3-319-11079-0|edition=3rd|author-link=Sheldon Axler}}
The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for [[Topological vector space|topological vector spaces]] – a large class of vector spaces including e.g. [[Hilbert space|Hilbert spaces]], [[Banach space|Banach spaces]], or [[Fréchet space|Fréchet spaces]].
* {{Cite book|last=Hefferon|first=Jim|year=2020|title=Linear Algebra|publisher=Orthogonal Publishing|isbn=978-1-944325-11-4|edition=4th|author-link=Jim Hefferon}}
* {{Cite book|last1=Lane|first1=Saunders Mac|last2=Birkhoff|first2=Garrett|year=1999|title=Algebra|publisher=[[American Mathematical Society|AMS Chelsea Publishing]]|isbn=978-0821816462|edition=3rd|author-link=Saunders Mac Lane|author-link2=Garrett Birkhoff|orig-year=1988}}
* {{Cite book|last=Roman|first=Steven|year=2005|title=Advanced Linear Algebra|publisher=[[Springer Science+Business Media|Springer]]|isbn=0-387-24766-1|edition=2nd|author-link=Steven Roman}}
* {{Cite book|last1=Rynne|first1=Brian P.|last2=Youngson|first2=Martin A.|year=2008|title=Linear Functional Analysis|location=|publisher=Springer|isbn=978-1848000049|pages=}}
* Lay, David C. (2021) ''Linear Algebra and Its Applications (6th Edition)''. Pearson.


=== Web ===
The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If ''X'' is an infinite-dimensional normed vector space that is [[Complete space|complete]] (i.e. ''X'' is a [[Banach space]]), then any Hamel basis of ''X'' is necessarily [[uncountable]]. This is a consequence of the [[Baire category theorem]]. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (''non-complete'') normed spaces that have countable Hamel bases. Consider {{nowrap|<math>c_{00}</math>,}} the space of the [[Sequence|sequences]] <math>x=(x_n)</math> of real numbers that have only finitely many non-zero elements, with the norm {{nowrap|<math display="inline">\|x\|=\sup_n |x_n|</math>.}} Its [[standard basis]], consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.


* {{cite web|last1=Lankham|first1=Isaiah|last2=Nachtergaele|first2=Bruno|author2-link=Bruno Nachtergaele|date=13 February 2010|title=Linear Algebra - As an Introduction to Abstract Mathematics|url=https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf|publisher=University of California, Davis|access-date=27 September 2011|last3=Schilling|first3=Anne|author3-link=Anne Schilling}}
==== Example ====
* {{Cite web|last=Weisstein|first=Eric Wolfgang|author-link=Eric W. Weisstein|title=Vector Space Span|url=https://mathworld.wolfram.com/VectorSpaceSpan.html|website=[[MathWorld]]|access-date=16 Feb 2021|ref=CITEREFMathWorld2021}}
In the study of [[Fourier series]], one learns that the functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions ''f'' satisfying<math display="block">\int_0^{2\pi} \left|f(x)\right|^2\,dx < \infty.</math>The functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are linearly independent, and every function ''f'' that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that<math display="block">\lim_{n\to\infty} \int_0^{2\pi} \biggl|a_0 + \sum_{k=1}^n \left(a_k\cos\left(kx\right)+b_k\sin\left(kx\right)\right)-f(x)\biggr|^2 dx = 0</math>for suitable (real or complex) coefficients ''a<sub>k</sub>'', ''b<sub>k</sub>''. But many<ref>Note that one cannot say "most" because the cardinalities of the two sets (functions that can and cannot be represented with a finite number of basis functions) are the same.</ref> square-integrable functions cannot be represented as ''finite'' linear combinations of these basis functions, which therefore ''do not'' comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas [[orthonormal bases]] of these spaces are essential in [[Fourier analysis]].
* {{Cite web|date=5 April 2020|title=Linear hull|url=https://encyclopediaofmath.org/wiki/Linear_hull|website=[[Encyclopedia of Mathematics]]|access-date=16 Feb 2021|ref=CITEREFEncyclopedia_of_Mathematics2020}}


=== Geometry ===
== External links ==
The geometric notions of an [[affine space]], [[projective space]], [[convex set]], and [[Cone (linear algebra)|cone]] have related notions of {{anchor|affine basis}} ''basis''.<ref>{{cite book|last=Rees|first=Elmer G.|year=2005|url=https://books.google.com/books?id=JkzPRaihGIYC&pg=PA7|title=Notes on Geometry|location=Berlin|publisher=Springer|isbn=978-3-540-12053-7|page=7}}</ref> An '''affine basis''' for an ''n''-dimensional affine space is <math>n+1</math> points in [[general linear position]]. A '''{{visible anchor|projective basis}}''' is <math>n+2</math> points in general position, in a projective space of dimension ''n''. A '''{{visible anchor|convex basis}}''' of a [[polytope]] is the set of the vertices of its [[convex hull]]. A '''{{visible anchor|cone basis}}'''<ref>{{cite journal|last=Kuczma|first=Marek|year=1970|title=Some remarks about additive functions on cones|journal=[[Aequationes Mathematicae]]|volume=4|issue=3|pages=303–306|doi=10.1007/BF01844160|s2cid=189836213}}</ref> consists of one point by edge of a polygonal cone. See also a [[Hilbert basis (linear programming)]].


* [https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span Linear Combinations and Span: Understanding linear combinations and spans of vectors], khanacademy.org.
=== Random basis ===
* {{Cite web|last=Sanderson|first=Grant|author-link=3Blue1Brown|date=August 6, 2016|title=Linear combinations, span, and basis vectors|url=https://www.youtube.com/watch?v=k7RM-ot2NWY&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3|series=Essence of Linear Algebra|archive-url=https://ghostarchive.org/varchive/youtube/20211211/k7RM-ot2NWY|archive-date=2021-12-11|via=[[YouTube]]|url-status=live}}{{cbignore}}
For a [[probability distribution]] in {{math|'''R'''<sup>''n''</sup>}} with a [[probability density function]], such as the equidistribution in an ''n''-dimensional ball with respect to Lebesgue measure, it can be shown that {{mvar|n}} randomly and independently chosen vectors will form a basis [[with probability one]], which is due to the fact that {{mvar|n}} linearly dependent vectors {{math|'''x'''<sub>1</sub>}}, ..., {{math|'''x'''<sub>''n''</sub>}} in {{math|'''R'''<sup>''n''</sup>}} should satisfy the equation {{math|1=det['''x'''<sub>1</sub> ⋯ '''x'''<sub>''n''</sub>] = 0}} (zero determinant of the matrix with columns {{math|'''x'''<sub>''i''</sub>}}), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.<ref>{{cite journal|last1=Igelnik|first1=B.|last2=Pao|first2=Y.-H.|year=1995|title=Stochastic choice of basis functions in adaptive function approximation and the functional-link net|journal=IEEE Trans. Neural Netw.|volume=6|issue=6|pages=1320–1329|doi=10.1109/72.471375|pmid=18263425}}</ref><ref name="GorbanTyukin2016">{{cite journal|last1=Gorban|first1=Alexander N.|last2=Tyukin|first2=Ivan Y.|last3=Prokhorov|first3=Danil V.|last4=Sofeikov|first4=Konstantin I.|year=2016|title=Approximation with Random Bases: Pro et Contra|journal=[[Information Sciences (journal)|Information Sciences]]|volume=364-365|pages=129–145|arxiv=1506.04631|doi=10.1016/j.ins.2015.09.021|author1-link=Aleksandr Gorban|s2cid=2239376}}</ref>
[[Berkas:Random_almost_orthogonal_sets.png|jmpl|270x270px|Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the ''n''-dimensional cube {{math|[−1, 1]<sup>''n''</sup>}} as a function of dimension, ''n''. Boxplots show the second and third quartiles of this data for each ''n'', red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound given by Eq. (1) and green curve shows a refined estimate.<ref name="GorbanTyukin2016" />]]
It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For [[Inner product space|spaces with inner product]], ''x'' is ε-orthogonal to ''y'' if <math>\left|\left\langle x,y \right\rangle\right| / \left(\left\|x\right\|\left\|y\right\|\right) < \varepsilon</math> (that is, cosine of the angle between {{mvar|x}} and {{mvar|y}} is less than {{mvar|ε}}).


{{Linear algebra}}
In high dimensions, two independent random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in ''n''-dimensional ball. Choose ''N'' independent random vectors from a ball (they are [[Independent and identically distributed random variables|independent and identically distributed]]). Let ''θ'' be a small positive number. Then for{{NumBlk||<math display="block">N\leq {\exp}\bigl(\tfrac14\varepsilon^2n\bigr)\sqrt{-\ln(1-\theta)}</math>|Eq. 1}}{{mvar|N}} random vectors are all pairwise ε-orthogonal with probability {{math|1 − ''θ''}}.<ref name="GorbanTyukin2016" /> This {{mvar|N}} growth exponentially with dimension {{mvar|n}} and <math>N\gg n</math> for sufficiently big {{mvar|n}}. This property of random bases is a manifestation of the so-called {{em|measure concentration phenomenon}}.<ref>{{cite journal|last=Artstein|first=Shiri|author-link=Shiri Artstein|year=2002|title=Proportional concentration phenomena of the sphere|url=http://www.tau.ac.il/~shiri/israelj/ISRAJ.pdf|journal=[[Israel Journal of Mathematics]]|volume=132|issue=1|pages=337–358|doi=10.1007/BF02784520|doi-access=free|citeseerx=10.1.1.417.2375|s2cid=8095719}}</ref>

The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the ''n''-dimensional cube {{math|[−1, 1]<sup>''n''</sup>}} as a function of dimension, ''n''. A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within {{math|π/2 ± 0.037π/2}} then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within {{math|π/2 ± 0.037π/2}} then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each ''n'', 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.

== Proof that every vector space has a basis ==
Let {{math|'''V'''}} be any vector space over some field {{math|'''F'''}}. Let {{math|'''X'''}} be the set of all linearly independent subsets of {{math|'''V'''}}.

The set {{math|'''X'''}} is nonempty since the empty set is an independent subset of {{math|'''V'''}}, and it is [[Partial order|partially ordered]] by inclusion, which is denoted, as usual, by {{math|⊆}}.

Let {{math|'''Y'''}} be a subset of {{math|'''X'''}} that is totally ordered by {{math|⊆}}, and let {{math|L<sub>'''Y'''</sub>}} be the union of all the elements of {{math|'''Y'''}} (which are themselves certain subsets of {{math|'''V'''}}).

Since {{math|('''Y''', ⊆)}} is totally ordered, every finite subset of {{math|L<sub>'''Y'''</sub>}} is a subset of an element of {{math|'''Y'''}}, which is a linearly independent subset of {{math|'''V'''}}, and hence {{math|L<sub>'''Y'''</sub>}} is linearly independent. Thus {{math|L<sub>'''Y'''</sub>}} is an element of {{math|'''X'''}}. Therefore, {{math|L<sub>'''Y'''</sub>}} is an upper bound for {{math|'''Y'''}} in {{math|('''X''', ⊆)}}: it is an element of {{math|'''X'''}}, that contains every element of {{math|'''Y'''}}.

As {{math|'''X'''}} is nonempty, and every totally ordered subset of {{math|('''X''', ⊆)}} has an upper bound in {{math|'''X'''}}, [[Zorn's lemma]] asserts that {{math|'''X'''}} has a maximal element. In other words, there exists some element {{math|L<sub>'''max'''</sub>}} of {{math|'''X'''}} satisfying the condition that whenever {{math|L<sub>'''max'''</sub> ⊆ L}} for some element {{math|L}} of {{math|'''X'''}}, then {{math|1=L = L<sub>'''max'''</sub>}}.

It remains to prove that {{math|L<sub>'''max'''</sub>}} is a basis of {{math|'''V'''}}. Since {{math|L<sub>'''max'''</sub>}} belongs to {{math|'''X'''}}, we already know that {{math|L<sub>'''max'''</sub>}} is a linearly independent subset of {{math|'''V'''}}.

If there were some vector {{math|'''w'''}} of {{math|'''V'''}} that is not in the span of {{math|L<sub>'''max'''</sub>}}, then {{math|'''w'''}} would not be an element of {{math|L<sub>'''max'''</sub>}} either. Let {{math|1=L<sub>'''w'''</sub> = L<sub>'''max'''</sub> ∪ {'''w'''}<nowiki/>}}. This set is an element of {{math|'''X'''}}, that is, it is a linearly independent subset of {{math|'''V'''}} (because '''w''' is not in the span of {{math|L<sub>'''max'''</sub>}}, and {{math|L<sub>'''max'''</sub>}} is independent). As {{math|L<sub>'''max'''</sub> ⊆ L<sub>'''w'''</sub>}}, and {{math|L<sub>'''max'''</sub> ≠ L<sub>'''w'''</sub>}} (because {{math|L<sub>'''w'''</sub>}} contains the vector {{math|'''w'''}} that is not contained in {{math|L<sub>'''max'''</sub>}}), this contradicts the maximality of {{math|L<sub>'''max'''</sub>}}. Thus this shows that {{math|L<sub>'''max'''</sub>}} spans {{math|'''V'''}}.

Hence {{math|L<sub>'''max'''</sub>}} is linearly independent and spans {{math|'''V'''}}. It is thus a basis of {{math|'''V'''}}, and this proves that every vector space has a basis.

This proof relies on Zorn's lemma, which is equivalent to the [[axiom of choice]]. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true.<ref>{{Harvnb|Blass|1984}}</ref> Thus the two assertions are equivalent.

== Catatan kaki ==
{{Reflist}}

== Referensi ==

=== Referensi umum ===

* {{Citation | last1=Blass | first1=Andreas | title=Axiomatic set theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Contemporary Mathematics volume 31 | mr=763890 | year=1984 | chapter=Existence of bases implies the axiom of choice | pages=31–33|isbn=978-0-8218-5026-8|chapter-url=http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf}}
* {{Citation | last1=Brown | first1=William A. | title=Matrices and vector spaces | publisher=M. Dekker | location=New York | isbn=978-0-8247-8419-5 | year=1991|url=https://books.google.com/books?id=pFQYKlnW5Z0C}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987}}

=== Referensi sejarah ===

* {{Citation | last1=Banach | first1=Stefan | author1-link=Stefan Banach | title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations) | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf | year=1922 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=3| pages=133–181 |language=fr| doi=10.4064/fm-3-1-133-181 }}
* {{Citation | last1=Bolzano | first1=Bernard | author1-link=Bernard Bolzano | title=Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) | url=http://dml.cz/handle/10338.dmlcz/400338 | year=1804|language=de}}
* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Éléments d'histoire des mathématiques (Elements of history of mathematics) | publisher=Hermann | location=Paris | year=1969|language=fr}}
* {{Citation | last1=Dorier | first1=Jean-Luc | title=A general outline of the genesis of vector space theory | mr=1347828 | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=227–261 | doi=10.1006/hmat.1995.1024| url=http://archive-ouverte.unige.ch/unige:16642 |doi-access=free}}
* {{Citation | last1=Fourier | first1=Jean Baptiste Joseph | author1-link=Joseph Fourier | title=Théorie analytique de la chaleur | url=https://books.google.com/books?id=TDQJAAAAIAAJ | publisher=Chez Firmin Didot, père et fils | year=1822|language=fr}}
* {{Citation | last1=Grassmann | first1=Hermann | author1-link=Hermann Grassmann | title=Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik | url=https://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1| year=1844|language=de}}, reprint: {{Citation | others=Kannenberg, L.C. | title=Extension Theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2031-5 | year=2000 | author=Hermann Grassmann. Translated by Lloyd C. Kannenberg.}}
* {{Citation|last=Hamel|first=Georg|author1-link=Georg Hamel|title=Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x+y)=f(x)+f(y)|journal=Mathematische Annalen|location=Leipzig|volume=60|pages=459–462|url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002260395|year=1905|issue=3|doi=10.1007/BF01457624|s2cid=120063569|language=de}}
* {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=Lectures on Quaternions | url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 | publisher=Royal Irish Academy | year=1853}}
* {{Citation |last1=Möbius |first1=August Ferdinand |author1-link=August Ferdinand Möbius |title=Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) |url=http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 |year=1827 |language=de |url-status=dead |archive-url=https://web.archive.org/web/20090412013616/http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0|archive-date=2009-04-12}}
* {{Citation | last1=Moore | first1=Gregory H. | title=The axiomatization of linear algebra: 1875–1940 | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=262–303 | doi=10.1006/hmat.1995.1025|doi-access=free}}
* {{Citation | last1=Peano | first1=Giuseppe | author1-link=Giuseppe Peano | title=Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva | year=1888 | location=Turin|language=it}}

== Pranala luar ==

* Video pembelajaran dari Khan Academy (bahasa Inggris)
** [https://web.archive.org/web/20120426050335/http://khanexercises.appspot.com/video?v=zntNi3-ybfQ ''Introduction to bases of subspaces'']
** [https://web.archive.org/web/20120426050418/http://khanexercises.appspot.com/video?v=Zn2K8UIT8r4 ''Proof that any subspace basis has same number of elements'']
* {{Cite web|date=August 6, 2016|title=Linear combinations, span, and basis vectors|url=https://www.youtube.com/watch?v=k7RM-ot2NWY&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3|work=Essence of linear algebra|archive-url=https://ghostarchive.org/varchive/youtube/20211117/k7RM-ot2NWY|archive-date=2021-11-17|via=[[YouTube]]|url-status=live}}
* {{springer|title=Basis|id=p/b015350}}
{{Aljabar linear}}
[[Kategori:Aljabar linear]]
[[Kategori:Aljabar linear]]

Revisi per 19 Maret 2024 04.28

The cross-hatched plane is the linear span of u and v in R3.

In mathematics, the linear span (also called the linear hull[1] or just span) of a set S of vectors (from a vector space), denoted span(S),[2] is defined as the set of all linear combinations of the vectors in S.[3] For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

To express that a vector space V is a linear span of a subset S, one commonly uses the following phrases—either: S spans V, S is a spanning set of V, V is spanned/generated by S, or S is a generator or generator set of V.

Definition

Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is defined to be the intersection W of all subspaces of V that contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W.

Alternatively, the span of S may be defined as the set of all finite linear combinations of elements (vectors) of S, which follows from the above definition.[4][5][6][7]

In the case of infinite S, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.

Examples

The real vector space has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a basis. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the canonical basis of .

Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, ½, 3), (1, 1, 1)}, but this set is not a basis, because it is linearly dependent.

The set {(1, 0, 0), (0, 1, 0), (1, 1, 0)} is not a spanning set of , since its span is the space of all vectors in whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not It can be identified with by removing the third components equal to zero.

The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in , and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of monomials xn, where n is a non-negative integer, spans the space of polynomials.

Theorems

Equivalence of definitions

The set of all linear combinations of a subset S of V, a vector space over K, is the smallest linear subspace of V containing S.

Proof. We first prove that span S is a subspace of V. Since S is a subset of V, we only need to prove the existence of a zero vector 0 in span S, that span S is closed under addition, and that span S is closed under scalar multiplication. Letting , it is trivial that the zero vector of V exists in span S, since . Adding together two linear combinations of S also produces a linear combination of S: , where all , and multiplying a linear combination of S by a scalar will produce another linear combination of S: . Thus span S is a subspace of V.
Suppose that W is a linear subspace of V containing S. It follows that , since every vi is a linear combination of S (trivially). Since W is closed under addition and scalar multiplication, then every linear combination must be contained in W. Thus, span S is contained in every subspace of V containing S, and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of S.

Size of spanning set is at least size of linearly independent set

Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V.

Proof. Let be a spanning set and be a linearly independent set of vectors from V. We want to show that .
Since S spans V, then must also span V, and must be a linear combination of S. Thus is linearly dependent, and we can remove one vector from S that is a linear combination of the other elements. This vector cannot be any of the wi, since W is linearly independent. The resulting set is , which is a spanning set of V. We repeat this step n times, where the resulting set after the pth step is the union of and m - p vectors of S.
It is ensured until the nth step that there will always be some vi to remove out of S for every adjoint of v, and thus there are at least as many vi's as there are wi's—i.e. . To verify this, we assume by way of contradiction that . Then, at the mth step, we have the set and we can adjoin another vector . But, since is a spanning set of V, is a linear combination of . This is a contradiction, since W is linearly independent.

Spanning set can be reduced to a basis

Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension. This also indicates that a basis is a minimal spanning set when V is finite-dimensional.

Generalizations

Generalizing the definition of the span of points in space, a subset X of the ground set of a matroid is called a spanning set if the rank of X equals the rank of the entire ground set[butuh rujukan].

The vector space definition can also be generalized to modules.[8][9] Given an R-module A and a collection of elements a1, ..., an of A, the submodule of A spanned by a1, ..., an is the sum of cyclic modules

consisting of all R-linear combinations of the elements ai. As with the case of vector spaces, the submodule of A spanned by any subset of A is the intersection of all submodules containing that subset.

Closed linear span (functional analysis)

In functional analysis, a closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set.

Suppose that X is a normed vector space and let E be any non-empty subset of X. The closed linear span of E, denoted by or , is the intersection of all the closed linear subspaces of X which contain E.

One mathematical formulation of this is

The closed linear span of the set of functions xn on the interval [0, 1], where n is a non-negative integer, depends on the norm used. If the L2 norm is used, then the closed linear span is the Hilbert space of square-integrable functions on the interval. But if the maximum norm is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the cardinality of the set of functions in the closed linear span is the cardinality of the continuum, which is the same cardinality as for the set of polynomials.

Notes

The linear span of a set is dense in the closed linear span. Moreover, as stated in the lemma below, the closed linear span is indeed the closure of the linear span.

Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, see Riesz's lemma).

A useful lemma

Let X be a normed space and let E be any non-empty subset of X. Then

  1. is a closed linear subspace of X which contains E,
  2. , viz. is the closure of ,

(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)

See also

Citations

  1. ^ (Encyclopedia of Mathematics 2020). Linear Hull.
  2. ^ (Axler 2015) pp. 29-30, §§ 2.5, 2.8
  3. ^ (Axler 2015) p. 29, § 2.7
  4. ^ (Hefferon 2020) p. 100, ch. 2, Definition 2.13
  5. ^ (Axler 2015) pp. 29-30, §§ 2.5, 2.8
  6. ^ (Roman 2005) pp. 41-42
  7. ^ (MathWorld 2021) Vector Space Span.
  8. ^ (Roman 2005) p. 96, ch. 4
  9. ^ (Lane & Birkhoff 1999) p. 193, ch. 6

Sources

Textbooks

Web

External links

Templat:Linear algebra

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