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{{Short description|RuangSifat vektor yang dihasilkanmatematis dari kombinasi linearstruktur elemen-elemenaljabar}}{{about|aljabar diabstrak|hukum suatufisika|Prinsip himpunanArchimedes}}
[[Berkas:Archimedean_property.png|jmpl|250x250px|Ilustrasi sifat Archimedes.]]
[[Berkas:Basis_for_a_plane.svg|jmpl|Bidang yang direntang oleh vektor '''u''' dan '''v''' di '''R'''<sup>3</sup>.]]
Dalam [[aljabar abstrak]] dan [[Analisis matematika|analisis]], '''Sifat Archimedes''', dinamai menurut ahli matematika Yunani kuno [[Archimedes]] dari [[Sirakusa]], adalah sifat yang dimiliki oleh beberapa [[struktur aljabar]], seperti [[Grup (matematika)|grup]], dan [[Medan (matematika)|medan]]. Secara kasar, ini adalah sifat yang tidak memiliki elemen '' jauh lebih besar '' atau '' jauh lebih kecil ''. Adalah [[Otto Stolz]] yang memberi nama pada aksioma Archimedes karena muncul sebagai [[Aksioma]] V dari Archimedes '' [[Pada Bola dan Tabung]] ''.<ref>G. Fisher (1994) in P. Ehrlich(ed.), Bilangan Riil, Generalisasi Realisasi, dan Teori Kontinua, 107-145, Kluwer Academic</ref>
Dalam [[aljabar linear]], '''rentang linear''' atau '''span''' dari sebarang [[Himpunan (matematika)|himpunan]] <math>S</math> berisi [[Vektor Euklides|vektor-vektor]] (yang berasal dari suatu ruang vektor) adalah himpunan semua [[kombinasi linear]] dari vektor-vektor di <math>S.</math><ref>{{Harvard citation text|Axler|2015}} p. 29, § 2.7</ref> Rentang linear dari <math>S</math> umum disimbolkan dengan <math>\text{span}(S).</math><ref name=":0">{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref> Sebagai contoh, dua vektor yang saling [[Kebebasan linear|bebas linear]] akan merentang suatu [[Bidang (geometri)|bidang]]. Rentang dapat dikarakterisasikan<!-- istilah 'dikarakterisasikan' secara praktis sama saja dengan istilah 'didefinisikan', namun saya ragu untuk menggunakan padanan ini. --kekavigi --> sebagai [[Irisan (teori himpunan)|irisan]] dari semua [[Subruang vektor|subruang (vektor)]] yang mengandung <math>S,</math> maupun sebagai subruang yang mengandung <math>S.</math> Alhasil, rentang dari himpunan vektor menghasilkan suatu ruang vektor. Rentang dapat diperumum untuk [[matroid]] dan [[Modul (matematika)|modul]].
 
Gagasan tersebut muncul dari teori [[Besaran (matematika)|besaran]] Yunani Kuno; itu masih memainkan peran penting dalam matematika modern seperti [[Aksioma Hilbert|aksioma]] [[David Hilbert]] untuk geometri, dan teori [[Grup terurut linear|grup terurut]], [[Bidang terurut|medan terurut]], dan [[medan lokal]].
Untuk menyatakan bahwa suatu ruang vektor <math>V</math> adalah rentang linear dari subset <math>S,</math> beberapa pernyataan berikut umum digunakan: <math>S</math> merentang <math>V,</math> <math>S</math> adalah ''himpunan merentang'' dari <math>V,</math> <math>V</math> direntang/dibangkitkan oleh <math>S,</math> atau <math>S</math> adalah [[Pembangkit (matematika)|pembangkit]] atau himpunan pembangkit dari <math>V.</math>
 
Struktur aljabar di mana dua elemen bukan nol adalah ''sebanding '', dalam arti bahwa tidak satu pun dari mereka [[sangat kecil]] dibandingkan dengan yang lain, dikatakan '''Archimedes'''. Suatu struktur yang memiliki sepasang elemen bukan nol, yang salah satunya sangat kecil terhadap yang lain, dikatakan sebagai '''tak-Archimedes'''. Misalnya, [[grup terurut linear]] yang merupakan Archimedes adalah [[Grup Archimedean|grup Archimedes]].
== Definisi ==
Untuk sebarang [[ruang vektor]] <math>V</math> atas [[Lapangan (matematika)|lapangan]] <math>K,</math> rentang dari suatu himpunan <math>S</math> yang beranggotakan vektor-vektor (tidak harus berhingga) didefinisikan sebagai irisan <math>W</math> dari semua [[Subruang vektor|subruang]] dari <math>V</math> yang mengandung <math>S.</math> Irisan <math>W</math> disebut sebagai subruang yang ''direntang oleh'' <math>S,</math> atau oleh vektor-vektor di <math>S.</math> Kebalikannya, <math>S</math> disebut ''himpunan merentang'' dari <math>W</math>, dan kita katakan <math>S</math> ''merentang <math>W.</math>''
 
Ini dapat dibuat tepat dalam berbagai konteks dengan rumusan yang sedikit berbeda.Misalnya, dalam konteks [[kolom terurut]], satu memiliki '''aksioma Archimedes''' yang merumuskan sifat ini, di mana medan [[bilangan riil]] adalah Archimedes, tetapi [[fungsi rasional]] dalam koefisien riil tidak.
Rentang dari <math>S</math> juga dapat didefinisikan sebagai himpunan dari semua [[kombinasi linear]] terhingga dari vektor-vektor di <math>S.</math><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> Secara matematis, ini dituliskan sebagai<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>Pada kasus <math>S</math> berukuran tak-hingga, syarat kombinasi linear yang tak-terhingga (yakni, keadaan ketika kombinasi menggunakan konsep penjumlahan tak-hingga, dengan mengasumsikan penjumlahan seperti itu dapat didefinisikan) tidak disertakan dalam definisi.
 
== Contoh ==
Ruang vektor [[Bilangan riil|riil]] <math>\mathbb R^3</math> dapat direntang oleh himpunan <math>\{(-1,0,0),\,(0,1,0),\,(0,0,1)\} </math>. Himpunan ini juga merupakan suatu [[Basis (aljabar linear)|basis]] dari <math>\mathbb R^3</math>. Jika <math>(-1,0,0)</math> digantikan dengan <math>(1,0,0)</math>, himpunan tersebut merupakan [[Basis (aljabar linear)|basis standar]] dari <math>\mathbb R^3</math>. Contoh himpunan pembangkit lain dari <math>\mathbb R^3</math> adalah <math>\{(1,2,3),\, (0, 1, 2),\, (-1, \tfrac{1}{2}, 3),\, (1, 1, 1)\}</math>, namun himpunan ini bukan basis karena bersifat [[Kebebasan linear|bergantung linear]].
 
In [[abstract algebra]] and [[Mathematical analysis|analysis]], the '''Archimedean property''', named after the ancient Greek mathematician [[Archimedes]] of [[Syracuse, Italy|Syracuse]], is a property held by some [[Algebraic structure|algebraic structures]], such as ordered or normed [[Group (algebra)|groups]], and [[Field (mathematics)|fields]]. The property, typically construed, states that given two positive numbers <math>x</math> and <math>y</math>, there is an integer <math>n</math> such that <math>nx > y</math>. It also means that the set of [[natural numbers]] is not bounded above.<ref>{{cite web|title=Math 2050C Lecture|url=https://www.math.cuhk.edu.hk/course_builder/2021/math2050c/MATH%202050C%20Lecture%204%20(Jan%2021).pdf|website=cuhk.edu.hk|access-date=3 September 2023}}</ref> Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was [[Otto Stolz]] who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''[[On the Sphere and Cylinder]]''.<ref>G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107-145, Kluwer Academic</ref>
Himpunan <math>\{(1, 0, 0),\,(0, 1, 0),\,(1,1,0)\}</math> bukan himpunan merentang dari <math>\mathbb R^3</math>, karena rentangnya adalah subruang semua vektor di <math>\mathbb R^3</math> yang komponen terakhirnya bernilai <math>0.</math> Subruang tersebut juga direntang oleh himpunan <math>\{(1,0,0),\,(0,1,0)\}, </math> karena <math>(1,1,0)</math> adalah kombinasi linear dari <math>(1,0,0)</math> dan <math>(0,1,0).</math>
 
The notion arose from the theory of [[Magnitude (mathematics)|magnitudes]] of Ancient Greece; it still plays an important role in modern mathematics such as [[David Hilbert]]'s [[Hilbert's axioms|axioms for geometry]], and the theories of [[Linearly ordered group|ordered groups]], [[Ordered field|ordered fields]], and [[local fields]].
Himpunan kosong adalah himpunan merentang dari <math>\{(0, 0, 0)\}, </math> karena himpunan kosong adalah subset dari semua subruang vektor yang mungkin di <math>\mathbb R^3,</math> dan <math>\{(0, 0, 0)\} </math> adalah irisan dari semua subruang tersebut.
 
An algebraic structure in which any two non-zero elements are ''comparable'', in the sense that neither of them is [[infinitesimal]] with respect to the other, is said to be '''Archimedean'''. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be '''non-Archimedean'''. For example, a [[linearly ordered group]] that is Archimedean is an [[Archimedean group]].
Himpunan semua [[monomial]] <math>x^n,</math> dengan <math>n</math> adalah bilangan bulat non-negatif, merentang ruang [[polinomial]].
 
This can be made precise in various contexts with slightly different formulations. For example, in the context of [[Ordered field|ordered fields]], one has the '''axiom of Archimedes''' which formulates this property, where the field of [[Real number|real numbers]] is Archimedean, but that of [[rational functions]] in real coefficients is not.
== Teorema ==
 
== Sejarah dan asal nama sifat Archimedes ==
=== Kesetaraan antar definisi ===
Konsep ini dinamai oleh [[Otto Stolz]] (pada tahun 1880-an) setelah ahli geografi dan fisikawan [[Yunani kuno]] [[Archimedes]] dari [[Sirakusa]].
Untuk sebarang ruang vektor <math>V</math> atas lapangan <math>K,</math> himpunan semua kombinasi linear dari subset <math>S</math> dari <math>V,</math> adalah subruang terkecil dari <math>V</math> yang mengandung <math>S.</math>
 
Sifat Archimedes muncul di Buku V dari [[Elemen Euklides]] sebagai Definisi 4:
: ''Bukti.'' Pertama kita tunjukkan bahwa <math>\text{span}(S)</math> adalah subruang dari <math>V.</math> Karena <math>S</math> adalah subset dari <math>V,</math> kita cukup membuktikan bahwa vektor <math>\mathbf 0</math> anggota dari <math>\text{span}(S), </math> bahwa <math>\text{span}(S)</math> dibawah penjumlahan, dan bahwa <math>\text{span}(S)</math> tertutup dibawah perkalian skalar. Misalkan <math>S = \{ \mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n \}</math>, mudah ditunjukkan bahwa vektor nol di <math>V</math> ada di <math>\text{span}(S), </math> karena <math>\mathbf 0 = 0 \mathbf v_1 + 0 \mathbf v_2 + \cdots + 0 \mathbf v_n. </math> Menjumlahkan sebarang dua kombinasi linear dari <math>S</math> akan menghasilkan kombinasi linear dari <math>S</math>: <math display="block">(\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>dengan semua <math>\lambda_i, \mu_i \in K</math>, dan mengalikan sebarang kombinasi linear dari <math>S</math> dengan sebarang skalar <math>c \in K</math> akan menghasilkan kombinasi linear dari <math>S</math>: <math display="block">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>Alhasil, <math>\text{span}(S)</math> adalah subruang dari <math>V.</math>
{{Quote|Besaran dikatakan memiliki rasio satu sama lain yang dapat, jika dikalikan, melebihi satu sama lain.}}
Karena Archimedes mengkreditkannya ke [[Eudoksos dari Knidos]] itu juga dikenal sebagai "Teorema Eudoxus" atau'' aksioma Eudoxus''<ref name="Knopp1951">{{cite book|last=Knopp|first=Konrad|year=1951|url=https://archive.org/details/theoryapplicatio00knop|title=Theory and Application of Infinite Series|location=London and Glasgow|publisher=Blackie & Son, Ltd.|isbn=0-486-66165-2|edition=English 2nd|page=[https://archive.org/details/theoryapplicatio00knop/page/7 7]|authorlink=Konrad Knopp|url-access=registration}}</ref>.
 
[[Archimedes menggunakan infinitesimal]] dalam argumen [[heuristik]], meskipun ia menyangkal bahwa argumen tersebut telah selesai [[bukti matematika]].
: Misalkan <math>W</math> adalah subruang <math>V</math> yang mengandung <math>S.</math> Perhatikan bahwa <math>S \subseteq \operatorname{span} S,</math> karena semua <math>\mathbf{v}_i</math> merupakan kombinasi linear dari <math>S</math> (secara langsung). Karena <math>W</math> tertutup dibawah penjumlahan dan perkalian skalar, maka setiap kombinasi linear <math>\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n</math> harus berada di <math>W.</math> Akibatnya, <math>\text{span}(S)</math> terkandung di semua subruang dari <math>V</math> yang mengandung <math>S.</math> Lebih lanjut, irisan semua subruang tersebut, yakni subruang terkecil, sama dengan himpunan semua kombinasi linear dari <math>S.</math>
 
== Definisi untuk grup terurut linear ==
=== Size of spanning set is at least size of linearly independent set ===
{{Main|Grup Archimedean}}
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}}.
Misalkan''{{mvar|x}}'' dan''{{mvar|y}}'' menjadi [[Grup berurutan linear#Definisi|elemen positif]] dari [[grup terurut linier]] '' G ''.
 
Kemudian'''''{{mvar|x}}'' inifintesimal terhadap ''{{mvar|y}}''''' (atau ekuivalen,'''''{{mvar|y}}'' takhingga terhadap''{{mvar|x}}''''') jika, untuk setiap [[bilangan asli]] ''{{mvar|n}}'', kelipatan{{math|'' nx ''}} kurang dari ''{{mvar|y}}'', yaitu, pertidaksamaan berikut berlaku:<blockquote><math> \underset{n \text{ suku}}{\underbrace{x + \dots + x}} < y </math></blockquote>Definisi ini dapat diperluas ke seluruh kelompok dengan mengambil [[Nilai absolut|nilai mutlak]].
: ''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>.
 
Grup ''{{mvar|G}}'' adalah '''Archimedes''' jika tidak ada pasangan {{math|(''x'', ''y'')}} sedemikian rupa sehingga ''{{mvar|x}}'' sangat kecil dibandingkan dengan ''{{mvar|y}}''.
: 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}}.
 
Selain itu, jika ''{{mvar|K}}'' adalah [[struktur aljabar]] dengan satuan (1) misalnya, [[Gelanggang (matematika)|gelanggang]] ,definisi serupa berlaku untuk ''{{mvar|K}}''. Jika ''{{mvar|x}}'' sangat kecil dibandingkan dengan 1, maka ''{{mvar|x}}'' adalah '''elemen yang sangat kecil'''. Demikian juga, jika''{{mvar|y}}'' tak hingga 1, maka ''{{mvar|y}}'' adalah '''elemen takhingga'''. Struktur aljabar ''{{mvar|K}}'' adalah Archimedes jika tidak memiliki elemen takhingga dan tidak memiliki elemen takhingga.
: 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.
 
== Medan terurut ==
=== Spanning set can be reduced to a basis ===
[[Medan terurut]] memiliki beberapa sifat tambahan:
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.
 
* [[Bilangan rasional]] adalah [[Pembenaman|terbenam]] di setiap medan terurut. Artinya, setiap kolom terurut memiliki [[Karakteristik (aljabar)|karakteristik]] nol.
== Generalizations ==
* Jika''{{mvar|x}}'' Infinitesimal, maka {{math|1/''x''}} tidak terbatas, dan sebaliknya. Oleh karena itu, untuk memverifikasi bahwa medan adalah Archimedes, cukup dengan memeriksa hanya bahwa tidak ada elemen yang sangat kecil, atau untuk memeriksa bahwa tidak ada elemen yang tak terbatas.
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}}.
* Jika''{{mvar|x}}'' sangat kecil dan<var> r </var>adalah bilangan rasional, maka {{math|'' rx ''}} juga sangat kecil. Akibatnya, diberi elemen umum ''{{mvar|c}}'', tiga bilangan {{math|''c''/2}}, ''{{mvar|c}}'', dan {{math|2''c''}} bisa jadi semua sangat kecil atau semua bukan sangat kecil.
 
Dalam setelan ini, medan terurut''{{mvar|K}}'' adalah Archimedes persis ketika pernyataan berikut, disebut '''aksioma Archimedes''', menyatakan:
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.
 
: "Misalkan''{{mvar|x}}'' adalah elemen apa pun dari ''{{mvar|K}}''. Kemudian ada bilangan asli ''{{mvar|n}}'' sehingga {{math|''n'' > ''x''}}."
== Closed linear span (functional analysis) ==
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.
 
Sebagai alternatif, seseorang dapat menggunakan karakterisasi berikut:<blockquote><math>\forall\, \varepsilon \in K\big(\varepsilon > 0 \implies \exists\ n \in N : 1/n < \varepsilon\big).</math></blockquote>
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}}.
 
== Definisi untuk medan ternorma ==
One mathematical formulation of this is
Kualifikasi "Archimedes" juga diformulasikan dalam teori [[Gelanggang penilaian|peringkat satu medan nilai]] dan ruang ternorma atas peringkat satu medan nilai sebagai berikut.
 
Misalkan ''{{mvar|F}}'' adalah medan yang diberkahi dengan fungsi nilai mutlak, yaitu fungsi yang mengaitkan bilangan real 0 dengan elemen medan 0 dan mengaitkan bilangan riil positif <math>|x|</math> dengan setiap bukan nol {{math|''x'' &isin; ''F''}} dan dirumuskan<blockquote><math> |xy|=|x| |y|</math> dan <math>|x+y| \le |x|+|y| </math>.</blockquote>Kemudian, ''{{mvar|F}}'' dikatakan '''Archimedes''' jika ada bukan nol {{math|''x'' &isin; ''F''}} ada [[bilangan asli]] ''{{mvar|n}}'' dirumuskan<blockquote><math> |\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. \, </math></blockquote>Demikian pula, ruang bernorma adalah Archimedes jika jumlah ''{{mvar|n}}'' suku, masing-masing sama dengan vektor bukan-nol ''{{mvar|x}}'', memiliki norma yang lebih besar dari satu untuk cukup besar ''{{mvar|n}}''.
: <math>\overline{\operatorname{Sp}}(E) = \{u\in X | \forall\varepsilon > 0\,\exists x\in\operatorname{Sp}(E) : \|x - u\|<\varepsilon\}.</math>
 
medan dengan nilai mutlak atau ruang bernorma adalah Archimedes atau memenuhi ketentuan yang lebih kuat, yang disebut sebagai [[ultrametrik]] [[pertidaksamaan segitiga]]<math>|x+y| \le \max(|x|,|y|)</math>,
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.
 
masing-masing. medan atau ruang bernorma yang memenuhi pertidaksamaan segitiga ultrametrik disebut '''tak-Archimedes'''.
=== 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.
 
Konsep ruang linier bernorma tak-Archimedes diperkenalkan oleh A. F. Monna.<ref name="monna1">Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74–84.</ref>
Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, see [[Riesz's lemma]]).
 
=== ALihat usefulpula lemma ===
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.)
 
* {{annotated link|0.999...#Infinitesimal|0.999...}}
== Catatan kaki ==
* {{annotated link|Ruang vektor berurutan Archimedean}}
<references responsive="" />
* {{annotated link|Konstruksi bilangan riil}}
 
== Daftar pustakaCatatan ==
<references responsive="1"></references>
 
=== BukuReferensi ===
{{refbegin}}
 
* {{Cite book|last=Schechter|first=Eric|year=1997|url=http://www.math.vanderbilt.edu/~schectex/ccc/|title=Handbook of Analysis and its Foundations|publisher=Academic Press|isbn=0-12-622760-8|ref=harv|postscript=.|authorlink=Eric Schechter|access-date=2009-01-30|archive-url=https://web.archive.org/web/20150307061351/http://www.math.vanderbilt.edu/%7Eschectex/ccc/|archive-date=2015-03-07|url-status=dead}}
* {{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}}
{{refend}}
* {{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.
 
== History and origin of the name of the Archimedean property ==
=== Situs ===
The concept was named by [[Otto Stolz]] (in the 1880s) after the [[Ancient Greece|ancient Greek]] geometer and physicist [[Archimedes]] of [[Syracuse, Italy|Syracuse]].
 
The Archimedean property appears in Book V of [[Euclid's Elements|Euclid's ''Elements'']] as Definition 4:
* {{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}}
{{Blockquote|Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.}}
* {{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}}
Because Archimedes credited it to [[Eudoxus of Cnidus]] it is also known as the "Theorem of Eudoxus" or the ''Eudoxus axiom''.<ref name="Knopp19512">{{cite book|last=Knopp|first=Konrad|year=1951|url=https://archive.org/details/theoryapplicatio00knop|title=Theory and Application of Infinite Series|location=London and Glasgow|publisher=Blackie & Son, Ltd.|isbn=0-486-66165-2|edition=English 2nd|page=[https://archive.org/details/theoryapplicatio00knop/page/7 7]|author-link=Konrad Knopp|url-access=registration}}</ref>
* {{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}}
 
[[Archimedes's use of infinitesimals|Archimedes used infinitesimals]] in [[heuristic]] arguments, although he denied that those were finished [[Mathematical proof|mathematical proofs]].
== Pranala luar ==
 
== Definition for linearly ordered groups ==
* [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.
{{Main|Archimedean group}}
* {{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}}
Let {{mvar|x}} and {{mvar|y}} be [[Linearly ordered group#Definitions|positive elements]] of a [[linearly ordered group]] ''G''. Then <math>x</math> '''is infinitesimal with respect to''' <math>y</math> (or equivalently, <math>y</math> '''is infinite with respect to''' <math>x</math>) if, for any [[natural number]] <math>n</math>, the multiple <math>nx</math> is less than <math>y</math>, that is, the following inequality holds:<math display="block"> \underbrace{x+\cdots+x}_{n\text{ terms}} < y. \, </math>This definition can be extended to the entire group by taking absolute values.
{{Aljabar linear}}
 
The group <math>G</math> is '''Archimedean''' if there is no pair <math>(x,y)</math> such that <math>x</math> is infinitesimal with respect to <math>y</math>.
 
Additionally, if <math>K</math> is an [[algebraic structure]] with a unit (1) — for example, a [[Ring (mathematics)|ring]] — a similar definition applies to <math>K</math>. If <math>x</math> is infinitesimal with respect to <math>1</math>, then <math>x</math> is an '''infinitesimal element'''. Likewise, if <math>y</math> is infinite with respect to <math>1</math>, then <math>y</math> is an '''infinite element'''. The algebraic structure <math>K</math> is Archimedean if it has no infinite elements and no infinitesimal elements.
 
=== Ordered fields ===
[[Ordered field|Ordered fields]] have some additional properties:
 
* The rational numbers are [[Embedding|embedded]] in any ordered field. That is, any ordered field has [[Characteristic (algebra)|characteristic]] zero.
* If <math>x</math> is infinitesimal, then <math>1/x</math> is infinite, and vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements.
* If <math>x</math> is infinitesimal and <math>r</math> is a rational number, then <math>rx</math> is also infinitesimal. As a result, given a general element <math>c</math>, the three numbers <math>c/2</math>, <math>c</math>, and <math>2c</math> are either all infinitesimal or all non-infinitesimal.
 
In this setting, an ordered field {{mvar|K}} is Archimedean precisely when the following statement, called the '''axiom of Archimedes''', holds:
 
: "Let <math>x</math> be any element of <math>K</math>. Then there exists a natural number <math>n</math> such that <math>n > x</math>."
 
Alternatively one can use the following characterization:<math display="block">\forall\, \varepsilon \in K\big(\varepsilon > 0 \implies \exists\ n \in N : 1/n < \varepsilon\big).</math>
 
== Definition for normed fields ==
The qualifier "Archimedean" is also formulated in the theory of [[Valuation ring|rank one valued fields]] and normed spaces over rank one valued fields as follows. Let <math>K</math> be a field endowed with an absolute value function, i.e., a function which associates the real number <math>0</math> with the field element 0 and associates a positive real number <math>|x|</math> with each non-zero <math>x \in K</math> and satisfies <math>|xy|=|x| |y|</math> and <math>|x+y| \le |x|+|y|</math>. Then, <math>K</math> is said to be '''Archimedean''' if for any non-zero <math>x \in K</math> there exists a [[natural number]] <math>n</math> such that<math display="block">|\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. </math>Similarly, a normed space is Archimedean if a sum of <math>n</math> terms, each equal to a non-zero vector <math>x</math>, has norm greater than one for sufficiently large <math>n</math>. A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the [[ultrametric]] [[triangle inequality]],<math display="block">|x+y| \le \max(|x|,|y|) ,</math>respectively. A field or normed space satisfying the ultrametric triangle inequality is called '''non-Archimedean'''.
 
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.<ref name="monna1943">{{cite journal|last1=Monna|first1=A. F.|date=1943|title=Over een lineaire ''P''-adische ruimte|journal=Nederl. Akad. Wetensch. Verslag Afd. Natuurk.|issue=52|pages=74–84|mr=15678}}</ref>
 
== Examples and non-examples ==
 
=== Archimedean property of the real numbers ===
The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function <math>|x|=1</math>, when <math>x \neq 0</math>, the more usual <math display="inline">|x| = \sqrt{x^2}</math>, and the <math>p</math>'''-adic absolute value''' functions. By [[Ostrowski's theorem]], every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some <math>p</math>-adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete. The completion with respect to the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.<ref>[[Neal Koblitz]], "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977.</ref> On the other hand, the completions with respect to the other non-trivial absolute values give the fields of [[P-adic number|p-adic numbers]], where <math>p</math> is a prime integer number (see below); since the <math>p</math>-adic absolute values satisfy the [[ultrametric]] property, then the <math>p</math>-adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).<!-- "by axiom" side -->In the [[axiomatic theory of real numbers]], the non-existence of nonzero infinitesimal real numbers is implied by the [[least upper bound property]] as follows. Denote by <math>Z</math> the set consisting of all positive infinitesimals. This set is bounded above by <math>1</math>. Now [[Proof by contradiction|assume for a contradiction]] that <math>Z</math> is nonempty. Then it has a [[least upper bound]] <math>c</math>, which is also positive, so <math>c/2 < c < 2c</math>. Since {{mvar|c}} is an [[upper bound]] of <math>Z</math> and <math>2c</math> is strictly larger than <math>c</math>, <math>2c</math> is not a positive infinitesimal. That is, there is some natural number <math>n</math> for which <math>1/n < 2c</math>. On the other hand, <math>c/2</math> is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal <math>x</math> between <math>c/2</math> and <math>c</math>, and if <math>1/k < c/2 \leq x</math> then <math>x</math> is not infinitesimal. But <math>1/(4n) < c/2</math>, so <math>c/2</math> is not infinitesimal, and this is a contradiction. This means that <math>Z</math> is empty after all: there are no positive, infinitesimal real numbers.
 
The Archimedean property of real numbers holds also in [[constructive analysis]], even though the least upper bound property may fail in that context.
 
=== Non-Archimedean ordered field ===
{{main article|Non-Archimedean ordered field}}
For an example of an [[ordered field]] that is not Archimedean, take the field of [[Rational function|rational functions]] with real coefficients. (A rational function is any function that can be expressed as one [[polynomial]] divided by another polynomial; we will assume in what follows that this has been done in such a way that the [[leading coefficient]] of the denominator is positive.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now <math>f > g</math> if and only if <math>f - g > 0</math>, so we only have to say which rational functions are considered positive. Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function <math>1/x</math> is positive but less than the rational function <math>1</math>. In fact, if <math>n</math> is any natural number, then <math>n(1/x) = n/x</math> is positive but still less than <math>1</math>, no matter how big <math>n</math> is. Therefore, <math>1/x</math> is an infinitesimal in this field.
 
This example generalizes to other coefficients. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say <math>y</math>, produces an example with a different [[order type]].
 
=== Non-Archimedean valued fields ===
The field of the rational numbers endowed with the p-adic metric and the [[p-adic number]] fields which are the completions, do not have the Archimedean property as fields with absolute values. <!-- Another example is the [[hyperreal numbers]] of [[nonstandard analysis]]. : (ed. I detest this, because the formal interpretation of the Axiom of Archimedes is indeed satisfied by hypernatural numbers in place of the "standard" natural numbers, which do not form a "hyperset" (or *-set, superset, whatever we call it) inside the system of the "hyperreal numbers".)--> All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.<ref name="shell1">Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990. {{ISBN|0-8247-8412-X}}</ref>
 
=== Equivalent definitions of Archimedean ordered field ===
Every linearly ordered field <math>K</math> contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit <math>1</math> of <math>K</math>, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered [[monoid]]<!-- semigroup -->. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in <math>K</math>. The following are equivalent characterizations of Archimedean fields in terms of these substructures.<ref name="Schechter">{{harvnb|Schechter|1997|loc=§10.3}}</ref>
 
# The natural numbers are [[Cofinal (mathematics)|cofinal]] in <math>K</math>. That is, every element of <math>K</math> is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound.
# Zero is the [[infimum]] in <math>K</math> of the set <math>\{1/2, 1/3, 1/4, \dots\}</math>. (If <math>K</math> contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.)
# The set of elements of <math>K</math> between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set <math>\{0\}</math> when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of infinitesimals is closed. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between. Consequently, any non-Archimedean ordered field is both incomplete and disconnected.
# For any <math>x</math> in <math>K</math> the set of integers greater than <math>x</math> has a least element. (If <math>x</math> were a negative infinite quantity every integer would be greater than it.)
# Every nonempty open interval of <math>K</math> contains a rational. (If <math>x</math> is a positive infinitesimal, the open interval <math>(x,2x)</math> contains infinitely many infinitesimals but not a single rational.)
# The rationals are [[Dense set|dense]] in <math>K</math> with respect to both sup and inf. (That is, every element of <math>K</math> is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.
 
== See also ==
 
* {{annotated link|0.999...#Infinitesimals|0.999...}}
* {{annotated link|Archimedean ordered vector space}}
* {{annotated link|Construction of the real numbers}}
 
== Notes ==
{{reflist}}
 
== References ==
{{refbegin}}
 
* {{Cite book|last=Schechter|first=Eric|year=1997|url=http://www.math.vanderbilt.edu/~schectex/ccc/|title=Handbook of Analysis and its Foundations|publisher=Academic Press|isbn=0-12-622760-8|author-link=Eric Schechter|access-date=2009-01-30|archive-url=https://web.archive.org/web/20150307061351/http://www.math.vanderbilt.edu/%7Eschectex/ccc/|archive-date=2015-03-07|url-status=dead}}
{{refend}}
----
----{{short description|Gambar invers nol di bawah homomorfisme}}
 
Dalam [[aljabar]], '''kernel''' dari [[homomorfisme]] (fungsi yang mempertahankan [[Struktur aljabar|struktur]]) umumnya [[gambar invers]] dari 0 (kecuali untuk [[Grup (matematika)|grup]] yang operasinya dilambangkan dengan multi, dimana kernel adalah kebalikan dari gambar 1). Kasus khusus yang penting adalah [[Kernel (aljabar linear)|kernel dari peta linear]]. [[Kernel (matriks)|kernel dari matriks]], juga disebut ''ruang nol'', adalah kernel dari peta linear yang ditentukan oleh matriks.
 
Kernel homomorfisme direduksi menjadi 0 (atau 1) jika dan hanya jika homomorfisme tersebut adalah [[Fungsi injeksi|injeksi]], Artinya jika gambar invers dari setiap elemen terdiri dari satu elemen. Ini berarti bahwa kernel dapat dilihat sebagai ukuran sejauh mana homomorfisme gagal untuk diinjeksi.<ref>See {{harvnb|Dummit|Foote|2004}} and {{harvnb|Lang|2002}}.</ref>
 
Untuk beberapa jenis struktur, seperti [[grup abelian]] dan [[ruang vektor]], kemungkinan kernel adalah substruktur dari jenis yang sama. Ini tidak selalu terjadi, dan terkadang, kemungkinan kernel telah menerima nama khusus, seperti [[subgrup normal]] untuk kelompok dan [[ideal dua sisi]] untuk [[Cincin (matematika)|cincin]].
 
Kernel memungkinkan untuk menentukan [[objek hasil bagi]] (juga disebut [[Hasil bagi (aljabar universal)|aljabar hasil bagi]] di [[aljabar universal]], dan [[kokernel]] di [[teori kategori]]). Untuk banyak jenis struktur aljabar, [[teorema fundamental homomorfisme]] (atau [[teorema isomorfisme pertama]]) menyatakan bahwa [[Galeri (matematika)|galeri]] dari homomorfisme adalah [[Isomorfisme|isomorfik]] terhadap hasil bagi oleh kernel.
 
Konsep kernel telah diperluas ke struktur sedemikian rupa sehingga gambar kebalikan dari satu elemen tidak cukup untuk memutuskan apakah homomorfisme adalah injeksi. Dalam kasus ini, kernel adalah [[hubungan kesesuaian]].
 
Artikel ini adalah survei untuk beberapa jenis kernel penting dalam struktur aljabar.
 
== Linear maps ==
{{Main|Kernel (aljabar linear)}}
Misalkan ''V'' dan ''W'' menjadi [[ruang vektor]] di atas [[Bidang (matematika)|bidang]] (atau lebih umum, [[Modul (matematika)|modul]] di atas [[Gelanggang (matematika)|gelanggang]] dan biarkan ''T'' menjadi [[peta liear]] dari ''V'' ke ''W''. Jika '''0'''<sub>''W''</sub> adalah [[vektor nol]] dari ''W'' , maka kernel ''T'' adalah [[preimage]] dari [[Ruang nol|nol subruang]] {'''0'''<sub>''W''</sub>}; that adalah, [[himpunan bagian]] dari ''V'' yang terdiri dari semua elemen ''V'' yang dipetakan oleh ''T'' ke elemen '''0'''<sub>''W''</sub>. Kernel biasanya dilambangkan sebagai {{math|ker '' T ''}}, atau variasinya:
 
: <math> \operatorname{ker} T = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}_{W}\}\text{.} </math>
 
Karena peta linier mempertahankan vektor nol, vektor nol '''0'''<sub>''V''</sub> dari ''V'' harus menjadi milik kernel. Transformasi ''T'' bersifat injeksi jika dan hanya jika kernelnya direduksi menjadi subruang nol.
 
Kernel ker ''T'' selalu merupakan [[subruang linier]] dari ''V'' . Jadi, masuk akal untuk membicarakan tentang [[Ruang hasil bagi (aljabar linear)|ruang hasil bagi]] ''V''/(ker ''T''). Teorema isomorfisme pertama untuk ruang vektor menyatakan bahwa ruang hasil bagi ini adalah [[Isomorfisme alami|isomorfis alami]] ke [[Citra (fungsi)|citra]] dari ''T'' (yang merupakan subruang dari ''W'' ). Akibatnya, [[Dimensi (aljabar linear)|dimensi]] dari ''V'' sama dengan dimensi kernel ditambah dimensi bayangan.
 
Jika ''V'' dan ''W'' adalah [[Ruang vektor berdimensi-hingga|dimensi-hingga]] dan [[Basis (aljabar linear)|basis]] telah dipilih, maka ''T'' dapat dijelaskan oleh [[Matriks (matematika)|matriks]] ''M'', dan kernel dapat dihitung dengan menyelesaikan [[sistem persamaan linear]] homogen {{nowrap|1=''M'''''v''' = '''0'''}}. Dalam hal ini, kernel ''T'' dapat diidentifikasi ke [[Kernel (matriks)|kernel matriks]] ''M'' , juga disebut "spasi nol" dari ''M'' . Dimensi ruang kosong, disebut nulitas ''M'' , diberikan oleh jumlah kolom ''M'' dikurangi [[Rank (teori matriks)|rank]] dari ''M'' , sebagai konsekuensi dari [[teori peringkat-nullity]].
 
Memecahkan [[persamaan diferensial homogen]] sering kali sama dengan menghitung kernel [[operator diferensial]] tertentu. Misalnya, untuk mencari semua dua kali - [[fungsi terdiferensiasi]] s ''f'' dari [[garis nyata]] ke dirinya sendiri sehingga
 
: <math>xf''(x) + 3f'(x) = f(x),</math>
 
biarkan ''V'' menjadi ruang dari semua fungsi yang dapat dibedakan dua kali, biarkan ''W'' menjadi ruang dari semua fungsi, dan tentukan operator linier ''T'' dari ''V'' menjadi ''W'' oleh
 
: <math>(Tf)(x) = xf''(x) + 3f'(x) - f(x)</math>
 
untuk ''f'' di ''V'' dan ''x'' sembarang [[bilangan real]]. Maka semua solusi persamaan diferensial ada di ker ''T'' .
 
Seseorang dapat mendefinisikan kernel untuk homomorfisme antara modul melalui [[Gelanggang (matematika)|gelanggang]] dengan cara yang analog. Ini termasuk kernel untuk homomorfisme antara [[grup abelian]] sebagai kasus khusus. Contoh ini menangkap esensi kernel secara umum [[kategori abelian]]; lihat [[Kernel (teori kategori)]].
 
== Aljabar dengan struktur nonaljabar ==
Kadang-kadang aljabar dilengkapi dengan struktur nonaljabar di samping operasi aljabar mereka. Misalnya, seseorang dapat mempertimbangkan [[grup topologi]] atau [[ruang vektor topologis]], dengan dilengkapi dengan [[Topologi (struktur)|topologi]]. Dalam hal ini, kita mengharapkan homomorfisme ''f'' untuk mempertahankan struktur tambahan ini; dalam contoh topologi, kita ingin ''f'' menjadi [[peta kontinu]]. Prosesnya mungkin mengalami hambatan dengan aljabar hasil bagi, yang mungkin tidak berperilaku baik. Dalam contoh topologi, kita dapat menghindari masalah dengan mensyaratkan bahwa struktur aljabar topologi menjadi [[Ruang Hausdorff|Hausdorff]] (seperti yang biasanya dilakukan); maka kernel (bagaimanapun itu dibangun) akan menjadi [[set tertutup]] dan [[Ruang hasil bagi (topologi)|ruang hasil bagi]] akan berfungsi dengan baik (dan juga Hausdorff).
 
== Kernel dalam teori kategori ==
Pengertian ''kernel'' dalam [[teori kategori]] adalah generalisasi dari kernel abelian aljabar; lihat [[Kernel (teori kategori)]]. Generalisasi kategorikal dari kernel sebagai hubungan kesesuaian adalah ''[[pasangan kernel]]'' . (Ada juga pengertian [[kernel perbedaan]], atau biner [[Equalizer (matematika)|equalizer]].)
 
== Lihat pula ==
 
* [[Kernel (aljabar linear)]]
* [[Himpunan nol]]
 
== Catatan ==
<references responsive="1"></references>
 
== Referensi ==
 
* {{Cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|year=2004|title=Abstract Algebra|publisher=[[John Wiley & Sons|Wiley]]|isbn=0-471-43334-9|edition=3rd|ref=harv}}
 
* {{Cite book|last=Lang|first=Serge|year=2002|title=Algebra|publisher=[[Springer Science+Business Media|Springer]]|isbn=0-387-95385-X|series=[[Graduate Texts in Mathematics]]|ref=harv|authorlink=Serge Lang}}
 
----
{{short description|Inverse image of zero under a homomorphism}}
In [[algebra]], the '''kernel''' of a [[homomorphism]] (function that preserves the [[Algebraic structure|structure]]) is generally the [[inverse image]] of 0 (except for [[Group (mathematics)|groups]] whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the [[Kernel (linear algebra)|kernel of a linear map]]. The [[Kernel (matrix)|kernel of a matrix]], also called the ''null space'', is the kernel of the linear map defined by the matrix.
 
The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is [[Injective function|injective]], that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.<ref>See {{harvp|Dummit|Foote|2004}} and {{harvp|Lang|2002}}.</ref>
 
For some types of structure, such as [[Abelian group|abelian groups]] and [[Vector space|vector spaces]], the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as [[normal subgroup]] for groups and [[Two-sided ideal|two-sided ideals]] for [[Ring (mathematics)|rings]].
 
Kernels allow defining [[Quotient object|quotient objects]] (also called [[Quotient (universal algebra)|quotient algebras]] in [[universal algebra]], and [[Cokernel|cokernels]] in [[category theory]]). For many types of algebraic structure, the [[fundamental theorem on homomorphisms]] (or [[first isomorphism theorem]]) states that [[Image (mathematics)|image]] of a homomorphism is [[Isomorphism|isomorphic]] to the quotient by the kernel.
 
The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a [[congruence relation]].
 
This article is a survey for some important types of kernels in algebraic structures.
 
== Survey of examples ==
 
=== Linear maps ===
{{Main|Kernel (linear algebra)}}
Let ''V'' and ''W'' be [[Vector space|vector spaces]] over a [[Field (mathematics)|field]] (or more generally, [[Module (mathematics)|modules]] over a [[Ring (mathematics)|ring]]) and let ''T'' be a [[linear map]] from ''V'' to ''W''. If '''0'''<sub>''W''</sub> is the [[zero vector]] of ''W'', then the kernel of ''T'' is the [[preimage]] of the [[Zero space|zero subspace]] {'''0'''<sub>''W''</sub>}; that is, the [[subset]] of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element '''0'''<sub>''W''</sub>. The kernel is usually denoted as {{nowrap|ker ''T''}}, or some variation thereof:
 
: <math> \ker T = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}_{W}\} . </math>
 
Since a linear map preserves zero vectors, the zero vector '''0'''<sub>''V''</sub> of ''V'' must belong to the kernel. The transformation ''T'' is injective if and only if its kernel is reduced to the zero subspace.
 
The kernel ker ''T'' is always a [[linear subspace]] of ''V''. Thus, it makes sense to speak of the [[Quotient space (linear algebra)|quotient space]] {{nowrap|''V'' / (ker ''T'')}}. The first isomorphism theorem for vector spaces states that this quotient space is [[Natural isomorphism|naturally isomorphic]] to the [[Image (function)|image]] of ''T'' (which is a subspace of ''W''). As a consequence, the [[Dimension (linear algebra)|dimension]] of ''V'' equals the dimension of the kernel plus the dimension of the image.
 
If ''V'' and ''W'' are [[Finite-dimensional vector space|finite-dimensional]] and [[Basis (linear algebra)|bases]] have been chosen, then ''T'' can be described by a [[Matrix (mathematics)|matrix]] ''M'', and the kernel can be computed by solving the homogeneous [[system of linear equations]] {{nowrap|1=''M'''''v''' = '''0'''}}. In this case, the kernel of ''T'' may be identified to the [[Kernel (matrix)|kernel of the matrix]] ''M'', also called "null space" of ''M''. The dimension of the null space, called the nullity of ''M'', is given by the number of columns of ''M'' minus the [[Rank (matrix theory)|rank]] of ''M'', as a consequence of the [[rank–nullity theorem]].
 
Solving [[Homogeneous differential equation|homogeneous differential equations]] often amounts to computing the kernel of certain [[Differential operator|differential operators]]. For instance, in order to find all twice-[[Differentiable function|differentiable functions]] ''f'' from the [[real line]] to itself such that
 
: <math>x f''(x) + 3 f'(x) = f(x),</math>
 
let ''V'' be the space of all twice differentiable functions, let ''W'' be the space of all functions, and define a linear operator ''T'' from ''V'' to ''W'' by
 
: <math>(Tf)(x) = x f''(x) + 3 f'(x) - f(x)</math>
 
for ''f'' in ''V'' and ''x'' an arbitrary [[real number]]. Then all solutions to the differential equation are in {{nowrap|ker ''T''}}.
 
One can define kernels for homomorphisms between modules over a [[Ring (mathematics)|ring]] in an analogous manner. This includes kernels for homomorphisms between [[Abelian group|abelian groups]] as a special case. This example captures the essence of kernels in general [[abelian categories]]; see [[Kernel (category theory)]].
 
=== Group homomorphisms ===
Let ''G'' and ''H'' be [[Group (mathematics)|groups]] and let ''f'' be a [[group homomorphism]] from ''G'' to ''H''. If ''e<sub>H</sub>'' is the [[identity element]] of ''H'', then the ''kernel'' of ''f'' is the preimage of the singleton set {''e<sub>H</sub>''}; that is, the subset of ''G'' consisting of all those elements of ''G'' that are mapped by ''f'' to the element ''e<sub>H</sub>''.
 
The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
 
: <math> \ker f = \{g \in G : f(g) = e_{H}\} .</math>
 
Since a group homomorphism preserves identity elements, the identity element ''e<sub>G</sub>'' of ''G'' must belong to the kernel.
 
The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e<sub>G</sub>''}. If ''f'' were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist {{nowrap|''a'', ''b'' &isin; ''G''}} such that {{nowrap|''a'' ≠ ''b''}} and {{nowrap|1=''f''(''a'') = ''f''(''b'')}}. Thus {{nowrap|1=''f''(''a'')''f''(''b'')<sup>−1</sup> = ''e''<sub>''H''</sub>}}. ''f'' is a group homomorphism, so inverses and group operations are preserved, giving {{nowrap|1=''f''(''ab''<sup>−1</sup>) = ''e''<sub>''H''</sub>}}; in other words, {{nowrap|''ab''<sup>−1</sup> &isin; ker ''f''}}, and ker ''f'' would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element {{nowrap|''g'' ≠ ''e''<sub>''G''</sub> &isin; ker ''f''}}, then {{nowrap|1=''f''(''g'') = ''f''(''e''<sub>''G''</sub>) = ''e''<sub>''H''</sub>}}, thus ''f'' would not be injective.
 
{{nowrap|ker ''f''}} is a [[subgroup]] of ''G'' and further it is a [[normal subgroup]]. Thus, there is a corresponding [[quotient group]] {{nowrap|''G'' / (ker ''f'')}}. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the [[Isomorphism theorems|first isomorphism theorem]] for groups.
 
In the special case of [[Abelian group|abelian groups]], there is no deviation from the previous section.
 
==== Example ====
Let ''G'' be the [[cyclic group]] on 6 elements {{nowrap|{{mset|0, 1, 2, 3, 4, 5}}}} with [[Modular arithmetic|modular addition]], ''H'' be the cyclic on 2 elements {{nowrap|{{mset|0, 1}}}} with modular addition, and ''f'' the homomorphism that maps each element ''g'' in ''G'' to the element ''g'' modulo 2 in ''H''. Then {{nowrap|ker ''f'' {{=}} {0, 2, 4}}}, since all these elements are mapped to 0<sub>''H''</sub>. The quotient group {{nowrap|''G'' / (ker ''f'')}} has two elements: {{nowrap|{{mset|0, 2, 4}}}} and {{nowrap|{{mset|1, 3, 5}}}}. It is indeed isomorphic to ''H''.
 
=== Ring homomorphisms ===
{{Ring theory sidebar}}
 
Let ''R'' and ''S'' be [[Ring (mathematics)|rings]] (assumed [[Unital algebra|unital]]) and let ''f'' be a [[ring homomorphism]] from ''R'' to ''S''. If 0<sub>''S''</sub> is the [[zero element]] of ''S'', then the ''kernel'' of ''f'' is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the [[zero ideal]] {{mset|0<sub>''S''</sub>}}, which is, the subset of ''R'' consisting of all those elements of ''R'' that are mapped by ''f'' to the element 0<sub>''S''</sub>. The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
 
: <math> \operatorname{ker} f = \{r \in R : f(r) = 0_{S}\} .</math>
 
Since a ring homomorphism preserves zero elements, the zero element 0<sub>''R''</sub> of ''R'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {{mset|0<sub>''R''</sub>}}. This is always the case if ''R'' is a [[Field (mathematics)|field]], and ''S'' is not the [[zero ring]].
 
Since ker ''f'' contains the multiplicative identity only when ''S'' is the zero ring, it turns out that the kernel is generally not a [[subring]] of ''R.'' The kernel is a sub[[Rng (algebra)|rng]], and, more precisely, a two-sided [[Ideal (ring theory)|ideal]] of ''R''. Thus, it makes sense to speak of the [[quotient ring]] {{nowrap|''R'' / (ker ''f'')}}. The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of ''f'' (which is a subring of ''S''). (Note that rings need not be unital for the kernel definition).
 
To some extent, this can be thought of as a special case of the situation for modules, since these are all [[Bimodule|bimodules]] over a ring ''R'':
 
* ''R'' itself;
* any two-sided ideal of ''R'' (such as ker ''f'');
* any quotient ring of ''R'' (such as {{nowrap|''R'' / (ker ''f'')}}); and
* the [[codomain]] of any ring homomorphism whose domain is ''R'' (such as ''S'', the codomain of ''f'').
 
However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.
 
This example captures the essence of kernels in general [[Mal'cev algebra|Mal'cev algebras]].
 
=== Monoid homomorphisms ===
Let ''M'' and ''N'' be [[Monoid (algebra)|monoids]] and let ''f'' be a [[monoid homomorphism]] from ''M'' to ''N''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''M'' × ''M''}} consisting of all those [[Ordered pair|ordered pairs]] of elements of ''M'' whose components are both mapped by ''f'' to the same element in ''N''. The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation thereof). In symbols:
 
: <math>\operatorname{ker} f = \left\{\left(m, m'\right) \in M \times M : f(m) = f\left(m'\right)\right\}.</math>
 
Since ''f'' is a [[Function (mathematics)|function]], the elements of the form {{nowrap|(''m'', ''m'')}} must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the [[Equality (mathematics)|diagonal set]] {{nowrap|{{mset|(''m'', ''m'') : ''m'' in ''M''}}}}.
 
It turns out that {{nowrap|ker ''f''}} is an [[equivalence relation]] on ''M'', and in fact a [[congruence relation]]. Thus, it makes sense to speak of the [[quotient monoid]] {{nowrap|''M'' / (ker ''f'')}}. The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of ''f'' (which is a [[submonoid]] of ''N''; for the congruence relation).
 
This is very different in flavour from the above examples. In particular, the preimage of the identity element of ''N'' is ''not'' enough to determine the kernel of ''f''.
 
== Universal algebra ==
All the above cases may be unified and generalized in [[universal algebra]].
 
=== General case ===
Let ''A'' and ''B'' be [[Algebraic structure|algebraic structures]] of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''A'' × ''A''}} consisting of all those [[Ordered pair|ordered pairs]] of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''. The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
 
: <math>\operatorname{ker} f = \left\{\left(a, a'\right) \in A \times A : f(a) = f\left(a'\right)\right\}\mbox{.}</math>
 
Since ''f'' is a [[Function (mathematics)|function]], the elements of the form {{nowrap|(''a'', ''a'')}} must belong to the kernel.
 
The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set {{nowrap|{{mset|(''a'', ''a'') : ''a'' &isin; ''A''}}}}.
 
It is easy to see that ker ''f'' is an [[equivalence relation]] on ''A'', and in fact a [[congruence relation]]. Thus, it makes sense to speak of the [[Quotient (universal algebra)|quotient algebra]] {{nowrap|''A'' / (ker ''f'')}}. The [[Isomorphism theorem#General|first isomorphism theorem]] in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a [[subalgebra]] of ''B'').
 
Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely [[Set (mathematics)|set]]-theoretic concept. For more on this general concept, outside of abstract algebra, see [[kernel of a function]].
 
== Algebras with nonalgebraic structure ==
Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider [[Topological group|topological groups]] or [[Topological vector space|topological vector spaces]], which are equipped with a [[Topology (structure)|topology]]. In this case, we would expect the homomorphism ''f'' to preserve this additional structure; in the topological examples, we would want ''f'' to be a [[continuous map]]. The process may run into a snag with the quotient algebras, which may not be well-behaved. In the topological examples, we can avoid problems by requiring that topological algebraic structures be [[Hausdorff space|Hausdorff]] (as is usually done); then the kernel (however it is constructed) will be a [[closed set]] and the [[Quotient space (topology)|quotient space]] will work fine (and also be Hausdorff).
 
== Kernels in category theory ==
The notion of ''kernel'' in [[category theory]] is a generalisation of the kernels of abelian algebras; see [[Kernel (category theory)]]. The categorical generalisation of the kernel as a congruence relation is the ''[[kernel pair]]''. (There is also the notion of [[difference kernel]], or binary [[Equalizer (mathematics)|equaliser]].)
 
== See also ==
 
* [[Kernel (linear algebra)]]
* [[Zero set]]
 
== Notes ==
{{reflist}}
 
== References ==
{{refbegin}}
* {{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|year=2004|title=Abstract Algebra|publisher=[[John Wiley & Sons|Wiley]]|isbn=0-471-43334-9|edition=3rd}}
* {{cite book|last=Lang|first=Serge|year=2002|title=Algebra|publisher=[[Springer Science+Business Media|Springer]]|isbn=0-387-95385-X|series=[[Graduate Texts in Mathematics]]|author-link=Serge Lang}}
{{refend}}
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