Pengguna:Kekavigi/bak pasir

"Vektor basis" beralih ke halaman ini. Untuk vektor basis dalam konteks kristal, lihat Struktur kristal. Untuk konsep yang lebih umum dalam fisika, lihat Kerangka acuan.

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 vektor B dalam suatu ruang vektor V disebut basis, jika setiap elemen di V dapat dituliskan sebagai kombinasi linear terhingga yang unik dari elemen-elemen di B. Koefisien-koefisien pada kombinasi linear tersebut disebut sebagai koordinat dari vektor atas B. Elemen-elemen dari basis disebut sebagai vektor basis. Basis juga dapat didefinisikan sebagai himpunan B yang elemen-elemennya saling bebas linear dan setiap elemen di V adalah kombinasi linear dari elemen-elemen di B. Dengan kata lain, basis adalah himpunan merentang (spanning) yang bebas linear.

Suatu ruang vektor dapat memiliki beberapa basis; namun semua basis tersebut akan memiliki jumlah elemen yang sama, yang disebut sebagai 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 formal

Basis untuk ruang vektor $V$ (atas medan $F$) adalah suatu himpunan bagian $B\subset V$ yang memenuhi:

1. Setiap $\mathbf {v} \in V$ dapat dituliskan sebagai $\mathbf {v} =\sum _{i=1}^{k}a_{i}\mathbf {b} _{i}$ dengan $k\in \mathbb {N} ,a_{1},\ldots ,a_{k}\in F,\mathbf {b} _{1},\ldots ,\mathbf {b} _{k}\in B$.
2. Jika $\mathbf {v} =\sum _{i=1}^{\tilde {k}}{\tilde {a}}_{i}{\tilde {\mathbf {b} }}_{i}$ representasi lain, maka $k={\tilde {k}}$ dan ada suatu permutasi $\iota :\{1,\ldots ,k\}\to \{1,\ldots ,k\}$ yang $a_{i}={\tilde {a}}_{\iota (i)}$ dan $\mathbf {b} _{i}={\tilde {\mathbf {b} }}_{\iota (i)}$.

A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. This means that a subset B of V is a basis if it satisfies the two following conditions:

linear independence

for every finite subset $\{\mathbf {v} _{1},\dotsc ,\mathbf {v} _{m}\}$ of B, if $c_{1}\mathbf {v} _{1}+\cdots +c_{m}\mathbf {v} _{m}=\mathbf {0}$ for some $c_{1},\dotsc ,c_{m}$ in F, then $c_{1}=\cdots =c_{m}=0$;

spanning property

for every vector v in V, one can choose $a_{1},\dotsc ,a_{n}$ in F and $\mathbf {v} _{1},\dotsc ,\mathbf {v} _{n}$ in B such that $\mathbf {v} =a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}$.

The scalars $a_{i}$ are called the coordinates of the vector v with respect to the basis B, and by the first property they are uniquely determined.

A vector space that has a finite basis is called finite-dimensional. In this case, the finite subset can be taken as B itself to check for linear independence in the above definition.

It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation, or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis, which is therefore not simply an unstructured set, but a sequence, an indexed family, or similar; see § Ordered bases and coordinates below.

Contoh

Gambar ini mengilustrasikan basis standar pada R². Vektor biru dan oranye adalah elemen dasarnya; vektor hijau dapat diberikan dalam istilah vektor basis, dan begitu juga bergantung linear padanya.

• Himpunan R² dari pasangan terurut dari bilangan riil adalah ruang vektor untuk penjumlahan berdasarkan komponen

$(a,b)+(c,d)=(a+c,b+d),$

dan perkalian skalar

$\lambda (a,b)=(\lambda a,\lambda b),$

dimana $\lambda$ adalah bilangan real apa pun. Basis sederhana dari ruang vektor ini, disebut basis standar terdiri dari dua vektor e₁ = (1,0) and e₂ = (0,1), karena vektor apapun v = (a, b) dari R² dapat ditulis secara unik sebagai

$v=ae_{1}+be_{2}.$

Pasangan vektor bebas linear lainnya R², seperti (1, 1) dan (−1, 2), bentuk menjadi dasar R².

• Lebih umum lagi, jika F adalah bidang, himpunan $F^{n}$ dari n-tupel dari elemen F adalah ruang vektor untuk penjumlahan dan perkalian skalar yang didefinisikan serupa. Karena

$e_{i}=(0,\ldots ,0,1,0,\ldots ,0)$

jadilah tupel n dengan semua komponen sama dengan 0, kecuali i yaitu 1. Kemudian $e_{1},\ldots ,e_{n}$ adalah basis dari $F^{n}$ yang disebut basis standar dari $F^{n}.$

• Jika F adalah bidang gelanggang polinomial F[X] dari polinomial dalam satu tak tentu memiliki basis B, yang disebut basis monomial, yang terdiri dari semua monomial:

$B=\{1,X,X^{2},\ldots \}.$

Kumpulan polinomial apa pun yang hanya ada satu polinomial pada setiap derajat juga merupakan basis. Kumpulan polinomial seperti itu disebut urutan polinomial. Contoh (di antara banyak) urutan polinomial tersebut adalah polinomial basis Bernstein, dan polinomial Chebyshev.

Koordinat

Misalkan V menjadi ruang vektor berdimensi berhingga n di atas bidang F, dan

$B=\{b_{1},\ldots ,b_{n}\}$

menjadi dasar dari V. Menurut definisi basis, setiap v pada V dapat ditulis, dengan cara yang unik, seperti

$v=\lambda _{1}b_{1}+\cdots +\lambda _{n}b_{n},$

dimana koefisiennya $\lambda _{1},\ldots ,\lambda _{n}$ adalah skalar (yaitu, elemen F), yang disebut koordinat dari v di atas B. Namun, jika seseorang berbicara tentang himpunan koefisien, seseorang kehilangan korespondensi antara koefisien dan elemen basis, dan beberapa vektor mungkin memiliki himpunan koefisien yang sama. Sebagai contoh, $3b_{1}+2b_{2}$ dan $2b_{1}+3b_{2}$ memiliki koefisien yang sama {2, 3}, dan berbeda. Oleh karena itu, sering kali nyaman untuk bekerja dengan dasar yang teratur; ini biasanya dilakukan oleh pengindeksan elemen dasar oleh bilangan asli pertama. Kemudian, koordinat vektor membentuk urutan dengan indeks serupa, dan vektor sepenuhnya dicirikan oleh urutan koordinat. Basis terurut juga disebut frame, kata yang biasa digunakan, dalam berbagai konteks, untuk merujuk ke urutan data yang memungkinkan penentuan koordinat.

Misalkan, seperti biasa, $F^{n}$ menjadi himpunan n-tupel dari elemen F. Himpunan ini adalah F ruang vektor, dengan penjumlahan dan perkalian skalar ditentukan berdasarkan komponen. Peta

$\varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}b_{1}+\cdots +\lambda _{n}b_{n}$

is a linear isomorphism from the vector space $F^{n}$ onto V. In other words, $F^{n}$ is the coordinate space of V, and the n-tuple $\varphi ^{-1}(v)$ is the coordinate vector of v.

Gambar invers oleh $\varphi$ pada $b_{i}$ adalah n-tupel $e_{i}$ semua yang komponennya 0, kecuali yang ke i yaitu 1. $e_{i}$ membentuk dasar terurut dari $F^{n},$ yang disebut standar dasar atau dasar kanonik. Dasar yang diurutkan B adalah gambar oleh $\varphi$ dari dasar kanonik $F^{n}$.

Ini mengikuti dari apa yang mendahului setiap basis terurut adalah gambar dengan isomorfisme linier dari basis kanonik $F^{n}$, dan bahwa setiap isomorfisme linier dari $F^{n}$ ke V dapat didefinisikan sebagai isomorfisme yang memetakan dasar kanonik $F^{n}$ ke urutan tertentu dasar dari V. Dengan kata lain, ini setara dengan mendefinisikan basis terurut dari V, atau isomorfisme linier dari $F^{n}$ ke V.

Perubahan basis

Artikel utama: Perubahan basis

Maka V jadilah ruang vektor berdimensi n di atas bidang F. Diberikan dua pangkalan (order) $B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})$ dan $B_{\mathrm {new} }=(w_{1},\ldots ,w_{n})$ dari V, sering kali berguna untuk menyatakan koordinat vektor x sehubungan dengan $B_{\mathrm {old} }$ dalam hal koordinat sehubungan dengan $B_{\mathrm {new} }.$ Ini dapat dilakukan dengan rumus perubahan-basis , yang dijelaskan di bawah ini. Subskrip "lama" dan "baru" telah dipilih karena biasa digunakan untuk merujuk $B_{\mathrm {old} }$ dan $B_{\mathrm {new} }$ sebagai dasar lama dan dasar baru . Ini berguna untuk menggambarkan koordinat lama dengan yang baru, karena, secara umum, seseorang memiliki 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.

Biasanya, vektor basis baru diberikan oleh koordinatnya di atas basis lama, yaitu

$w_{j}=\sum _{i=1}^{n}a_{i,j}v_{i}.$

If $(x_{1},\ldots ,x_{n})$ and $(y_{1},\ldots ,y_{n})$ are the coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is

$x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},$

for i = 1, ..., n.

Rumus ini dapat ditulis secara ringkas dalam notasi matriks. Misalkan A adalah matriks dari $a_{i,j},$ dan

$X={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\quad$ dan $\quad Y={\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}}$

jadilah vektor kolom dari koordinat v di basis lama dan basis baru, maka rumus untuk mengubah koordinat adalah

$X=AY.$

Rumusnya dapat dibuktikan dengan mempertimbangkan dekomposisi vektor x pada dua basa: satu memiliki

$x=\sum _{i=1}^{n}x_{i}v_{i},$

dan

${\begin{aligned}x&=\sum _{j=1}^{n}y_{j}w_{j}\\&=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}v_{i}\\&=\sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{i,j}y_{j}\right)v_{i}.\end{aligned}}$

Rumus perubahan basis kemudian dari keunikan dekomposisi vektor atas basis, di sini $B_{\mathrm {old} };$ adalah

$x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},$

untuk i = 1, ..., n.

Lihat pula

• Perubahan basis
• Bingkai ruang vektor
• Basis bola

Referensi

Referensi umum

• Blass, Andreas (1984), "Existence of bases implies the axiom of choice", Axiomatic set theory, Contemporary Mathematics volume 31, Providence, R.I.: American Mathematical Society, hlm. 31–33, ISBN 978-0-8218-5026-8, MR 0763890 
• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5 
• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6 

Referensi sejarah

• Banach, Stefan (1922), "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathematicae (dalam bahasa Prancis), 3: 133–181, doi:10.4064/fm-3-1-133-181, ISSN 0016-2736 
• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (dalam bahasa Jerman) 
• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (dalam bahasa Prancis), Paris: Hermann 
• Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", Historia Mathematica, 22 (3): 227–261, doi:10.1006/hmat.1995.1024, MR 1347828 
• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (dalam bahasa Prancis), Chez Firmin Didot, père et fils 
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (dalam bahasa Jerman) , reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Extension Theory, Kannenberg, L.C., Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5 
• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy 
• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (dalam bahasa Jerman), diarsipkan dari versi asli tanggal 2009-04-12 
• Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940", Historia Mathematica, 22 (3): 262–303, doi:10.1006/hmat.1995.1025 
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (dalam bahasa Italia), Turin 

Examples

This picture illustrates the standard basis in R². The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is linearly dependent upon them.

The set R² of the ordered pairs of real numbers is a vector space under the operations of component-wise addition

$$(a,b)+(c,d)=(a+c,b+d)$$

and scalar multiplication

$$\lambda (a,b)=(\lambda a,\lambda b),$$

where $\lambda$ is any real number. A simple basis of this vector space consists of the two vectors e₁ = (1, 0) and e₂ = (0, 1). These vectors form a basis (called the standard basis) because any vector v = (a, b) of R² may be uniquely written as

$$\mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.$$

Any other pair of linearly independent vectors of R², such as (1, 1) and (−1, 2), forms also a basis of R². More generally, if F is a field, the set $F^{n}$ of n-tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let

$$\mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)$$

be the n-tuple with all components equal to 0, except the ith, which is 1. Then $\mathbf {e} _{1},\ldots ,\mathbf {e} _{n}$ is a basis of $F^{n},$ which is called the standard basis of $F^{n}.$ A different flavor of example is given by polynomial rings. If F is a field, the collection F[X] of all polynomials in one indeterminate X with coefficients in F is an F-vector space. One basis for this space is the monomial basis B, consisting of all monomials:

$$B=\{1,X,X^{2},\ldots \}.$$

Any set of polynomials such that there is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials) is also a basis. (Such a set of polynomials is called a polynomial sequence.) But there are also many bases for F[X] that are not of this form.

Properties

Many properties of finite bases result from the Steinitz exchange lemma, which states that, for any vector space V, given a finite spanning set S and a linearly independent set L of n elements of V, one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L, having its other elements in S, and having the same number of elements as S.

Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require the axiom of choice or a weaker form of it, such as the ultrafilter lemma.

If V is a vector space over a field F, then:

• If L is a linearly independent subset of a spanning set S ⊆ V, then there is a basis B such that
  $$L\subseteq B\subseteq S.$$
• V has a basis (this is the preceding property with L being the empty set, and S = V).
• All bases of V have the same cardinality, which is called the dimension of V. This is the dimension theorem.
• A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.
• A linearly independent set L is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.

If V is a vector space of dimension n, then:

• A subset of V with n elements is a basis if and only if it is linearly independent.
• A subset of V with n elements is a basis if and only if it is a spanning set of V.

Let V be a vector space of finite dimension n over a field F, and

$$B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}$$

be a basis of V. By definition of a basis, every v in V may be written, in a unique way, as

$$\mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},$$

where the coefficients $\lambda _{1},\ldots ,\lambda _{n}$ are scalars (that is, elements of F), which are called the coordinates of v over B. However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, $3\mathbf {b} _{1}+2\mathbf {b} _{2}$ and $2\mathbf {b} _{1}+3\mathbf {b} _{2}$ have the same set of coefficients {2, 3}, and are different. It is therefore often convenient to work with an ordered basis; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis is also called a frame, a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.

Let, as usual, $F^{n}$ be the set of the n-tuples of elements of F. This set is an F-vector space, with addition and scalar multiplication defined component-wise. The map

$$\varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}$$

is a linear isomorphism from the vector space $F^{n}$ onto V. In other words, $F^{n}$ is the coordinate space of V, and the n-tuple $\varphi ^{-1}(\mathbf {v} )$ is the coordinate vector of v.

The inverse image by $\varphi$ of $\mathbf {b} _{i}$ is the n-tuple $\mathbf {e} _{i}$ all of whose components are 0, except the ith that is 1. The $\mathbf {e} _{i}$ form an ordered basis of $F^{n}$, which is called its standard basis or canonical basis. The ordered basis B is the image by $\varphi$ of the canonical basis of $F^{n}$.

It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of $F^{n}$, and that every linear isomorphism from $F^{n}$ onto V may be defined as the isomorphism that maps the canonical basis of $F^{n}$ onto a given ordered basis of V. In other words, it is equivalent to define an ordered basis of V, or a linear isomorphism from $F^{n}$ onto V.

Change of basis

Artikel utama: Change of basis

Let V be a vector space of dimension n over a field F. Given two (ordered) bases $B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})$ and $B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})$ of V, it is often useful to express the coordinates of a vector x with respect to $B_{\mathrm {old} }$ in terms of the coordinates with respect to $B_{\mathrm {new} }.$ 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 $B_{\mathrm {old} }$ and $B_{\mathrm {new} }$ 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 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.

Typically, the new basis vectors are given by their coordinates over the old basis, that is,

$$\mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.$$

If $(x_{1},\ldots ,x_{n})$ and $(y_{1},\ldots ,y_{n})$ are the coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is

$$x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},$$

for i = 1, ..., n.

This formula may be concisely written in matrix notation. Let A be the matrix of the $a_{i,j}$, and

$$X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}$$

be the column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is

$$X=AY.$$

The formula can be proven by considering the decomposition of the vector x on the two bases: one has

$$\mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},$$

and

$$\mathbf {x} =\sum _{j=1}^{n}y_{j}\mathbf {w} _{j}=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}=\sum _{i=1}^{n}{\biggl (}\sum _{j=1}^{n}a_{i,j}y_{j}{\biggr )}\mathbf {v} _{i}.$$

The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here $B_{\text{old}}$; that is

$$x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},$$

for i = 1, ..., n.

Related notions

Free module

Artikel utama: Free module dan Free abelian group

If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of a module. For modules, linear independence and spanning sets are defined exactly as for vector spaces, although "generating set" is more commonly used than that of "spanning set".

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 resolutions.

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 G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then there is a basis $\mathbf {e} _{1},\ldots ,\mathbf {e} _{n}$ of H and an integer 0 ≤ k ≤ n such that $a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}$ is a basis of G, for some nonzero integers $a_{1},\ldots ,a_{k}$. For details, see Free abelian group § Subgroups.

Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel) 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 bases on Hilbert spaces, Schauder bases, and Markushevich bases on 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 $2^{\aleph _{0}}$, where $\aleph _{0}$ (aleph-nought) is the smallest infinite cardinal, the cardinal of the integers.

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 spaces – a large class of vector spaces including e.g. Hilbert spaces, Banach spaces, or Fréchet spaces.

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 (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 $c_{00}$, the space of the sequences $x=(x_{n})$ of real numbers that have only finitely many non-zero elements, with the norm ${\textstyle \|x\|=\sup _{n}|x_{n}|}$. Its standard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.

Example

In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } 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

$$\int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .$$

The functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } 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

$$\lim _{n\to \infty }\int _{0}^{2\pi }{\biggl |}a_{0}+\sum _{k=1}^{n}\left(a_{k}\cos \left(kx\right)+b_{k}\sin \left(kx\right)\right)-f(x){\biggr |}^{2}dx=0$$

for suitable (real or complex) coefficients a_k, b_k. But many 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.

Geometry

The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis. An affine basis for an n-dimensional affine space is $n+1$ points in general linear position. A projective basis is $n+2$ points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).

Random basis

For a probability distribution in Rⁿ with a probability density function, such as the equidistribution in an n-dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that n linearly dependent vectors x₁, ..., x_n in Rⁿ should satisfy the equation det[x₁ ⋯ x_n] = 0 (zero determinant of the matrix with columns x_i), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.

Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1, 1]ⁿ 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.

It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal to y if $\left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon$ (that is, cosine of the angle between x and y is less than ε).

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). Let θ be a small positive number. Then for

$$N\leq {\exp }{\bigl (}{\tfrac {1}{4}}\varepsilon ^{2}n{\bigr )}{\sqrt {-\ln(1-\theta )}}$$

(Eq. 1)

N random vectors are all pairwise ε-orthogonal with probability 1 − θ. This N growth exponentially with dimension n and $N\gg n$ for sufficiently big n. This property of random bases is a manifestation of the so-called measure concentration phenomenon.

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 [−1, 1]ⁿ 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 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 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 V be any vector space over some field F. Let X be the set of all linearly independent subsets of V.

The set X is nonempty since the empty set is an independent subset of V, and it is partially ordered by inclusion, which is denoted, as usual, by ⊆.

Let Y be a subset of X that is totally ordered by ⊆, and let L_Y be the union of all the elements of Y (which are themselves certain subsets of V).

Since (Y, ⊆) is totally ordered, every finite subset of L_Y is a subset of an element of Y, which is a linearly independent subset of V, and hence L_Y is linearly independent. Thus L_Y is an element of X. Therefore, L_Y is an upper bound for Y in (X, ⊆): it is an element of X, that contains every element of Y.

As X is nonempty, and every totally ordered subset of (X, ⊆) has an upper bound in X, Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L_max of X satisfying the condition that whenever L_max ⊆ L for some element L of X, then L = L_max.

It remains to prove that L_max is a basis of V. Since L_max belongs to X, we already know that L_max is a linearly independent subset of V.

If there were some vector w of V that is not in the span of L_max, then w would not be an element of L_max either. Let L_w = L_max ∪ {w}. This set is an element of X, that is, it is a linearly independent subset of V (because w is not in the span of L_max, and L_max is independent). As L_max ⊆ L_w, and L_max ≠ L_w (because L_w contains the vector w that is not contained in L_max), this contradicts the maximality of L_max. Thus this shows that L_max spans V.

Hence L_max is linearly independent and spans V. It is thus a basis of 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. Thus the two assertions are equivalent.

See also

• Basis of a matroid
• Basis of a linear program
• Change of basis
• Frame of a vector space
• Spherical basis

References

General references

• Blass, Andreas (1984), "Existence of bases implies the axiom of choice" (PDF), Axiomatic set theory, Contemporary Mathematics volume 31, Providence, R.I.: American Mathematical Society, hlm. 31–33, ISBN 978-0-8218-5026-8, MR 0763890 
• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5 
• Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6 

Historical references

• Banach, Stefan (1922), "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathematicae (dalam bahasa Prancis), 3: 133–181, doi:10.4064/fm-3-1-133-181, ISSN 0016-2736 
• Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (dalam bahasa Jerman) 
• Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (dalam bahasa Prancis), Paris: Hermann 
• Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", Historia Mathematica, 22 (3): 227–261, doi:10.1006/hmat.1995.1024 , MR 1347828 
• Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (dalam bahasa Prancis), Chez Firmin Didot, père et fils 
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (dalam bahasa Jerman) , reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Extension Theory, Kannenberg, L.C., Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5 
• Hamel, Georg (1905), "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x+y)=f(x)+f(y)", Mathematische Annalen (dalam bahasa Jerman), Leipzig, 60 (3): 459–462, doi:10.1007/BF01457624 
• Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy 
• Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (dalam bahasa Jerman), diarsipkan dari versi asli tanggal 2009-04-12 
• Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940", Historia Mathematica, 22 (3): 262–303, doi:10.1006/hmat.1995.1025  
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (dalam bahasa Italia), Turin 

External links

• Instructional videos from Khan Academy
  • Introduction to bases of subspaces
  • Proof that any subspace basis has same number of elements
• "Linear combinations, span, and basis vectors". Essence of linear algebra. August 6, 2016. Diarsipkan dari versi asli tanggal 2021-11-17 – via YouTube. 
• Hazewinkel, Michiel, ed. (2001) [1994], "Basis", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4