In

_{''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

^{''n''} with a _{1}, …, x_{''n''} in R^{''n''} should satisfy the equation (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.
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 $\backslash left,\; \backslash left\backslash langle\; x,y\; \backslash right\backslash rangle\backslash \; /\; \backslash left(\backslash left\backslash ,\; x\backslash right\backslash ,\; \backslash left\backslash ,\; y\backslash right\backslash ,\; \backslash right)\; <\; \backslash varepsilon$ (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

_{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, _{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} ∪ . 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

''Existence of bases implies the Axiom of Choice''

Contemporary Mathematics. 31. pp. 31-33. Thus the two assertions are equivalent.

Introduction to bases of subspaces

Proof that any subspace basis has same number of elements

* * {{DEFAULTSORT:Basis (Linear Algebra) Linear algebra Articles containing proofs Axiom of choice

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, a set of vectors in a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is called a basis if every element of may be written in a unique way as a finite linear combination
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called .
Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

.
A vector space can have several bases; however all the bases have the same number of elements, called the dimension
In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...

of the vector space.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
Definition

A basis of avector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

over a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

(such as the real numbers
Real may refer to:
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or the complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s ) is a linearly independent
In the theory of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of that spans . This means that a subset of is a basis if it satisfies the two following conditions:
* the ''linear independence'' property:
*: for every finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

subset $\backslash $ of , if $c\_1\; \backslash mathbf\; v\_1\; +\; \backslash cdots\; +\; c\_m\; \backslash mathbf\; v\_m\; =\; \backslash mathbf\; 0$ for some $c\_1,\backslash dotsc,c\_m$ in , then and
* the ''spanning'' property:
*: for every vector in , one can choose $a\_1,\backslash dotsc,a\_n$ in and $\backslash mathbf\; v\_1,\; \backslash dotsc,\; \backslash mathbf\; v\_n$ in such that
The scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

s $a\_i$ are called the coordinates of the vector with respect to the basis , and by the first property they are uniquely determined.
A vector space that has a finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

basis is called finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...

. In this case, the finite subset can be taken as itself to check for linear independence in the above definition.
It is often convenient or even necessary to have an ordering
Order or ORDER or Orders may refer to:
* Orderliness, a desire for organization
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements hav ...

on the basis vectors, for example, when discussing orientation
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, 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
In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

, an indexed family
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, or similar; see below.
Examples

*The set of theordered pair
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s of real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s is a vector space for the following properties:
*: component-wise addition
*::$(a,\; b)\; +\; (c,\; d)\; =\; (a\; +\; c,\; b+d),$
*:and scalar multiplication
*::$\backslash lambda\; (a,b)\; =\; (\backslash lambda\; a,\; \backslash lambda\; b),$
*:where $\backslash lambda$ is any real number. A simple basis of this vector space, called the standard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

consists of the two vectors and , since, any vector of may be uniquely written as
*::$\backslash mathbf\; v\; =\; a\; \backslash mathbf\; e\_1\; +\; b\; \backslash mathbf\; e\_2.$
*:Any other pair of linearly independent vectors of , such as and , forms also a basis of .
*More generally, if is a field
Field may refer to:
Expanses of open ground
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* Battlefield
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* Meadow, a grassl ...

, the set $F^n$ of -tuples of elements of is a vector space for similarly defined addition and scalar multiplication. Let
*::$\backslash mathbf\; e\_i\; =\; (0,\; \backslash ldots,\; 0,1,0,\backslash ldots,\; 0)$
*: be the -tuple with all components equal to 0, except the th, which is 1. Then $\backslash mathbf\; e\_1,\; \backslash ldots,\; \backslash mathbf\; e\_n$ is a basis of $F^n,$ which is called the ''standard basis'' of $F^n.$
*If is a field, the polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s in one indeterminate has a basis , called the monomial basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, consisting of all monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one summand, term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of Variable (mathematics), variables w ...

s:
*::$B=\backslash .$
*:Any set of polynomials such that there is exactly one polynomial of each degree is also a basis. Such a set of polynomials is called a polynomial sequence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Examples (among many) of such polynomial sequences are Bernstein basis polynomials, and Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined several equivalent ways; in this article the polynomials are defined ...

.
Properties

Many properties of finite bases result from the Steinitz exchange lemma, which states that, for any vector space , given a finitespanning set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

and a linearly independent
In the theory of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

set of elements of , one may replace well-chosen elements of by the elements of to get a spanning set containing , having its other elements in , and having the same number of elements as .
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
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

or a weaker form of it, such as the ultrafilter lemma.
If is a vector space over a field , then:
* If is a linearly independent subset of a spanning set , then there is a basis such that
*:$L\backslash subseteq\; B\backslash subseteq\; S.$
* has a basis (this is the preceding property with being the empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

, and ).
* All bases of have the same cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which is called the dimension
In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...

of . This is the dimension theorem.
* A generating set is a basis of if and only if it is minimal, that is, no proper subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of is also a generating set of .
* A linearly independent set is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.
If is a vector space of dimension , then:
* A subset of with elements is a basis if and only if it is linearly independent.
* A subset of with elements is a basis if and only if it is spanning set of .
Coordinates

Let be a vector space of finite dimension over a field , and :$B\; =\; \backslash $ be a basis of . By definition of a basis, every in may be written, in a unique way, as :$\backslash mathbf\; v\; =\; \backslash lambda\_1\; \backslash mathbf\; b\_1\; +\; \backslash cdots\; +\; \backslash lambda\_n\; \backslash mathbf\; b\_n,$ where the coefficients $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_n$ are scalars (that is, elements of ), which are called the ''coordinates'' of over . 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, $3\; \backslash mathbf\; b\_1\; +\; 2\; \backslash mathbf\; b\_2$ and $2\; \backslash mathbf\; b\_1\; +\; 3\; \backslash mathbf\; b\_2$ have the same set of coefficients , 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 asequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 -tuples of elements of . This set is an -vector space, with addition and scalar multiplication defined component-wise. The map
:$\backslash varphi:\; (\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_n)\; \backslash mapsto\; \backslash lambda\_1\; \backslash mathbf\; b\_1\; +\; \backslash cdots\; +\; \backslash lambda\_n\; \backslash mathbf\; b\_n$
is a linear isomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

from the vector space $F^n$ onto . In other words, $F^n$ is the coordinate space of , and the -tuple $\backslash varphi^(\backslash mathbf\; v)$ is the coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...

of .
The inverse image
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

by $\backslash varphi$ of $\backslash mathbf\; b\_i$ is the -tuple $\backslash mathbf\; e\_i$ all of whose components are 0, except the th that is 1. The $\backslash mathbf\; e\_i$ form an ordered basis of $F^n$, which is called its standard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

or canonical basisIn mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kr ...

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

Let be a vector space of dimension over a field . Given two (ordered) bases $B\_\backslash text\; =\; (\backslash mathbf\; v\_1,\; \backslash ldots,\; \backslash mathbf\; v\_n)$ and $B\_\backslash text\; =\; (\backslash mathbf\; w\_1,\; \backslash ldots,\; \backslash mathbf\; w\_n)$ of , it is often useful to express the coordinates of a vector with respect to $B\_\backslash mathrm$ in terms of the coordinates with respect to $B\_\backslash mathrm\; .$ 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 $B\_\backslash mathrm$ and $B\_\backslash mathrm$ 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 hasexpressions
Expression may refer to:
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* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

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,
:$\backslash mathbf\; w\_j\; =\; \backslash sum\_^n\; a\_\; \backslash mathbf\; v\_i.$
If $(x\_1,\; \backslash ldots,\; x\_n)$ and $(y\_1,\; \backslash ldots,\; y\_n)$ are the coordinates of a vector over the old and the new basis respectively, the change-of-basis formula is
:$x\_i\; =\; \backslash sum\_^n\; a\_y\_j,$
for .
This formula may be concisely written in matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

notation. Let be the matrix of the and
:$X=\; \backslash beginx\_1\backslash \backslash \backslash vdots\backslash \backslash x\_n\backslash end\; \backslash quad\; \backslash text\; \backslash quad\; Y=\; \backslash beginy\_1\backslash \backslash \backslash vdots\backslash \backslash y\_n\backslash end$
be the column vector
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

s of the coordinates of 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 on the two bases: one has
:$\backslash mathbf\; x\; =\; \backslash sum\_^n\; x\_i\; \backslash mathbf\; v\_i,$
and
:$\backslash begin\; \backslash mathbf\; x\; \&=\backslash sum\_^n\; y\_j\; \backslash mathbf\; w\_j\; \backslash \backslash \; \&=\backslash sum\_^n\; y\_j\backslash sum\_^n\; a\_\backslash mathbf\; v\_i\backslash \backslash \; \&=\backslash sum\_^n\; \backslash left(\backslash sum\_^n\; a\_y\_j\backslash right)\backslash mathbf\; v\_i.\; \backslash end$
The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here that is
:$x\_i\; =\; \backslash sum\_^n\; a\_y\_j,$
for .
Related notions

Free module

If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of amodule
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modula ...

. For modules, linear independence
In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vect ...

and spanning set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s are defined exactly as for vector spaces, although "generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set (mathematics), set of objects, together with a set of Operation (mathe ...

" 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 resolution In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

s.
A module over the integers is exactly the same thing as an abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. 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 is a subgroup of a finitely generated free abelian group (that is an abelian group that has a finite basis), there is a basis $\backslash mathbf\; e\_1,\; \backslash ldots,\; \backslash mathbf\; e\_n$ of and an integer such that $a\_1\; \backslash mathbf\; e\_1,\; \backslash ldots,\; a\_k\; \backslash mathbf\; e\_k$ is a basis of , for some nonzero integers For details, see .
Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term (named afterGeorg Hamel
Georg Karl Wilhelm Hamel (12 September 1877 – 4 October 1954) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of ...

) 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 space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s, Schauder bases, and Markushevich bases on normed linear space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the continuum, which is the cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

where $\backslash aleph\_0$ 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 space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s – a large class of vector spaces including e.g. Hilbert space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s, Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, or Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are Complete space, complete with ...

s.
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 which is complete (i.e. ''X'' is a Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

), then any Hamel basis of ''X'' is necessarily uncountable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. This is a consequence of the Baire category theorem
The Baire category theorem (BCT) is an important result in general topology
, a useful example in point-set topology. It is connected but not path-connected.
In mathematics, general topology is the branch of topology that deals with the basic Set t ...

. 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 which have countable Hamel bases. Consider the space of the sequence
In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

s $x=(x\_n)$ of real numbers which have only finitely many non-zero elements, with the norm Its standard basis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.
Example

In the study ofFourier series
In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...

, one learns that the functions are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval , 2πthat are square-integrable on this interval, i.e., functions ''f'' satisfying
:$\backslash int\_0^\; \backslash left,\; f(x)\backslash ^2\backslash ,dx\; <\; \backslash infty.$
The functions are linearly independent, and every function ''f'' that is square-integrable on , 2πis an "infinite linear combination" of them, in the sense that
:$\backslash lim\_\; \backslash int\_0^\; \backslash left,\; a\_0\; +\; \backslash sum\_^n\; \backslash left(a\_k\backslash cos\backslash left(kx\backslash right)+b\_k\backslash sin\backslash left(kx\backslash right)\backslash right)-f(x)\backslash ^2\; dx\; =\; 0$
for suitable (real or complex) coefficients ''a''Fourier analysis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

.
Geometry

The geometric notions of anaffine space
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...

, projective space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set o ...

, and cone
A cone is a three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

have related notions of ''basis''. An affine basis for an ''n''-dimensional affine space is $n+1$ points in general linear position
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commu ...

. A is $n+2$ points in general position, in a projective space of dimension ''n''. A of a polytope
In elementary , a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional . Polytopes may exist in any general number of dimensions ''n'' as an ''n''-dimensional polytope or ...

is the set of the vertices of its convex hull
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

. A consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).
Random basis

For aprobability distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

in Rprobability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...

, 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 xindependent and identically distributed
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...

). Let ''θ'' be a small positive number. Then for
''N'' random vectors are all pairwise ε-orthogonal with probability . This ''N'' growth exponentially with dimension ''n'' and $N\backslash gg\; n$ 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 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 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 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 ispartially ordered
250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable.
In mathematics, especially order the ...

by inclusion, which is denoted, as usual, by .
Let Y be a subset of X that is totally ordered by , and let LZorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...

asserts that X has a maximal element. In other words, there exists some element Laxiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true.Blass, Andreas (1984)''Existence of bases implies the Axiom of Choice''

Contemporary Mathematics. 31. pp. 31-33. Thus the two assertions are equivalent.

See also

* * * * Basis of a matroidNotes

References

General references

* * *Historical references

* * * * * * , reprint: * * * *External links

* Instructional videos from Khan AcademyIntroduction to bases of subspaces

Proof that any subspace basis has same number of elements

* * {{DEFAULTSORT:Basis (Linear Algebra) Linear algebra Articles containing proofs Axiom of choice