HOME

TheInfoList




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 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 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). ...
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: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
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 ...

complex number
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). ...

subset
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 \ of , if c_1 \mathbf v_1 + \cdots + c_m \mathbf v_m = \mathbf 0 for some c_1,\dotsc,c_m in , then and * the ''spanning'' property: *: for every vector in , one can choose a_1,\dotsc,a_n in and \mathbf v_1, \dotsc, \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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
, 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 ...

sequence
, 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 the
ordered 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 ...

ordered pair
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 *::\lambda (a,b) = (\lambda a, \lambda b), *:where \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 *::\mathbf v = a \mathbf e_1 + b \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 * 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 ...
, the set F^n of -tuples of elements of is a vector space for similarly defined addition and scalar multiplication. Let *::\mathbf e_i = (0, \ldots, 0,1,0,\ldots, 0) *: be the -tuple with all components equal to 0, except the th, 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. *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 ...

polynomial
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=\. *: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 finite
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 ...
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 ...

axiom of choice
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\subseteq B\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 ...

empty set
, 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). ...

proper subset
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 = \ be a basis of . By definition of a basis, every in 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 ), 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 \mathbf b_1 + 2 \mathbf b_2 and 2 \mathbf b_1 + 3 \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 a
sequence 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 :\varphi: (\lambda_1, \ldots, \lambda_n) \mapsto \lambda_1 \mathbf b_1 + \cdots + \lambda_n \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 \varphi^(\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 ...

inverse image
by \varphi of \mathbf b_i is the -tuple \mathbf e_i all of whose components are 0, except the th that is 1. The \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 \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_\text = (\mathbf v_1, \ldots, \mathbf v_n) and B_\text = (\mathbf w_1, \ldots, \mathbf w_n) of , it is often useful to express the coordinates of a vector with respect to B_\mathrm in terms of the coordinates with respect to B_\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_\mathrm and B_\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 has
expressions Expression may refer to: Linguistics * 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, :\mathbf w_j = \sum_^n a_ \mathbf v_i. If (x_1, \ldots, x_n) and (y_1, \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 = \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= \beginx_1\\\vdots\\x_n\end \quad \text \quad Y= \beginy_1\\\vdots\\y_n\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 :\mathbf x = \sum_^n x_i \mathbf v_i, and :\begin \mathbf x &=\sum_^n y_j \mathbf w_j \\ &=\sum_^n y_j\sum_^n a_\mathbf v_i\\ &=\sum_^n \left(\sum_^n a_y_j\right)\mathbf v_i. \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 = \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 a
module 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 \mathbf e_1, \ldots, \mathbf e_n of and an integer such that a_1 \mathbf e_1, \ldots, a_k \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 after
Georg 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 ...

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 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 \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 ...

uncountable
. 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 ...

sequence
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 of
Fourier 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 :\int_0^ \left, f(x)\^2\,dx < \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 :\lim_ \int_0^ \left, a_0 + \sum_^n \left(a_k\cos\left(kx\right)+b_k\sin\left(kx\right)\right)-f(x)\^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 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 an
affine 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 ...

convex set
, 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 ...

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


Random basis

For a
probability 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 R''n'' with a
probability 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 x1, …, 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 \left, \left\langle x,y \right\rangle\ / \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 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\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 is
partially 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 LY 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 LY is a subset of an element of Y, which is a linearly independent subset of V, and hence LY is linearly independent. Thus LY is an element of X. Therefore, LY 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 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 Lmax of X satisfying the condition that whenever Lmax ⊆ L for some element L of X, then L = Lmax. It remains to prove that Lmax is a basis of V. Since Lmax belongs to X, we already know that Lmax is a linearly independent subset of V. If there were some vector w of V that is not in the span of Lmax, then w would not be an element of Lmax either. Let Lw = Lmax ∪ . 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 Lmax, and Lmax is independent). As Lmax ⊆ Lw, and Lmax ≠ Lw (because Lw contains the vector w that is not contained in Lmax), this contradicts the maximality of Lmax. Thus this shows that Lmax spans V. Hence Lmax 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 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 ...

axiom of choice
. 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 matroid


Notes


References


General references

* * *


Historical references

* * * * * * , reprint: * * * *


External links

* Instructional videos from Khan Academy
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