In mathematics, a set  of vectors in a vector space  is called a basis (: bases) if every element of  may be written in a unique way as a finite linear combination 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.

A vector space can have several bases; however all the bases have the same number of elements, called the dimension 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  over a field  (such as the real numbers  or the complex numbers ) is a linearly independent subset of  that spans . This means that a subset  of  is a basis if it satisfies the two following conditions:
linear independence
 for every finite subset  of , if  for some  in , then 
spanning property
 for every vector  in , one can choose  in  and  in  such that 

The scalars  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 basis is called finite-dimensional. 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 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  below.

Examples 

The set  of the ordered pairs of real numbers is a vector space under the operations of component-wise addition
 
and scalar multiplication
 
where  is any real number. A simple basis of this vector space consists of the two vectors  and .  These vectors form a basis (called the standard basis) because any vector  of  may be uniquely written as  Any other pair of linearly independent vectors of , such as  and , forms also a basis of .

More generally, if  is a field, the set  of -tuples of elements of  is a vector space for similarly defined addition and scalar multiplication. Let  be the -tuple with all components equal to 0, except the th, which is 1. Then  is a basis of  which is called the standard basis of 

A different flavor of example is given by polynomial rings.  If  is a field, the collection  of all polynomials in one indeterminate  with coefficients in  is an -vector space.  One basis for this space is the monomial basis , consisting of all monomials:  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  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 , given a finite spanning set  and a linearly independent 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 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 
  has a basis (this is the preceding property with  being the empty set, and ).
 All bases of  have the same cardinality, which is called the dimension 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 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 a spanning set of .

Coordinates  

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

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

where the coefficients  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,  and  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 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,  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 

is a linear isomorphism from the vector space  onto . In other words,  is the coordinate space of , and the -tuple  is the coordinate vector of .

The inverse image by  of  is the -tuple  all of whose components are 0, except the th that is 1. The  form an ordered basis of , which is called its standard basis or canonical basis. The ordered basis  is the image by  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  onto  may be defined as the isomorphism that maps the canonical basis of  onto a given ordered basis of . In other words, it is equivalent to define an ordered basis of , or a linear isomorphism from  onto .

Change of basis 

Let  be a vector space of dimension  over a field . Given two (ordered) bases  and  of , it is often useful to express the coordinates of a vector  with respect to  in terms of the coordinates with respect to  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  and  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, 

If  and  are the coordinates of a vector  over the old and the new basis respectively, the change-of-basis formula is 

for .

This formula may be concisely written in matrix notation. Let  be the matrix of the  and

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

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

and

The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here  that is

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. 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  is a subgroup of a finitely generated free abelian group  (that is an abelian group that has a finite basis), then there is a basis  of  and an integer  such that  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) 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  where  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 which 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 which have countable Hamel bases. Consider  the space of the sequences  of real numbers which have only finitely many non-zero elements, with the norm  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  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

The functions  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

for suitable (real or complex) coefficients ak, bk. 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  points in general linear position. A  is  points in general position, in a projective space of dimension n. A  of a polytope is the set of the vertices of its 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  with a probability density function, such as the equidistribution in an n-dimensional ball with respect to Lebesgue measure, it can be shown that  randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that  linearly dependent vectors , ...,  in  should satisfy the equation  (zero determinant of the matrix with columns ), 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  (that is, cosine of the angle between  and  is less than ).

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

 random vectors are all pairwise ε-orthogonal with probability . This  growth exponentially with dimension  and  for sufficiently big . This property of random bases is a manifestation of the so-called .

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  be any vector space over some field . Let  be the set of all linearly independent subsets of .

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

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

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

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

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

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

Hence  is linearly independent and spans . It is thus a basis of , 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

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
 
 

Articles containing proofs
Axiom of choice
Linear algebra