Basis (linear algebra)


In mathematics, a set of elements of a vector space is called a basis if every element of can 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.
Basis vectors find applications in the study of crystal structures and frames of reference.

Definition

A basis of a vector space over a field 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 ; and
  • the spanning property: for every vector in, one can choose in and in such that In other words, can be represented as a linear combination of some vectors in.
The first property may be equivalently phrased as follows: if a linear combination of vectors in is equal to the zero vector, then all the scalars coefficients of the combination must be zero.
If is a basis for, then every vector in can be written as a linear combination of vectors in ; it follows from linear independence that this can be done in exactly one way. Thus, the scalar coefficients that appear in this combination are uniquely determined; they are called the coordinates of with respect to the basis.
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 with 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 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 is also a basis. 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.
  • 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, 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, especially when used in conjunction with an origin, is also called a coordinate frame or simply a frame.
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 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 , 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 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 that is complete, 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 normed spaces that have countable Hamel bases. Consider the space of the sequences of real numbers that 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.