basis vector

Basis vector redirects here. For basis vector in the context of crystals, see crystal structure.
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set.

Definition

Enlarge picture
This picture illustrates the standard basis in R2. The red and blue vectors are the elements of the basis


A basis B of a vector space V is a linearly independent subset of V that spans (or generates) V.

In more detail, suppose that B = { v1, …, vn } is a finite subset of a vector space V over a field F (such as the real or complex numbers R or C). Then B is a basis if it satisfies the following conditions:
  • the linear independence property,
: for all a1, …, anF, if a1v1 + … + anvn = 0, then necessarily a1 = … = an = 0; and
  • the spanning property,
: for every x in V it is possible to choose a1, …, anF such that x = a1v1 + … + anvn.


A vector space that has a finite basis is called finite-dimensional. To deal with infinite dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) BV is a basis, if
  • every finite subset B0B obeys the independence property shown above; and
  • for every x in V it is possible to choose a1, …, anF and v1, …, vnB such that x = a1v1 + … + anvn.
The axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. That is why the sums in the above definition are all finite. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see ''Related notions below.

It is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span V: see Ordered bases and coordinates below.

Properties

Again, B denotes a subset of a vector space V. Then, B is a basis if and only if any of the following equivalent conditions are met:
  • B is a minimal generating set of V, i.e., it is a generating set but no proper subset of B is.
  • B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no other linearly independent set contains it as a proper subset.
  • Every vector in V can be expressed as a linear combination of vectors in B in a unique way. If the basis is ordered (see Ordered bases and coordinates below) then the coefficients in this linear combination provide coordinates of the vector relative to the basis.
The theorem that every vector space has a basis is implied by the well-ordering theorem, or any other equivalent of the axiom of choice. (Proof: Well-order the elements of the vector space. Create the subset of all elements not linearly dependent on their predecessors. This is easily shown to be a basis). The converse is also true. All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space. The latter result is known as the dimension theorem, and requires the ultrafilter lemma, a strictly weaker form of the axiom of choice.

Examples

  • Consider R2, the vector space of all co-ordinates (a, b) where both a and b are real numbers. Then a very natural and simple basis is simply the vectors e1 = (1,0) and e2 = (0,1): suppose that v = (a, b) is a vector in R2, then v = a (1,0) + b (0,1). But any two linearly independent vectors, like (1,1) and (−1,2), will also form a basis of R2 (see the section Proving that a set is a basis further down).
  • More generally, the vectors e1, e2, ..., en are linearly independent and generate Rn. Therefore, they form a basis for Rn and the dimension of Rn is n. This basis is called the standard basis.
  • Let V be the real vector space generated by the functions et and e2t. These two functions are linearly independent, so they form a basis for V.
  • Let R[x] denote the vector space of real polynomials; then (1, x, x2, ...) is a basis of R[x]. The dimension of R[x] is therefore equal to aleph-0.

Basis extension

Between any linearly independent set and any generating set there is a basis. More formally: if L is a linearly independent set in the vector space V and G is a generating set of V containing L, then there exists a basis of V that contains L and is contained in G. In particular (taking G = V), any linearly independent set L can be "extended" to form a basis of V. These extensions are not unique.

Proving that a set is a basis

To prove that a set B is a basis for a (finite-dimensional) vector space V, it is sufficient to show that the number of elements in B equals the dimension of V, and one of the following:
  • B is linearly independent, or
  • span(B) = V.

Example of alternative proofs

Often, a mathematical result can be proven in more than one way. Here, using three different proofs, we show that the vectors (1,1) and (-1,2) form a basis for R2.

From the definition of basis

We have to prove that these two vectors are linearly independent and that they generate R2.

Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that:
Then:
  and  
  and  
Subtracting the first equation from the second, we obtain:
  so  
And from the first equation then:


Part II: To prove that these two vectors generate R2, we have to let (a,b) be an arbitrary element of R2, and show that there exist numbers x,y such that:
Then we have to solve the equations:
Subtracting the first equation from the second, we get:
          and then
        and finally

By the dimension theorem

Since (-1,2) is clearly not a multiple of (1,1) and since (1,1) is not the zero vector, these two vectors are linearly independent. Since the dimension of R2 is 2, the two vectors already form a basis of R2 without needing any extension.

By the invertible matrix theorem

Simply compute the determinant
Since the above matrix has a nonzero determinant, its columns form a basis of R2. See: invertible matrix.

Ordered bases and coordinates

A basis is just a set of vectors with no given ordering. For many purposes it is convenient to work with an ordered basis. For example, when working with a coordinate representation of a vector it is customary to speak of the "first" or "second" coordinate, which makes sense only if an ordering is specified for the basis. For finite-dimensional vector spaces one typically indexes a basis {vi} by the first n integers. An ordered basis is also called a frame.

Suppose V is an n-dimensional vector space over a field F. A choice of an ordered basis for V is equivalent to a choice of a linear isomorphism φ from the coordinate space Fn to V.

Proof. The proof makes use of the fact that the standard basis of Fn is an ordered basis.

Suppose first that
φ : FnV
is a linear isomorphism. Define an ordered basis {vi} for V by
vi = φ(ei) for 1 ≤ in
where {ei} is the standard basis for Fn.

Conversely, given an ordered basis, consider the map defined by
φ(x) = x1v1 + x2v2 + ... + xnvn,
where x = x1e1 + x2e2 + ... + xnen is an element of Fn. It is not hard to check that φ is a linear isomorphism.

These two constructions are clearly inverse to each other. Thus ordered bases for V are in 1-1 correspondence with linear isomorphisms FnV.

The inverse of the linear isomorphism φ determined by an ordered basis {vi} equips V with coordinates: if, for a vector vV, φ-1(v) = (a1, a2,...,an) ∈ Fn, then the components aj = aj(v) are the coordinates of v in the sense that v = a1(v) v1 + a2(v) v2 + ... + an(v) vn.

The maps sending a vector v to the components aj(v) are linear maps from V to F, because of φ-1 is linear. Hence they are linear functionals. They form a basis for the dual space of V, called the dual basis.

Related notions

The phrase Hamel basis (named after Georg Hamel, or algebraic basis) is sometimes used to refer to a basis as defined in this article, where the number of terms in the linear combination a1v1 + … + anvn is always finite.

In Hilbert spaces and other Banach spaces, there is a need to work with linear combinations of infinitely many vectors. In an infinite-dimensional Hilbert space, a set of vectors orthogonal to each other can never span the whole space via their finite linear combinations. What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations. Except in the finite-dimensional case, this concept is not purely algebraic, and is distinct from a Hamel basis; it is also more generally useful. An orthonormal basis of an infinite-dimensional Hilbert space is therefore not a Hamel basis.

In topological vector spaces, quite generally, one may define infinite sums (infinite series) and express elements of the space as certain infinite linear combinations of other elements. To keep clear the distinction of bases using finite and infinite combination, the former ones are called Hamel bases and the latter ones Schauder bases, if the context requires it. The corresponding dimensions are also known as Hamel dimension and Schauder dimension.

Example

In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are an "orthonormal 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 {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are linearly independent, and every function f that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that



for suitable (real or complex) coefficients ak, bk. But most 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 of little (if any) interest, whereas orthonormal bases of these spaces are essential in Fourier analysis.

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