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Encyclopedia > Basis (linear algebra)

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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

Contents

Definition

This picture illustrates the standard basis in R2. The red and blue vectors are the elements of the basis
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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In Abstract Algebra, a generator is defined as follows: Let G be a group and , then a is called a generator and G is a cyclic group. ...


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: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Please refer to Real vs. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

  • 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 admits 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 In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

  • 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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a series is a sum of a sequence of terms. ...


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. In linear algebra, linear transformations can be represented by matrices. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...


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: It has been suggested that this article or section be merged with Logical biconditional. ...

  • 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. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... The well-ordering theorem (not to be confused with the well-ordering axiom) states that every set can be well-ordered. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the dimension theorem for vector spaces states that a vector space has a definite, well-defined number of dimensions. ... In mathematics, the Ultrafilter Lemma states that every filter is a subset of some ultrafilter, i. ...


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.

In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ...

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:

a(1,1)+b(-1,2)=(0,0). ,

Then:

(a-b,a+2b)=(0,0) ,
  and  
a-b=0 ;
  and  
a+2b=0. ,

Subtracting the first equation from the second, we obtain:

3b=0 ;
  so  
b=0. ,

And from the first equation then:

a=0. ,

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:

x(1,1)+y(-1,2)=(a,b). ,

Then we have to solve the equations:

x-y=a ,
x+2y=b. ,

Subtracting the first equation from the second, we get:

3y=b-a, ,
          and then
y=(b-a)/3, ,
        and finally
x=y+a=((b-a)/3)+a=(b+2a)/3. ,

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. In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...


By the invertible matrix theorem

Simply compute the determinant In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

detbegin{bmatrix}1&-11&2end{bmatrix}=3neq0.

Since the above matrix has a nonzero determinant, its columns form a basis of R2. See: invertible matrix. In linear algebra, a column vector is an m × 1 matrix, i. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...


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. In mathematics, an index set is another name for a function domain. ...


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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. Definition Let F denote an arbitrary field (such as the real numbers R or the complex numbers C). ...


Proof. The proof makes use of the fact that the standard basis of Fn is an ordered basis. In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...


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. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...


Related notions

The phrase Hamel basis (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 mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, an orthonormal basis of an inner product space V(i. ...


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. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis. ...


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 Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

int_0^{2pi} left|f(x)right|^2,dx<infty.

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

lim_{nrightarrowinfty}int_0^{2pi}biggl|a_0+sum_{k=1}^n bigl(a_kcos(kx)+b_ksin(kx)bigr)-f(x)biggr|^2,dx=0

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.


See also


  Results from FactBites:
 
Basis (linear algebra) - Wikipedia, the free encyclopedia (1187 words)
In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space.
More precisely, a basis of a vector space is a set of linearly independent vectors that span the whole space.
What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations.
PlanetMath: linear algebra (482 words)
Linear algebra is the branch of mathematics devoted to the theory of linear structure.
From the geometric point of view, “linear” is synonymous with “straight”, and consequently linear algebra can be regarded as the branch of mathematics dealing with lines and planes, as well as with transformations of space that preserve “straightness”, e.g.
Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering.
  More results at FactBites »

 
 

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