Basis (linear algebra)

Basis (linear algebra)
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Basis_(linear_algebra)".

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.

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1 Definition
2 Properties
3 Examples
4 Basis extension
5 Proving that a finite set is a basis
6 Example of alternative proofs
7 Ordered bases and coordinates
8 Related notions
- 8.1 Banach Spaces
- 8.2 Example
9 See also
10 External links

Definition

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, …, v_n } 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, …, a_n ∈ F, if a₁v₁ + … + a_nv_n = 0, then necessarily a₁ = … = a_n = 0; and

the spanning property,

for every x in V it is possible to choose a₁, …, a_n ∈ F such that x = a₁v₁ + … + a_nv_n.

The numbers a_i are called the coordinates of the vector x with respect to the basis B, and by the first property they are uniquely determined.

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) B ⊂ V is a basis, if

every finite subset B₀ ⊆ B obeys the independence property shown above; and
for every x in V it is possible to choose a₁, …, a_n ∈ F and v₁, …, v_n ∈ B such that x = a₁v₁ + … + a_nv_n.

The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. 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.

Every vector space has a basis. The proof of this requires the axiom of choice. All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space. This result is known as the dimension theorem, and requires the ultrafilter lemma, a strictly weaker form of the axiom of choice.

Also many vector sets can be attributed a standard basis which comprises both spanning and linearly independent vectors.

Standard bases for example:

In Rⁿ {E1,...,En} where En is the n-th column of the identity matrix which consists of all ones in the main diagonal and zeros everywhere else. This is because the columns of the identity matrix are linerly independent can always span a vector set by expressing it as a linear combination.

In P₂ where P₂ is the set of all polynomials of degree at most 2 {1,x,x²} is the standard basis.

In M₂₂ {M_1,1,M_1,2,M_2,1,M_2,2} where M₂₂ is the set of all 2x2 matrices. and M_m,n is the 2x2 matrix with a 1 in the m,n position and zeros everywhere else. This again is a standard basis since it is linearly independent and spanning.

Examples

Consider R², the vector space of all coordinates (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 finite set is a basis further down).

More generally, the vectors e₁, e₂, ..., e_n 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 e^t and e^2t. 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 finite 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.

This does not work for infinite-dimensional vector spaces.

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:

$3b=0 \;$

$b=0. \,$

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.

By the invertible matrix theorem

Simply compute the determinant

$\det\begin{bmatrix}1&-1\\1&2\end{bmatrix}=3\neq0.$

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 {v_i} 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 : Fⁿ → V

is a linear isomorphism. Define an ordered basis {v_i} for V by

v_i = φ(e_i) for 1 ≤ i ≤ n

where {e_i} is the standard basis for Fⁿ.

Conversely, given an ordered basis, consider the map defined by

φ(x) = x₁v₁ + x₂v₂ + ... + x_nv_n,

where x = x₁e₁ + x₂e₂ + ... + x_ne_n 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 Fⁿ → V.

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

The maps sending a vector v to the components a_j(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 + … + a_nv_n 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.

Banach Spaces

As a result of the Baire category theorem if a Banach space has infinite Hamel dimension, it must have an uncountable Hamel basis. The Banach space could not be written as a countable union of the finite spanning sets. Thus any Banach space X of cardinality strictly greater than the continuum ( $\mathfrak c = 2^{\aleph_0}$ ) must have an uncountable Hamel basis. An example of such a space is the set of bounded functions from [0,1] to $\mathbb{R}$ equipped with the uniform norm. This is a Banach space that contains the subset $\left \{\chi_{A}\right \}_{A \subseteq [0,1]}$ (see characteristic function). This last subset has cardinality $2^{\mathfrak c}$ , so the Banach space has cardinality strictly greater than ${\mathfrak c}$ and hence has infinite Hamel dimension. From the existence of such a Banach space X, one can take the guaranteed infinite Hamel basis, normalize and then construct a linear bijection T from X onto itself so that the graph of T is not closed (even though the image of T being X must be closed). Such T could not be bounded and would have to be unbounded on say a countable sub-collection of the Hamel basis. This is an instance in which the closed graph theorem is not applicable as T would be unbounded.

Example

In the study of Fourier series, one learns that the functions {1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } 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

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

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

$\lim_{n\rightarrow\infty}\int_0^{2\pi}\biggl|a_0+\sum_{k=1}^n \bigl(a_k\cos(kx)+b_k\sin(kx)\bigr)-f(x)\biggr|^2\,dx=0$

for suitable (real or complex) coefficients a_k, b_k. 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.

External links

MIT Linear Algebra Lecture on Bases at Google Video, from MIT OpenCourseWare

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