Basis (linear algebra)


Basis (linear algebra) - Definition

In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions:

B is both a set of linearly independent vectors and a generating set of V.
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 proper superset is.
every vector in V can be expressed as a linear combination of vectors in B in a unique way.

Recall that a set B is a generating set of V if every vector in V is a linear combination of vectors in B. This definition includes a finiteness condition: a linear combination is always a finite sum of the form a₁v₁ + ... + a_nv_n.

Importantly, one can show that every vector space has a basis. For spaces that cannot be finitely generated, Zorn's lemma is needed for the proof. Also, 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 for vector spaces.

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Examples

Example I: Show that the vectors (1,1) and (-1,2) form a basis for R².

Proof: We have to prove that these 2 vectors are both 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:

<math>

(a-b,a+2b)=(0,0) \,<math>

and

and

Subtracting the first equation from the second, we obtain:

<math>

3b=0 \;<math> 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:

<math>

3y=b-a, \,<math> and then

<math>

y=(b-a)/3, \,<math> and finally

Example II: It is easy to show that 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.

Example III: Let W be the real vector space generated by the functions e^t, e^2t. The two functions are linearly independent, and therefore form a basis for W.

Example IV: 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.

Other notions

The phrase Hamel basis is sometimes used to denote a basis as defined above, in which the fact that all linear combinations are finite is crucial. A set B is a Hamel basis of a vector space V if every member of V is a linear combination of just finitely many members of B.

However, in Hilbert spaces and other Banach spaces, one often considers 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 finite linear combinations, but what is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations. More generally, in topological vector spaces, one may define infinite sums (or series) and express elements of the space as infinite linear combinations of other elements. To better distinguish these notions, vector space bases are also called Hamel bases and the vector space dimension is also known as Hamel dimension.

An "orthonormal basis" of an infinite-dimensional Hilbert space is not a Hamel basis

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 set of all complex-valued functions that are quadratically integrable on the interval [0, 2π], i.e., functions f satisfying

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

These functions are linearly independent, and every function that is quadratically integrable on that interval is an "infinite linear combination" of them. That means that

<math>\lim_{n\rightarrow\infty}\int_0^{2\pi}\left|\left(a_0+\sum_{k=1}^n a_k\cos(kx)+b_k\sin(kx)\right)-f(x)\right|^2\,dx=0<math>

for suitable coefficients a_k, b_k. But most quadratically 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; orthonormal bases of these spaces are important to Fourier analysis.