基 (代數)

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在線性代數中,向量空間V的基是一個線性獨立向量的集合並能夠線性擴展為V。

F為一域，V為一F-向量空間，B為V之一子集合。若以下任一條件成立，則稱B為V之基，:

1. B是一組線性獨立向量的集合並能線性擴展為V。
2. B是V的最小生產集合。
3. B是一個線性獨立向量的最大集合。
4. 每個V裡的向量都能夠以B中的向量用唯一方式線性組合起來。

以上四條件等價。

這個定義包含有限的條件:一個線性組合通常是個有限的總和。即: a₁v₁ + ... + a_nv_n. 其中a_i在F中，v_i在B中，B本身不必為有限集。

每個向量空間都有一個基。若該空間並非有限生成，欲證明此斷言須用佐恩引理（Zorn's Lemma）。同一向量空間的任何基都擁有相同序數(元素數量),稱為维数。這就是向量空間的维数理論。

目录 1 例子 2 基的延伸 3 排序基 4 相關記號 4.1 例子 5 參看

[编辑] 例子

例子I: (1,1) 及(-1,2) 組成R²的基。

證明: 這兩個向量是線性獨立並能產生R²。

第一部分：證明他們是線性獨立，假如有a,b 兩個數字:

因此:

  及  

(1)

  及  

(2)

把方程2減去方程1,得出:

  因此  

由方程1得出:

第二部分：證明他們能產生R², 我們先假設(a,b)R²中的任意元素。這裡存在x,y 能夠:

求解:

(1)

(2)

把方程2減去方程1,得出:

          因此

        最後

例子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.

例子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.

例子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.

[编辑] 基的延伸

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.

[编辑] 排序基

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 a ordering is specificed for the basis. For finite-dimensional vector spaces one typically indexes a basis {v_i} by the first n integers and orders the basis elements so that v₁ < v₂ < … .

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 from the coordinate space Fⁿ, with its standard basis, to V. To see this, let

A : Fⁿ → V

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

v_i = A(e_i) for 1 ≤ i ≤ n

where {e_i} is the standard basis for Fⁿ. Conversely, given any ordered basis {v_i} for V define a linear map A : Fⁿ → V by

It is not hard to check that A is an isomorphism. Thus ordered bases for V are in 1-1 correspondence with linear isomorphisms Fⁿ → V.

[编辑] 相關記號

The phrase Hamel basis is sometimes used to refer to 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.

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, vector space bases are called Hamel bases if the context requires it, and the vector space dimension is also known as Hamel dimension.

[编辑] 例子

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

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

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.

[编辑] 參看

• 線性代數
• 線性組合