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In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors. Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset of V that spans V. This means that a subset B of V is a basis if it satisfies the two following conditions: linear independence for every finite subset { v 1 , … , v m } {\displaystyle \{\mathbf {v} _{1},\dotsc ,\mathbf {v} _{m}\}} of B, if c 1 v 1 + ⋯ + c m v m = 0 {\displaystyle c_{1}\mathbf {v} _{1}+\cdots +c_{m}\mathbf {v} _{m}=\mathbf {0} } for some c 1 , … , c m {\displaystyle c_{1},\dotsc ,c_{m}} in F, then c 1 = ⋯ = c m = 0 {\displaystyle c_{1}=\cdots =c_{m}=0} ; spanning property for every vector v in V, one can choose a 1 , … , a n {\displaystyle a_{1},\dotsc ,a_{n}} in F and v 1 , … , v n {\displaystyle \mathbf {v} _{1},\dotsc ,\mathbf {v} _{n}} in B such that v = a 1 v 1 + ⋯ + a n v n {\displaystyle \mathbf {v} =a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}} . The scalars a i {\displaystyle a_{i}} are called the coordinates of the vector v 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. In this case, the finite subset can be taken as B itself to check for linear independence in the above definition. It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation, or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis, which is therefore not simply an unstructured set, but a sequence, an indexed family, or similar; see § Ordered bases and coordinates below. Examples The set R2 of the ordered pairs of real numbers is a vector space under the operations of component-wise addition and scalar multiplication where λ {\displaystyle \lambda } is any real number. A simple basis of this vector space consists of the two vectors e1 = (1, 0) and e2 = (0, 1). These vectors form a basis (called the standard basis) because any vector v = (a, b) of R2 may be uniquely written as Any other pair of linearly independent vectors of R2, such as (1, 1) and (−1, 2), forms also a basis of R2. More generally, if F is a field, the set F n {\displaystyle F^{n}} of n-tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let be the n-tuple with all components equal to 0, except the ith, which is 1. Then e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} is a basis of F n , {\displaystyle F^{n},} which is called the standard basis of F n . {\displaystyle F^{n}.} A different flavor of example is given by polynomial rings. If F is a field, the collection F[X] of all polynomials in one indeterminate X with coefficients in F is an F-vector space. One basis for this space is the monomial basis B, consisting of all monomials: Any set of polynomials such that there is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials) is also a basis. (Such a set of polynomials is called a polynomial sequence.) But there are also many bases for F[X] that are not of this form. Properties Many properties of finite bases result from the Steinitz exchange lemma, which states that, for any vector space V, given a finite spanning set S and a lin.... Discover the B B Hamel popular books. Find the top 100 most popular B B Hamel books.