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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 . DEFINITION A basis ''B'' of a Vector Space ''V'' is a Linearly Independent subset of ''V'' that Span s (or Generates ) ''V''. In more detail, suppose that ''B'' = { ''v''1, …, ''v''''n'' } is a finite subset of a vector space ''V'' over a Field F (such as the Real or Complex Number s '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:
:: for all ''a''1, …, ''a''''n'' ∈ F, if ''a''1''v''1 + … + ''a''''n''''v''''n'' = 0, then necessarily ''a''1 = … = ''a''''n'' = 0; and
:: for every ''x'' in ''V'' it is possible to choose ''a''1, …, ''a''''n'' ∈ F such that ''x'' = ''a''1''v''1 + … + ''a''''n''''v''''n''. 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
The axioms of a below. It is often convenient to list the basis vectors in a specific ''order'', for example, when considering the '' 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:
The theorem that every vector space has a basis is implied by the . 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. EXAMPLES
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:
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: : Then: : and and Subtracting the first equation from the second, we obtain: : 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: : and then : and finally : 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 : Since the above matrix has a nonzero determinant, its . 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 F''n'' to ''V''. ''Proof''. The proof makes use of the fact that the Standard Basis of F''n'' is an ordered basis. Suppose first that φ 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 '''F'''''n''. Conversely, given an ordered basis, consider the map defined by : ''φ''(''x'') = ''x''1''v''1 + ''x''2''v''2 + ... + ''x''''n''''v''''n'', where ''x'' = ''x''1e1 + ''x''2e2 + ... + ''x''''n''e''n'' is an element of '''F'''''n''. 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 F''n'' → ''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'') = (''a''1, ''a''2,...,''a''''n'') ∈ F''n'', then the components ''a''''j'' = ''a''''j''(''v'') are the coordinates of ''v'' in the sense that ''v'' = ''a''1(''v'') ''v''1 + ''a''2(''v'') ''v''2 + ... + ''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 Functional s. 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 ''a''1''v''1 + … + ''a''''n''''v''''n'' is always finite. In Hilbert Space s and other Banach Space s, 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 Space s, 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.'' 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 2π that are square-integrable on this interval, i.e., functions ''f'' satisfying | ||
| :<math>\lim {n Ightarrow\infty}\int 0^{2\pi}\biggla 0+\sum {k | 1}^n \bigl(a_k\cos(kx)+b_k\sin(kx)\bigr)-f(x)\biggr^2\,dx=0</math> |
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