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| [[File:3d two bases same vector.svg|130px|thumb|The same vector can be represented in two different bases (purple and red arrows).]]
| | If Americans love anything, it's [http://www.ractiv.com/ gadgets]. Unfortunately, that is really a large reason why our landfills are as full ensuring your company are. Gadgets break or become obsolete, and batteries die. Fortunately, gadget makers are finally learning in the solar-powered calculator and entering into more eco-friendly offerings.<br><br>Once are usually done, save the completed song to ones computer by pressing the "save" key. Some applications automatically save your recorded file in MP3 format and most will prompt you to decide which format you prefer in saving the lodge.<br><br><br><br>India does have a mighty army and became one of the major superpowers, though they will never show keep in mind this. The defence technology is so advanced that many discoveries are been sold to foreign countries. The Arihant submarine is probably the most advanced in the ominous landscape. The Arjun battletank peerless. The LCA so small and mavouvrabe that may truly stealthy. The LCA-tejas is associated with a secret composite materials and cannot been shot down.<br><br>The father of new york state India Gandhi but also Nelson Mandela both Practised the Indian dharma and forgave their enemies. Both were succesfull in liberating their country without lack of control.<br><br>You may have long weeks. you will have temporary defeat. and you may definitely be laughed at by spouse and children members at the beginning of your building.<br><br>This System works along with a 5 pip trailing stop and a 20 pip take gain. If the signal falls between 5 pips and 20 pips, Forex Ambush 2.0 gives a trade stick. If it falls far removed from this range, Forex Ambush 2.0 advises you in order to trade. The program does not use hard stop loss technique.<br><br>At the peak of Mr. Lee's not so veiled diatribe against the Bush administration for all of the blunders that followed Katrina much more he was suddenly and summarily overshadowed by another of his counterparts. Stand up Mr. Nagin. Perhaps this pair believes that get this unit foot on mouth episode deserves nevertheless another. Near to the anniversary of Mayor Nagin's famous "Chocolate City" remarks he rose to trump himself yet once when.<br><br>Basic mobile features haven't been this good. Along with BlackBerry Curve 8520 in Pink along with the BlackBerry Curve 8520 in White, you'll get to enjoy beauty in simplicity. |
| : ''Basis vector redirects here. For basis vector in the context of crystals, see [[crystal structure]]. For a more general concept in physics, see [[frame of reference]].''
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| In [[linear algebra]], a '''basis''' is a [[set (mathematics)|set]] of [[linear independence|linearly independent]] [[vector space|vector]]s that, in a [[linear combination]], can represent every vector in a given [[vector space]] or [[free module]], or, more simply put, which define a "coordinate system" (as long as the basis is given a definite order).<ref>Halmos, Paul Richard (1987) ''Finite-dimensional vector spaces'' (4th edition) Springer-Verlag, New York, [http://books.google.co.uk/books?id=mdWeEhA17scC&pg=PA10 page 10], ISBN 0-387-90093-4</ref> In more general terms, a basis is a linearly independent [[spanning set]].
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| Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors, whose coefficients are referred to as vector '''components'''. The statement that every vector space has a basis is equivalent to the [[axiom of choice]]. All bases of a vector space have the same number of elements, called the [[dimension (vector space)|dimension]] of the vector space.
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| == Definition ==
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| [[File:Basis graph (no label).svg|thumb|400px|This picture illustrates the [[standard basis]] in '''''R'''<sup>2</sup>''. The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is [[linearly dependent]] upon them.]]
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| A '''basis''' ''B'' of a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''F'' is a [[Linear independence|linearly independent]] subset of ''V'' that [[linear span|span]]s ''V''.
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| In more detail, suppose that ''B'' = { ''v''<sub>1</sub>, …, ''v''<sub>''n''</sub> } is a finite subset of a vector space ''V'' over a [[field (mathematics)|field]] '''F''' (such as the [[real numbers|real]] or [[complex number]]s '''R''' or '''C'''). Then ''B'' is a basis if it satisfies the following conditions:
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| * the ''linear independence'' property,
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| :: for all ''a''<sub>1</sub>, …, ''a''<sub>''n''</sub> ∈ '''F''', if ''a''<sub>1</sub>''v''<sub>1</sub> + … + ''a''<sub>''n''</sub>''v''<sub>''n''</sub> = 0, then necessarily ''a''<sub>1</sub> = … = ''a''<sub>''n''</sub> = 0; and
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| * the ''spanning'' property,
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| :: for every ''x'' in ''V'' it is possible to choose ''a''<sub>1</sub>, …, ''a''<sub>''n''</sub> ∈ '''F''' such that ''x'' = ''a''<sub>1</sub>''v''<sub>1</sub> + … + ''a''<sub>''n''</sub>''v''<sub>''n''</sub>.
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| The numbers ''a''<sub>i</sub> are called the coordinates of the vector ''x'' with respect to the basis ''B'', and by the first property they are uniquely determined.
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| A vector space that has a [[finite set|finite]] basis is called [[Dimension (vector space)|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
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| * every finite subset ''B''<sub>0</sub> ⊆ ''B'' obeys the independence property shown above; and
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| * for every ''x'' in ''V'' it is possible to choose ''a''<sub>1</sub>, …, ''a''<sub>''n''</sub> ∈ '''F''' and ''v''<sub>1</sub>, …, ''v''<sub>''n''</sub> ∈ ''B'' such that ''x'' = ''a''<sub>1</sub>''v''<sub>1</sub> + … + ''a''<sub>''n''</sub>''v''<sub>''n''</sub>.
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| 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|Related notions]] below.
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| 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 (mathematics)|set]]) of linearly independent vectors that span ''V'': see ''[[#Ordered bases and coordinates|Ordered bases and coordinates]]'' below.
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| ==Expression of a basis==
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| [[Image:Eulerangles.svg|right|thumb|300px|Basis defined by Euler angles - The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labeled N, is shown in green.]]
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| There are several ways to describe a basis for the space. Some are made ''[[ad hoc]]'' for a specific dimension. For example, there are [[orientation (geometry)|several ways]] to give a basis in dim 3, like [[Euler angles]].
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| The general case is to give a matrix with the components of the new basis vectors in columns. This is also the more general method because it can express any possible set of vectors even if it is not a basis. This matrix can be seen as three things: | |
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| '''Basis Matrix''': Is a matrix that represents the basis, because its columns are the components of vectors of the basis. This matrix represents any vector of the new basis as linear combination of the current basis.
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| '''Rotation operator''': When [[orthonormal]] bases are used, any other orthonormal basis can be defined by a [[rotation matrix]]. This matrix represents the [[rotation operator]] that rotates the vectors of the basis to the new one. It is exactly the same matrix as before because the rotation matrix multiplied by the identity matrix I has to be the new basis matrix.
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| '''Change of basis matrix''': This matrix can be used to change different objects of the space to the new basis. Therefore is called "[[change of basis]]" matrix. It is important to note that some objects change their components with this matrix and some others, like vectors, with its inverse.
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| == Properties ==
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| 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:
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| * ''B'' is a minimal generating set of ''V'', i.e., it is a generating set and no [[subset|proper subset]] of ''B'' is also a generating set.
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| * ''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.
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| * 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|Ordered bases and coordinates]]'' below) then the coefficients in this linear combination provide ''coordinates'' of the vector relative to the basis.
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| Every vector space has a basis. The proof of this requires the [[axiom of choice]]. All bases of a vector space have the same [[cardinal number|cardinality]] (number of elements), called the [[Dimension (vector space)|dimension]] of the vector space. This result is known as the [[dimension theorem for vector spaces|dimension theorem]], and requires the [[ultrafilter lemma]], a strictly weaker form of the axiom of choice.
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| Also many vector sets can be attributed a [[standard basis]] which comprises both spanning and linearly independent vectors.
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| Standard bases for example:
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| In R<sup>n</sup> {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 linearly independent can always span a vector set by expressing it as a linear combination.
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| In P<sub>2</sub> where P<sub>2</sub> is the set of all polynomials of degree at most 2 {1,x,x<sup>2</sup>} is the standard basis.
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| In M<sub>22</sub> {M<sub>1,1</sub>,M<sub>1,2</sub>,M<sub>2,1</sub>,M<sub>2,2</sub>} where M<sub>22</sub> is the set of all 2×2 matrices. and M<sub>m,n</sub> is the 2×2 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.
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| == Examples ==
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| *Consider '''R'''<sup>2</sup>, 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 '''e'''<sub>1</sub> = (1,0) and '''e'''<sub>2</sub> = (0,1): suppose that ''v'' = (''a'', ''b'') is a vector in '''R'''<sup>2</sup>, 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 '''R'''<sup>2</sup>.
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| *More generally, the vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, ..., '''e'''<sub>''n''</sub> are linearly independent and generate '''R'''<sup>''n''</sup>. Therefore, they form a basis for '''R'''<sup>''n''</sup> and the dimension of '''R'''<sup>''n''</sup> is ''n''. This basis is called the ''[[standard basis]]''.
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| *Let ''V'' be the [[real number|real]] vector space generated by the functions ''e''<sup>''t''</sup> and ''e''<sup>2''t''</sup>. These two functions are linearly independent, so they form a basis for ''V''.
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| *Let '''R'''[x] denote the vector space of real [[polynomial]]s; then (1, x, x<sup>2</sup>, ...) is a basis of '''R'''[x]. The dimension of '''R'''[x] is therefore equal to [[aleph number#Aleph-naught|aleph-0]].
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| == Extending to a basis ==
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| Let ''S'' be a subset of a vector space ''V''. To extend ''S'' to a basis means to find a basis ''B'' that contains ''S'' as a subset. This can be done if and only if ''S'' is linearly independent. Almost always, there is more than one such ''B'', except in rather special circumstances (i.e. ''S'' is already a basis, or ''S'' is empty and ''V'' has two elements).
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| A similar question is when does a subset ''S'' contain a basis. This occurs if and only if ''S'' spans ''V''. In this case, ''S'' will usually contain several different bases.
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| == Example of alternative proofs ==
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| Often, a mathematical result can be proven in more than one way.
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| Here, using three different proofs, we show that the vectors (1,1) and (−1,2) form a basis for '''R'''<sup>2</sup>.
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| === From the definition of ''basis'' ===
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| We have to prove that these two vectors are linearly independent and that they generate '''R'''<sup>2</sup>.
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| Part I: If two vectors v,w are linearly independent, then <math> av + bw = 0 </math> (a and b scalars) implies <math>a = 0, b = 0. </math>
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| To prove that they are linearly independent, suppose that there are numbers a,b such that:
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| :<math> a(1,1)+b(-1,2)=(0,0) \,</math>
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| (i.e., they are linearly dependent). Then:<br>
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| :<div style="vertical-align: 10%;display:inline;"><math>
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| (a-b,a+2b)=(0,0) \,</math></div> and <div style="vertical-align: 10%;display:inline;"><math>
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| a-b=0 \;</math></div> and <div style="vertical-align: 10%;display:inline;"><math>
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| a+2b=0. \,</math></div>
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| Subtracting the first equation from the second, we obtain:<br>
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| :<div style="vertical-align: 10%;display:inline;"><math>
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| 3b=0 \;</math></div> so <div style="vertical-align: 10%;display:inline;"><math>
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| b=0. \,</math></div>
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| Subtracting this equation from the first equation then:<br>
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| :<math> a=0. \,</math>
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| Hence we have linear independence.
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| Part II: To prove that these two vectors generate '''R'''<sup>2</sup>, we have to let (a,b) be an arbitrary element of '''R<sup>2</sup>''', and show that there exist numbers r,s ∈ '''R''' such that:<br>
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| :<math> r(1,1)+s(-1,2)=(a,b). \,</math>
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| Then we have to solve the equations:<br>
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| :<math> r-s=a \,</math>
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| :<math> r+2s=b. \,</math>
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| Subtracting the first equation from the second, we get:<br>
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| :<div style="vertical-align: 10%;display:inline;"><math>
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| 3s=b-a, \,</math></div> and then
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| :<div style="vertical-align: 10%;display:inline;"><math>
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| s=(b-a)/3, \,</math></div> and finally
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| :<math> r=s+a=((b-a)/3)+a=(b+2a)/3. \,</math>
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| === By the dimension theorem ===
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| Since (−1,2) is clearly not a multiple of (1,1) and since (1,1) is not the [[null vector (vector space)|zero vector]], these two vectors are linearly independent. Since the dimension of '''R'''<sup>2</sup> is 2, the two vectors already form a basis of '''R'''<sup>2</sup> without needing any extension.
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| === By the invertible matrix theorem ===
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| Simply compute the [[determinant]]
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| :<math>\det\begin{bmatrix}1&-1\\1&2\end{bmatrix}=3\neq0.</math>
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| Since the above matrix has a nonzero determinant, its [[column vector|columns]] form a basis of '''R'''<sup>2</sup>. See: [[invertible matrix]].
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| == Ordered bases and coordinates ==
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| A basis is just a linearly independent ''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 [[index set|indexes]] a basis {''v''<sub>''i''</sub>} by the first ''n'' integers. An ordered basis is also called a '''frame'''.
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| Suppose ''V'' is an ''n''-dimensional vector space over a [[field (mathematics)|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'''<sup>''n''</sup> to ''V''.
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| ''Proof''. The proof makes use of the fact that the [[standard basis]] of '''F'''<sup>''n''</sup> is an ordered basis.
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| Suppose first that
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| :''φ'' : '''F'''<sup>''n''</sup> → ''V''
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| is a linear isomorphism. Define an ordered basis {''v''<sub>''i''</sub>} for ''V'' by
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| : ''v''<sub>''i''</sub> = ''φ''('''e'''<sub>''i''</sub>) for 1 ≤ ''i'' ≤ ''n''
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| where {'''e'''<sub>''i''</sub>} is the standard basis for '''F'''<sup>''n''</sup>.
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| Conversely, given an ordered basis, consider the map defined by
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| : ''φ''(''x'') = ''x''<sub>1</sub>''v''<sub>1</sub> + ''x''<sub>2</sub>''v''<sub>2</sub> + ... + ''x''<sub>''n''</sub>''v''<sub>''n''</sub>,
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| where ''x'' = ''x''<sub>1</sub>'''e'''<sub>1</sub> + ''x''<sub>2</sub>'''e'''<sub>2</sub> + ... + ''x''<sub>''n''</sub>'''e'''<sub>''n''</sub> is an element of '''F'''<sup>''n''</sup>. It is not hard to check that ''φ'' is a linear isomorphism.
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| These two constructions are clearly inverse to each other. Thus ordered bases for ''V'' are in 1-1 correspondence with linear isomorphisms '''F'''<sup>''n''</sup> → ''V''.
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| The inverse of the linear isomorphism ''φ'' determined by an ordered basis {''v''<sub>''i''</sub>} equips ''V'' with ''coordinates'': if, for a vector ''v'' ∈ ''V'', ''φ''<sup>−1</sup>(''v'') = (''a''<sub>1</sub>, ''a''<sub>2</sub>,...,''a''<sub>''n''</sub>) ∈ '''F'''<sup>''n''</sup>, then the components ''a''<sub>''j''</sub> = ''a''<sub>''j''</sub>(''v'') are the coordinates of ''v'' in the sense that ''v'' = ''a''<sub>1</sub>(''v'') ''v''<sub>1</sub> + ''a''<sub>2</sub>(''v'') ''v''<sub>2</sub> + ... + ''a''<sub>''n''</sub>(''v'') ''v''<sub>''n''</sub>.
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| The maps sending a vector ''v'' to the components ''a''<sub>''j''</sub>(''v'') are linear maps from ''V'' to '''F''', because of ''φ''<sup>−1</sup> is linear. Hence they are [[linear functional]]s. They form a basis for the '''[[dual space]]''' of ''V'', called the '''dual basis'''.
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| == Related notions ==
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| === Analysis ===
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| In the context of infinite-dimensional vector spaces over the real or complex numbers, the term '''''Hamel basis''''' (named after [[Georg Hamel]]) or '''''algebraic basis''''' can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are [[orthogonal basis|orthogonal bases]] on [[Hilbert space]]s, [[Schauder basis|Schauder bases]] and [[Markushevich basis|Markushevich bases]] on [[normed linear space]]s.
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| The common feature of the other notions is that they permit the taking of infinite linear combinations of the basic vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for [[topological vector space]]s – a large class of vector spaces including e.g. [[Hilbert space]]s, [[Banach space]]s or [[Fréchet space]]s.
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| The preference of other types of bases for infinite dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If ''X'' is an infinite dimensional normed vector space which is [[complete space|complete]] (i.e. ''X'' is a [[Banach space]]), then any Hamel basis of ''X'' is necessarily [[uncountable]]. This is a consequence of the [[Baire category theorem]]. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite dimensional spaces have by definition finite bases and there are infinite dimensional (''non-complete'') normed spaces which have countable Hamel bases. Consider <math>c_{00}</math>, the space of the [[sequence]]s <math>x=(x_n)</math> of real numbers which have only finitely many non-zero coordinates, with the norm <math>\|x\|=\sup_n |x_n|.</math> The [[standard basis]] is its countable Hamel basis.
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| ====Example====
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| 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
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| :<math>\int_0^{2\pi} \left|f(x)\right|^2\,dx<\infty.</math>
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| 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
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| :<math>\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</math>
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| for suitable (real or complex) coefficients ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>. 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 typically not useful, whereas [[orthonormal bases]] of these spaces are essential in [[Fourier analysis]].
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| ===Affine geometry===
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| The related notions of an [[affine space]], [[projective space]], [[convex set]], and [[Cone (linear algebra)|cone]] have related notions of '''{{visible anchor|affine basis}}'''<ref>Notes on geometry, by Elmer G. Rees, [http://books.google.com/books?id=JkzPRaihGIYC&pg=PA7 p. 7]</ref> (a basis for an ''n''-dimensional affine space is <math>n+1</math> points in [[general linear position]]), '''{{visible anchor|projective basis}}''' (essentially the same as an affine basis, this is <math>n+1</math> points in general linear position, here in projective space), '''{{visible anchor|convex basis}}''' (the vertices of a polytope), and '''{{visible anchor|cone basis}}'''<ref>[http://www.springerlink.com/content/v8110k8n2864g32g/ Some remarks about additive functions on cones], Marek Kuczma</ref> (points on the edges of a polygonal cone); see also a [[Hilbert basis (linear programming)]].
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| ==See also==
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| * [[Change of basis]]
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| * [[Frame of a vector space]]
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| * [[Spherical basis]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| ===General references===
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| * {{Citation | last1=Blass | first1=Andreas | title=Axiomatic set theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Contemporary Mathematics volume 31 | mr=763890 | year=1984 | chapter=Existence of bases implies the axiom of choice | pages=31–33|isbn=0-8218-5026-1}}
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| * {{Citation | last1=Brown | first1=William A. | title=Matrices and vector spaces | publisher=M. Dekker | location=New York | isbn=978-0-8247-8419-5 | year=1991}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96412-6 | year=1987}}
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| ===Historical references===
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| * {{fr icon}} {{Citation | last1=Banach | first1=Stefan | author1-link=Stefan Banach | title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations) | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf | year=1922 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=3}}
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| * {{de icon}} {{Citation | last1=Bolzano | first1=Bernard | author1-link=Bernard Bolzano | title=Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) | url=http://dml.cz/handle/10338.dmlcz/400338 | year=1804}}
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| * {{fr icon}} {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Éléments d'histoire des mathématiques (Elements of history of mathematics) | publisher=Hermann | location=Paris | year=1969}}
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| * {{Citation | last1=Dorier | first1=Jean-Luc | title=A general outline of the genesis of vector space theory | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WG9-45NJHDR-C&_user=1634520&_coverDate=12%2F31%2F1995&_rdoc=2&_fmt=high&_orig=browse&_srch=doc-info(%23toc%236817%231995%23999779996%23308480%23FLP%23display%23Volume)&_cdi=6817&_sort=d&_docanchor=&_ct=9&_acct=C000054038&_version=1&_urlVersion=0&_userid=1634520&md5=fd995fe2dd19abde0c081f1e989af006 | mr=1347828 | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=227–261 | doi=10.1006/hmat.1995.1024}}
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| * {{fr icon}} {{Citation | last1=Fourier | first1=Jean Baptiste Joseph | author1-link=Joseph Fourier | title=Théorie analytique de la chaleur | url=http://books.google.com/books?id=TDQJAAAAIAAJ | publisher=Chez Firmin Didot, père et fils | year=1822}}
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| * {{de icon}} {{Citation | last1=Grassmann | first1=Hermann | author1-link=Hermann Grassmann | title=Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik | url=http://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1&dq=Die+Lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik | year=1844}}, reprint: {{Citation | other1-last=Kannenberg | other1-first=L.C. | title=Extension Theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2031-5 | year=2000 | author=Hermann Grassmann. Translated by Lloyd C. Kannenberg.}}
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| * {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=Lectures on Quaternions | url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 | publisher=Royal Irish Academy | year=1853}}
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| * {{de icon}} {{Citation | last1=Möbius | first1=August Ferdinand | author1-link=August Ferdinand Möbius | title=Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) | url=http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_MOBIUS__1_1_0 | year=1827}}
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| * {{Citation | last1=Moore | first1=Gregory H. | title=The axiomatization of linear algebra: 1875–1940 | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WG9-45NJHDR-D&_user=1634520&_coverDate=12%2F31%2F1995&_rdoc=3&_fmt=high&_orig=browse&_srch=doc-info(%23toc%236817%231995%23999779996%23308480%23FLP%23display%23Volume)&_cdi=6817&_sort=d&_docanchor=&_ct=9&_acct=C000054038&_version=1&_urlVersion=0&_userid=1634520&md5=4327258ef37b4c293b560238058e21ad | year=1995 | journal=[[Historia Mathematica]] | volume=22 | issue=3 | pages=262–303 | doi=10.1006/hmat.1995.1025}}
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| * {{it icon}} {{Citation | last1=Peano | first1=Giuseppe | author1-link=Giuseppe Peano | title=Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva | year=1888 | location=Turin}}
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| ==External links==
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| * Instructional videos from Khan Academy
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| **[http://khanexercises.appspot.com/video?v=zntNi3-ybfQ Introduction to bases of subspaces]
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| **[http://khanexercises.appspot.com/video?v=Zn2K8UIT8r4 Proof that any subspace basis has same number of elements]
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| * {{springer|title=Basis|id=p/b015350}}
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| {{linear algebra}}
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| {{DEFAULTSORT:Basis (Linear Algebra)}}
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| [[Category:Linear algebra]]
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| [[Category:Articles containing proofs]]
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| [[Category:Matroid theory]]
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