Editing Vector space (section)

== Bases, vector coordinates, and subspaces ==
[[File:Vector components and base change.svg|class=skin-invert-image|A vector {{math|'''v'''}} in {{math|'''R'''<sup>2</sup>}} (blue) expressed in terms of different bases: using the [[standard basis]] of {{Math|'''R'''<sup>2</sup>}}: {{math|1='''v''' = ''x'''''e'''<sub>1</sub> + ''y'''''e'''<sub>2</sub>}} (black), and using a different, non-[[orthogonal vector|orthogonal]] basis: {{math|1='''v''' = '''f'''<sub>1</sub> + '''f'''<sub>2</sub>}} (red).|thumb|200px]]

;[[Linear combination]]
: Given a set {{mvar|G}} of elements of a {{mvar|F}}-vector space {{mvar|V}}, a linear combination of elements of {{mvar|G}} is an element of {{mvar|V}} of the form <math display=block> a_1 \mathbf{g}_1 + a_2 \mathbf{g}_2 + \cdots + a_k \mathbf{g}_k,</math> where <math>a_1, \ldots, a_k\in F</math> and <math>\mathbf{g}_1, \ldots, \mathbf{g}_k\in G.</math> The scalars <math>a_1, \ldots, a_k</math> are called the ''coefficients'' of the linear combination.{{sfn|Brown|1991|p=94}}
;[[Linear independence]]
:The elements of a subset {{mvar|G}} of a {{mvar|F}}-vector space {{mvar|V}} are said to be ''linearly independent'' if no element of {{mvar|G}} can be written as a linear combination of the other elements of {{mvar|G}}. Equivalently, they are linearly independent if two linear combinations of elements of {{mvar|G}} define the same element of {{mvar|V}} if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.{{sfn|Brown|1991|pages=99-101}}
;[[Linear subspace]]
:A ''linear subspace'' or ''vector subspace'' {{mvar|W}} of a vector space {{mvar|V}} is a non-empty subset of {{mvar|V}} that is [[closure (mathematics)|closed]] under vector addition and scalar multiplication; that is, the sum of two elements of {{mvar|W}} and the product of an element of {{mvar|W}} by a scalar belong to {{mvar|W}}.{{sfn|Brown|1991|p=92}} This implies that every linear combination of elements of {{mvar|W}} belongs to {{mvar|W}}. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.{{sfn|Stoll|Wong|1968|p=[https://books.google.com/books?id=gLbiBQAAQBAJ&pg=PA14 14]}}<br>The closure property also implies that ''every [[intersection (set theory)|intersection]] of linear subspaces is a linear subspace.''{{sfn|Stoll|Wong|1968|p=[https://books.google.com/books?id=gLbiBQAAQBAJ&pg=PA14 14]}}
;[[Linear span]]
:Given a subset {{mvar|G}} of a vector space {{mvar|V}}, the ''linear span'' or simply the ''span'' of {{mvar|G}} is the smallest linear subspace of {{mvar|V}} that contains {{mvar|G}}, in the sense that it is the intersection of all linear subspaces that contain {{mvar|G}}. The span of {{mvar|G}} is also the set of all linear combinations of elements of {{mvar|G}}.<br> If {{mvar|W}} is the span of {{mvar|G}}, one says that {{mvar|G}} ''spans'' or ''generates'' {{mvar|W}}, and that {{mvar|G}} is a ''[[spanning set]]'' or a ''generating set'' of {{mvar|W}}.{{sfn|Roman|2005|pages=41-42}}
;[[Basis (linear algebra)|Basis]] and [[dimension (vector space)|dimension]]
:A subset of a vector space is a ''basis'' if its elements are linearly independent and span the vector space.{{sfnm
 | 1a1 = Lang | 1y = 1987 | 1p = 10–11
 | 2a1 = Anton | 2a2 = Rorres | 2y = 2010 | 2p = [https://books.google.com/books?id=1PJ-WHepeBsC&pg=PA212 212]}} Every vector space has at least one basis, or many in general (see {{slink|Basis (linear algebra)|Proof that every vector space has a basis}}).{{sfn|Blass|1984}} Moreover, all bases of a vector space have the same [[cardinality]], which is called the ''dimension'' of the vector space (see [[Dimension theorem for vector spaces]]).{{sfn|Joshi|1989|p=[https://books.google.com/books?id=RM1D3mFw2u0C&pg=PA450 450]}} This is a fundamental property of vector spaces, which is detailed in the remainder of the section.

<span id=label1>''Bases''</span> are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called [[Hamel bases]], depends on the [[axiom of choice]]. It follows that, in general, no base can be explicitly described.{{sfn|Heil|2011|p=[https://books.google.com/books?id=prfuUT0Sw-AC&pg=PA126 126]}} For example, the [[real number]]s form an infinite-dimensional vector space over the [[rational number]]s, for which no specific basis is known. 

Consider a basis <math>(\mathbf{b}_1, \mathbf{b}_2 , \ldots, \mathbf{b}_n)</math> of a vector space {{mvar|V}} of dimension {{mvar|n}} over a field {{mvar|F}}. The definition of a basis implies that every <math>\mathbf v \in V</math> may be written 
<math display=block>\mathbf v = a_1 \mathbf b_1 + \cdots + a_n \mathbf b_n,</math> 
with <math>a_1,\dots, a_n</math> in {{mvar|F}}, and that this decomposition is unique. The scalars <math>a_1, \ldots, a_n</math> are called the ''coordinates'' of {{math|'''v'''}} on the basis. They are also said to be the ''coefficients'' of the decomposition of {{math|'''v'''}} on the basis. One also says that the {{mvar|n}}-[[tuple]] of the coordinates is the [[coordinate vector]] of {{math|'''v'''}} on the basis, since the set <math>F^n</math> of the {{mvar|n}}-tuples of elements of {{mvar|F}} is a vector space for [[componentwise operation|componentwise]] addition and scalar multiplication, whose dimension is {{mvar|n}}.

The [[one-to-one correspondence]] between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a [[vector space isomorphism]], which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.{{sfn|Halmos|1948|p=[https://books.google.com/books?id=1hzYCwAAQBAJ&pg=PA12 12]}}