Editing Basis (linear algebra) (section)

== Coordinates {{anchor|Ordered bases and coordinates}} ==

Let {{mvar|V}} be a vector space of finite dimension {{mvar|n}} over a field {{mvar|F}}, and 
<math display="block">B = \{\mathbf b_1, \ldots, \mathbf b_n\}</math>
be a basis of {{mvar|V}}. By definition of a basis, every {{math|'''v'''}} in {{mvar|V}} may be written, in a unique way, as
<math display="block">\mathbf v = \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n,</math>
where the coefficients <math>\lambda_1, \ldots, \lambda_n</math> are scalars (that is, elements of {{mvar|F}}), which are called the ''coordinates'' of {{math|'''v'''}} over {{mvar|B}}. However, if one talks of the ''set'' of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same ''set'' of coefficients. For example, <math>3 \mathbf b_1 + 2 \mathbf b_2</math> and <math>2 \mathbf b_1 + 3 \mathbf b_2</math> have the same set of coefficients {{math|{2, 3}<nowiki/>}}, and are different. It is therefore often convenient to work with an '''ordered basis'''; this is typically done by [[index set|indexing]] the basis elements by the first natural numbers. Then, the coordinates of a vector form a [[sequence (mathematics)|sequence]] similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an [[origin (mathematics)|origin]], is also called a '''''coordinate frame''''' or simply a ''frame'' (for example, a [[Cartesian frame]] or an [[affine frame]]).

Let, as usual, <math>F^n</math> be the set of the [[tuple|{{mvar|n}}-tuples]] of elements of {{mvar|F}}. This set is an {{mvar|F}}-vector space, with addition and scalar multiplication defined component-wise. The map 
<math display="block">\varphi: (\lambda_1, \ldots, \lambda_n) \mapsto \lambda_1 \mathbf b_1 + \cdots + \lambda_n \mathbf b_n</math>
is a [[linear isomorphism]] from the vector space <math>F^n</math> onto {{mvar|V}}. In other words, <math>F^n</math> is the [[coordinate space]] of {{mvar|V}}, and the {{mvar|n}}-tuple <math>\varphi^{-1}(\mathbf v)</math> is the [[coordinate vector]] of {{math|'''v'''}}.

The [[inverse image]] by <math>\varphi</math> of <math>\mathbf b_i</math> is the {{mvar|n}}-tuple <math>\mathbf e_i</math> all of whose components are 0, except the {{mvar|i}}th that is 1. The <math>\mathbf e_i</math> form an ordered basis of <math>F^n</math>, which is called its [[standard basis]] or [[canonical basis]]. The ordered basis {{mvar|B}} is the image by <math>\varphi</math> of the canonical basis of {{nowrap|<math>F^n</math>.}}

It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of {{nowrap|<math>F^n</math>,}} and that every linear isomorphism from <math>F^n</math> onto {{mvar|V}} may be defined as the isomorphism that maps the canonical basis of <math>F^n</math> onto a given ordered basis of {{mvar|V}}. In other words, it is equivalent to define an ordered basis of {{mvar|V}}, or a linear isomorphism from <math>F^n</math> onto {{mvar|V}}.