Editing Vector space (section)

==Linear maps and matrices==
{{Main|Linear map}}
The relation of two vector spaces can be expressed by ''linear map'' or ''[[linear transformation]]''. They are [[function (mathematics)|functions]] that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
<math display=block>
\begin{align}
 f(\mathbf{v} + \mathbf{w}) &= f(\mathbf{v}) + f(\mathbf{w}), \\ 
 f(a \cdot \mathbf{v}) &= a \cdot f(\mathbf{v})
\end{align}
</math> 
for all <math>\mathbf{v}</math> and <math>\mathbf{w}</math> in <math>V,</math> all <math>a</math> in <math>F.</math>{{sfn|Roman|2005|loc=ch. 2, p. 45}}

An ''[[isomorphism]]'' is a linear map {{math|''f'' : ''V'' → ''W''}} such that there exists an [[inverse map]] {{math|''g'' : ''W'' → ''V''}}, which is a map such that the two possible [[function composition|compositions]] {{math|''f'' ∘ ''g'' : ''W'' → ''W''}} and {{math|''g'' ∘ ''f'' : ''V'' → ''V''}} are [[Identity function|identity maps]]. Equivalently, {{math|''f''}} is both one-to-one ([[injective]]) and onto ([[surjective]]).{{sfn|Lang|1987|loc=ch. IV.4, Corollary, p. 106}} If there exists an isomorphism between {{math|''V''}} and {{math|''W''}}, the two spaces are said to be ''isomorphic''; they are then essentially identical as vector spaces, since all identities holding in {{math|''V''}} are, via {{math|''f''}}, transported to similar ones in {{math|''W''}}, and vice versa via {{math|''g''}}.

[[File:Vector components.svg|class=skin-invert-image|180px|right|thumb|Describing an arrow vector {{math|'''v'''}} by its coordinates {{math|''x''}} and {{math|''y''}} yields an isomorphism of vector spaces.]]
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see {{slink||Examples}}) are isomorphic: a planar arrow {{math|'''v'''}} departing at the [[origin (mathematics)|origin]] of some (fixed) [[coordinate system]] can be expressed as an ordered pair by considering the {{math|''x''}}- and {{math|''y''}}-component of the arrow, as shown in the image at the right. Conversely, given a pair {{math|(''x'', ''y'')}}, the arrow going by {{math|''x''}} to the right (or to the left, if {{math|''x''}} is negative), and {{math|''y''}} up (down, if {{math|''y''}} is negative) turns back the arrow {{math|'''v'''}}.{{sfn|Nicholson|2018|loc=ch. 7.3}}

Linear maps {{math|''V'' → ''W''}} between two vector spaces form a vector space {{math|Hom<sub>''F''</sub>(''V'', ''W'')}}, also denoted {{math|L(''V'', ''W'')}}, or {{math|𝓛(''V'', ''W'')}}.{{sfn|Lang|1987|loc=Example IV.2.6}} The space of linear maps from {{math|''V''}} to {{math|''F''}} is called the ''[[dual vector space]]'', denoted {{math|''V''<sup>∗</sup>}}.{{sfn|Lang|1987|loc=ch. VI.6}} Via the injective [[natural (category theory)|natural]] map {{math|''V'' → ''V''<sup>∗∗</sup>}}, any vector space can be embedded into its ''bidual''; the map is an isomorphism if and only if the space is finite-dimensional.{{sfn|Halmos|1974|loc=p. 28, Ex. 9}}

Once a basis of {{math|''V''}} is chosen, linear maps {{math|''f'' : ''V'' → ''W''}} are completely determined by specifying the images of the basis vectors, because any element of {{math|''V''}} is expressed uniquely as a linear combination of them.{{sfn|Lang|1987|loc=Theorem IV.2.1, p. 95}} If {{math|1=dim ''V'' = dim ''W''}}, a [[bijection|1-to-1 correspondence]] between fixed bases of {{math|''V''}} and {{math|''W''}} gives rise to a linear map that maps any basis element of {{math|''V''}} to the corresponding basis element of {{math|''W''}}. It is an isomorphism, by its very definition.{{sfn|Roman|2005|loc=Th. 2.5 and 2.6, p. 49}} Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is ''completely classified'' ([[up to]] isomorphism) by its dimension, a single number. In particular, any ''n''-dimensional {{math|''F''}}-vector space {{math|''V''}} is isomorphic to {{math|''F''<sup>''n''</sup>}}. However, there is no "canonical" or preferred isomorphism; an isomorphism {{math|''φ'' : ''F''<sup>''n''</sup> → ''V''}} is equivalent to the choice of a basis of {{math|''V''}}, by mapping the standard basis of {{math|''F''<sup>''n''</sup>}} to {{math|''V''}}, via {{math|''φ''}}. 

===Matrices===
{{Main|Matrix (mathematics)|l1=Matrix|Determinant}}
[[Image:Matrix.svg|class=skin-invert-image|right|thumb|200px|A typical matrix]]
''Matrices'' are a useful notion to encode linear maps.{{sfn|Lang|1987|loc=ch. V.1}} They are written as a rectangular array of scalars as in the image at the right. Any {{math|''m''}}-by-{{math|''n''}} matrix <math>A</math> gives rise to a linear map from {{math|''F''<sup>''n''</sup>}} to {{math|''F''<sup>''m''</sup>}}, by the following
<math display=block>\mathbf x = (x_1, x_2, \ldots, x_n) \mapsto \left(\sum_{j=1}^n a_{1j}x_j, \sum_{j=1}^n a_{2j}x_j, \ldots, \sum_{j=1}^n a_{mj}x_j \right),</math> 
where <math display="inline">\sum</math> denotes [[summation]], or by using the [[matrix multiplication]] of the matrix <math>A</math> with the coordinate vector <math>\mathbf{x}</math>:
<div id=equation2><math display=block>\mathbf{x} \mapsto A \mathbf{x}.</math></div>
Moreover, after choosing bases of {{math|''V''}} and {{math|''W''}}, ''any'' linear map {{math|''f'' : ''V'' → ''W''}} is uniquely represented by a matrix via this assignment.{{sfn|Lang|1987|loc=ch. V.3., Corollary, p. 106}}

[[Image:Determinant parallelepiped.svg|class=skin-invert-image|200px|right|thumb|The volume of this [[parallelepiped]] is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors {{math|'''r'''<sub>1</sub>}}, {{math|'''r'''<sub>2</sub>}}, and {{math|'''r'''<sub>3</sub>}}.]]
The [[determinant]] {{math|det (''A'')}} of a [[square matrix]] {{math|''A''}} is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.{{sfn|Lang|1987|loc=Theorem VII.9.8, p. 198}} The linear transformation of {{math|'''R'''<sup>''n''</sup>}} corresponding to a real ''n''-by-''n'' matrix is [[Orientation (vector space)|orientation preserving]] if and only if its determinant is positive.

===Eigenvalues and eigenvectors===
{{Main|Eigenvalues and eigenvectors}}
[[Endomorphism]]s, linear maps {{math|''f'' : ''V'' → ''V''}}, are particularly important since in this case vectors {{math|'''v'''}} can be compared with their image under {{math|''f''}}, {{math|''f''('''v''')}}. Any nonzero vector {{math|'''v'''}} satisfying {{math|1=''λ'''''v''' = ''f''('''v''')}}, where {{math|''λ''}} is a scalar, is called an ''eigenvector'' of {{math|''f''}} with ''eigenvalue'' {{math|''λ''}}.{{sfn|Roman|2005||loc=ch. 8, p. 135–156}} Equivalently, {{math|'''v'''}} is an element of the [[Kernel (linear algebra)|kernel]] of the difference {{math|''f'' − ''λ'' · Id}} (where Id is the [[identity function|identity map]] {{math|''V'' → ''V'')}}. If {{math|''V''}} is finite-dimensional, this can be rephrased using determinants: {{math|''f''}}  having eigenvalue {{math|''λ''}} is equivalent to
<math display=block>\det(f - \lambda \cdot \operatorname{Id}) = 0.</math>
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in {{math|''λ''}}, called the [[characteristic polynomial]] of {{math|''f''}}.{{sfn||Lang|1987|loc=ch. IX.4}} If the field {{math|''F''}} is large enough to contain a zero of this polynomial (which automatically happens for {{math|''F''}} [[algebraically closed field|algebraically closed]], such as {{math|1=''F'' = '''C'''}}) any linear map has at least one eigenvector. The vector space {{math|''V''}} may or may not possess an [[eigenbasis]], a basis consisting of eigenvectors. This phenomenon is governed by the [[Jordan canonical form]] of the map.{{sfn|Roman|2005|loc=ch. 8, p. 140}} The set of all eigenvectors corresponding to a particular eigenvalue of {{math|''f''}} forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and {{math|''f''}}) in question.