Editing Vector space (section)

===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.