Editing Real coordinate space (section)

===Matrix notation===
{{main|Matrix (mathematics)}}
In standard [[matrix (mathematics)|matrix]] notation, each element of {{math|'''R'''<sup>''n''</sup>}} is typically written as a [[column vector]]
<math display="block">\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>
and sometimes as a [[row vector]]:
<math display="block">\mathbf x = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}.</math>

The coordinate space {{math|'''R'''<sup>''n''</sup>}} may then be interpreted as the space of all {{math|''n'' × 1}} [[column vector]]s, or all {{math|1 × ''n''}} [[row vector]]s with the ordinary matrix operations of addition and [[scalar multiplication]].

[[Linear transformation]]s from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} may then be written as {{math|''m'' × ''n''}} matrices which act on the elements of {{math|'''R'''<sup>''n''</sup>}} via [[left and right (algebra)|left]] multiplication (when the elements of {{math|'''R'''<sup>''n''</sup>}} are column vectors) and on elements of {{math|'''R'''<sup>''m''</sup>}} via right multiplication (when they are row vectors). The formula for left multiplication, a special case of [[matrix multiplication]], is:
<math display="block">(A{\mathbf x})_k = \sum_{l=1}^n A_{kl} x_l</math>

{{anchor|continuity of linear maps}}Any linear transformation is a [[continuous function]] (see [[#Topological properties|below]]). Also, a matrix defines an [[open map]] from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} if and only if the [[rank (matrix theory)|rank of the matrix]] equals to {{mvar|m}}.