Editing Real coordinate space (section)

== Vector space ==
The coordinate space {{math|'''R'''<sup>''n''</sup>}} forms an {{mvar|n}}-dimensional [[vector space]] over the [[field (mathematics)|field]] of real numbers with the addition of the structure of [[linearity]], and is often still denoted {{math|'''R'''<sup>''n''</sup>}}.  The operations on {{math|'''R'''<sup>''n''</sup>}} as a vector space are typically defined by
<math display="block">\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)</math>
<math display="block">\alpha \mathbf x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n).</math>
The [[additive identity|zero vector]] is given by
<math display="block">\mathbf 0 = (0, 0, \ldots, 0)</math>
and the [[additive inverse]] of the vector {{math|'''x'''}} is given by
<math display="block">-\mathbf x = (-x_1, -x_2, \ldots, -x_n).</math>

This structure is important because any {{mvar|n}}-dimensional real vector space is isomorphic to the vector space {{math|'''R'''<sup>''n''</sup>}}.

===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}}.

===Standard basis===
{{main|Standard basis}}
The coordinate space {{math|'''R'''<sup>''n''</sup>}} comes with a standard basis:
<math display="block">\begin{align}
\mathbf e_1 & = (1, 0, \ldots, 0) \\
\mathbf e_2 & = (0, 1, \ldots, 0) \\
& {}\;\;  \vdots \\
\mathbf e_n & = (0, 0, \ldots, 1)
\end{align}</math>

To see that this is a basis, note that an arbitrary vector in {{math|'''R'''<sup>''n''</sup>}} can be written uniquely in the form
<math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>