Editing Linear subspace (section)

===Linear parametric equations===
The subset of ''K''<sup>''n''</sup> described by a system of homogeneous linear [[parametric equations]] is a subspace:

:<math>\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right]  \in K^n : \begin{alignat}{7}
x_1 &&\; = \;&& a_{11} t_1 &&\; + \;&& a_{12} t_2 &&\; + \cdots + \;&& a_{1m} t_m &    \\
x_2 &&\; = \;&& a_{21} t_1 &&\; + \;&& a_{22} t_2 &&\; + \cdots + \;&& a_{2m} t_m &    \\
&& \vdots\;\; &&       &&       &&            &&                &&            &    \\
x_n &&\; = \;&& a_{n1} t_1 &&\; + \;&& a_{n2} t_2 &&\; + \cdots + \;&& a_{nm} t_m &    \\
\end{alignat} \text{ for some } t_1,\ldots,t_m\in K \right\}. </math>

For example, the set of all vectors (''x'',&nbsp;''y'',&nbsp;''z'') parameterized by the equations

:<math>x = 2t_1 + 3t_2,\;\;\;\;y = 5t_1 - 4t_2,\;\;\;\;\text{and}\;\;\;\;z = -t_1 + 2t_2</math>

is a two-dimensional subspace of ''K''<sup>3</sup>, if ''K'' is a [[number field]] (such as real or rational numbers).<ref name="fields" group="note">Generally, ''K'' can be any field of such [[characteristic (algebra)|characteristic]] that the given integer matrix has the appropriate [[rank (matrix theory)|rank]] in it. All fields include [[integer]]s, but some integers may equal to zero in some fields.</ref>