Editing Linear subspace (section)

===Systems of linear equations===
The solution set to any [[homogeneous system of linear equations]] with ''n'' variables is a subspace in the [[coordinate space]] ''K''<sup>''n''</sup>:
<math display="block">\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right]  \in K^n : \begin{alignat}{6}
a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0&    \\
a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0&    \\
&&     &&            &&              &&                && \vdots\quad& \\
a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0&
\end{alignat} \right\}. </math>

For example, the set of all vectors {{math|(''x'', ''y'', ''z'')}} (over real or [[rational number]]s) satisfying the equations
<math display="block">x + 3y + 2z = 0 \quad\text{and}\quad 2x - 4y + 5z = 0</math>
is a one-dimensional subspace. More generally, that is to say that given a set of ''n'' independent functions, the dimension of the subspace in ''K''<sup>''k''</sup> will be the dimension of the [[null set]] of ''A'', the composite matrix of the ''n'' functions.