Euclidean Subspace - Linear Parametric Equations

Linear Parametric Equations

The subset of Rn described by a system of homogeneous linear parametric equations is a subspace:

$\left\{ \left \in \textbf{R}^n : \begin{alignat}{7} x_1 &&\; = \;&& a_{11} t_1 &&\; + \;&& a_{12} t_2 &&\; + \cdots + \;&& a_{1m} t_m & \\ x_2 &&\; = \;&& a_{21} t_1 &&\; + \;&& a_{22} t_2 &&\; + \cdots + \;&& a_{2m} t_m & \\ \vdots \,&& && \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; & \\ x_n &&\; = \;&& a_{n1} t_1 &&\; + \;&& a_{n2} t_2 &&\; + \cdots + \;&& a_{nm} t_m & \\ \end{alignat} \text{ for some } t_1,\ldots,t_m\in\textbf{R} \right\}.$

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

is a two-dimensional subspace of R3.

Read more about this topic: Euclidean Subspace