Set of vectors defined by an equation Let $W$ be a set of vectors given by an equation $2x_1 + x_2 - 2x_3 + 3x_4 = 5$. I need to write this into a affine subspace form like $W = p + span(s)$ and I have absolutely no idea how should I proceed with this kind of definition.
 A: Note that
\begin{align}
\textsf{W} &= \left\{ 
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} \in \mathbb{R}^4 :\, 2x_1 + x_2 - 2x_3 + 3x_4 = 5
\right\} \\
&= \left\{ 
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} \in \mathbb{R}^4 :\, x_2 = 5 - 2x_1 - 2x_3 - 3x_4
\right\} \\
&= \left\{ 
\begin{pmatrix} x_1 \\ 5 - 2x_1 + 2x_3 - 3x_4 \\ x_3 \\ x_4 \end{pmatrix} :\, x_1,x_3,x_4 \in \mathbb{R}
\right\} \\
&= \left\{ 
\begin{pmatrix} 0 \\ 5 \\ 0 \\ 0 \end{pmatrix} + x_1\begin{pmatrix} 1 \\ -2 \\ 0 \\ 0 \end{pmatrix} + x_3\begin{pmatrix} 0 \\ 2 \\ 1 \\ 0 \end{pmatrix} + x_4\begin{pmatrix} 0 \\ -3 \\ 0 \\ 1 \end{pmatrix} :\, t_1,t_2,t_3 \in \mathbb{R}
\right\}
\end{align}
so, putting
$$p = \begin{pmatrix} 0 \\ 5 \\ 0 \\ 0 \end{pmatrix} \textrm{ and } S = \left\{ \begin{pmatrix} 1 \\ -2 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ -3 \\ 0 \\ 1 \end {pmatrix} \right\}$$
we have $\textsf{W} = p + \operatorname{span}(S)$.
A: You should start by noticing that
\begin{align*}
S = \{\textbf{x}\in\textbf{R}^{4}\mid 2x_{1} + x_{2} - 2x_{3} + 3x_{4} = 0\}
\end{align*}
Based on this, you can consider $p$ to be the point $(0,5,0,0)$, which results from setting $x_{1} = x_{3} = x_{4} = 0$. Thus $W = p + \text{span}(S)$. Hopefully this helps.
