Consider again the solution set to the linear system in
Example 2.3.3, which we described using parametric equations as
\begin{equation*}
x_1=7-2s-3t, x_2=s, x_3=1, x_4=t, x_5=2, \ s,t \in\R\text{,}
\end{equation*}
and using set building notation as
\begin{equation*}
S=\left\{(7-2s-3t, s, 1, t, 2)\colon s, t\in\R\right\}\text{.}
\end{equation*}
Using the vector space structure of \(\R^5\text{,}\) we can give yet another description of the general solution:
\begin{equation*}
\boldx=(7,0,1,0,2)+s(-2,1,0,0,0)+t(-3,0,0,1,0), \ s,t\in \R\text{.}
\end{equation*}
We will call this the vector parametrization description of the solution set. This description is obtained in steps by first breaking up the expression \((7-2s-3t, s, 1, t, 2)\) into a constant vector plus a vector involving the parameters \(s\) and \(t\text{,}\) and then rewriting the latter as a linear combination of two vectors with \(s\) and \(t\) as the coefficients:
\begin{align*}
(7-2s-3t, s, 1, t, 2) \amp = (7,0,1,0,2)+(-2s-3t,s,0,t,0) \\
\amp = (7,0,1,0,2)+(-2s,s,0,0,0)+(-3t,0,0,t,0)\\
\amp =(7,0,1,0,2)+s(-2,1,0,0,0)+t(-3,0,0,1,0)\text{.}
\end{align*}
The solution set of any consistent linear system can be described in this manner. In general, given a consistent system in \(n\) unknowns with \(k\) free variables, the general solution of the system is given by a vector parametrization of the form
\begin{equation}
(a_1,a_2,\dots, a_n)+t_1\boldv_1+t_2\boldv_2+\cdots +t_k\boldv_k\text{.}\tag{2.16}
\end{equation}
The vector parametrization
(2.16) leads to a useful geometric understanding of the set of solutions to a linear system. The point
\(P=(a_1,a_2,\dots, a_n)\) can be thought of as a
particular solution to the equation: the solution you obtain by choosing
\(t_1=t_2=\cdots =t_k=0\) in
(2.16). The general solution is then obtained by adding to the point
\(P\) any linear combination of the vectors
\(\boldv_1,\boldv_2,\dots, \boldv_k\text{.}\)
