Parametric form of the solution set of a system of linear equations

Definition: The parametric vector form expresses the solution set of a linear system as a vector equation with free parameters:

$$x = p + s_1 v_1 + s_2 v_2 + \cdots + s_k v_k$$

where:

• $p$ is a particular solution (a fixed vector)
• $v_1, v_2, \ldots, v_k$ are direction vectors (one per free variable)
• $s_1, s_2, \ldots, s_k$ are free parameters in $\mathbb{R}$

Example: For the system with solution $x_1 = 3 + 2x_3$, $x_2 = 1 - x_3$, $x_3$ free:

$$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$$

Geometrically, this describes a line through $(3, 1, 0)$ in the direction $(2, -1, 1)$.