Definition of free/independent variable in a linear system

Definition: In the echelon form of a linear system, a free variable (also called an independent variable) is a variable that does not correspond to any pivot column.

Free variables are not determined by the system — they can be assigned any value. Once free variables are assigned values, the basic variables are uniquely determined through back-substitution.

Properties:

• The number of free variables = total variables $-$ number of pivots = $n - \text{rank}(A)$
• Each free variable adds one dimension to the solution space
• Free variables are used as parameters in the parametric form of the solution

Example: In the echelon form:
$$\begin{bmatrix} \boxed{2} & 3 & 1 & | & 5 \\ 0 & \boxed{1} & -2 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$

$x_3$ is a free variable (no pivot in column 3). It can take any value.