Matrix Algebra

General Solutions

Basic and Free Variables. Suppose that the following REF is obtained.

$\displaystyle \left[\begin{array}{rrrr} 1 & 0 & -5 & 1 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The variables and corresponding to the pivot columns $\begin{bmatrix}1 0 0 \end{bmatrix}$ and $\begin{bmatrix}0 1 0 \end{bmatrix}$ are called basic variables. And the variable corresponding to the column $\begin{bmatrix}-5 1 0 \end{bmatrix}$ is called a free variable.

General Solutions. In the previous example a general solution can be expressed as

$\displaystyle \left\{\begin{array}{l} x_1 = 1 + 5 t \\ x_2 = 4 - t \\ x_3 = t \end{array}\right.$

where

is a parameter and corresponds to the free variable

EXAMPLE 3. Find the general solution of the linear system whose augmented matrix has been reduced to

$\displaystyle \left[\begin{array}{rrrrrr} 1 & 6 & 2 & -5 & -2 & -4 \\ 0 & 0 & 2 & -8 & -1 & 3 \\ 0 & 0 & 0 & 0 & 1 & 7 \end{array}\right]$