Homogeneous systems

Linear systems of the form $A\mathbf{x}=0$ are homogeneous. Linear systems of the form $A\mathbf{x}=\mathbf{b}$ where $\mathbf{b}\ne 0,$ are inhomogeneous.

Here the trivial/nontrivial refers to $\mathbf{x}$. If $$\begin{gathered} \mathbf{x}=\left[\text{matrix}0 \\ 0 \\ ⋮ \\ 0\right] \end{gathered}$$ it is trivial solution. Otherwise it is nontrivial

Observations

Example

Identify the free variables, and the solution set, of the system. In Echelon Form: $\left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

x_1 + -2x_3 = 0 \\ x_2+x_3=0 \\ 0=0 } x_1 = 2x_3 \\ x_2=-x_3 \\ x_3 = x_3 }

$$\begin{gathered} \text{Solution set}=\text{x}=\left[\text{array}2x_3 \\ -x_3 \\ {x}_3\right]=x_3\left[\text{array}2 \\ -1 \\ 1\right] \end{gathered}$$