Homogeneous Systems

What is it?

In Mathematics and Linear Algebra, a homogeneous system is a linear system which the right hand of every linear equation is equal to $0$. This can be defined mathematically as $Ax=b/b=0$, where $A$ is the matrix of coefficients, $x$ is the unknown vector, and $b$ is the *known vector, the value after the equal sign.

$$\{\begin{array}{l}6x-y+5.25z=0\\ -2x+y+10z=0\\ +9x-5y-4z=0\end{array}\text{rewritten as}\left(\begin{array}{cccc}6& -1& 5.25& |0\\ -2& 1& 10& |0\\ 9& -5& -4& |0\end{array}\right)$$

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Solutions of homogeneous systems

A homogeneous system can have have two types of solutions, the trivial and non-trivial solution.

• Trivial solution

  The system always has a trivial solution, where the value of each variable is equal to $0$. This solution can happen to every homogeneous system.

• Non-trivial solution

  The value of a least one variable is not equal to $0$, even if it depends on the value of another variable. It can only happen when the number of equations is less than the number of variables.