A [[system of linear equations]] is called **homogeneous** if the vector of constants is the [[zero vector]] $\begin{cases} a_{11} x_{1}&+&\cdots&+&a_{1 n} x_{n}&&=0\\ a_{21} x_{1}&+&\cdots&+&a_{2 n} x_{n}&&=0\\ &&\ \ \vdots&&&&\ \vdots\\ a_{m 1} x_{1}&+&\cdots&+&a_{m n} x_{n}&&=0\end{cases}$ Homogeneous systems always have a trivial solution vector, the [[zero vector]] A non-trivial solution to a homogeneous system expresses a linear dependence of the column vectors $x_{1}\left(\begin{array}{c}a_{11} \\ \vdots \\ a_{m 1}\end{array}\right)+\cdots+x_{n}\left(\begin{array}{c}a_{1 n} \\ \vdots \\ a_{m n}\end{array}\right)=\left(\begin{array}{c}0 \\ \vdots \\ 0\end{array}\right)$ $x_1 \mathbf a_1+\cdots+x_n \mathbf a_n=\mathbf 0$ A homogeneous system with more unknowns than column vectors are guaranteed to have a non-trivial solution because [[larger sets of vectors than the maximal are dependent]]