How do I show that a homogeneous system of linear equations, where all of the coefficients of one of the unknowns is zero, has a non-zero solution?
Say among the variables $\displaystyle x_1,x_2,...,x_n$ the $\displaystyle x_k$ variable have all its cofficients to be zero. Then you can let $\displaystyle x_k$ be whatever you wish it to be i.e. let $\displaystyle x_1=x_2= ... = x_{k-1} = 0, x_k = t, x_{k+1} = ... = x_n = 0$ where $\displaystyle t\in \mathbb{R}$.