# Condition for a Linear Equation System to have non-trivial Solution

I have this Theorem in my book:

For a Homogeneous System of $$m$$ Linear Equations in $$n$$ unknowns, if $$m \lt n$$, then the system has a non-trivial solution.

I have a confusion about the condition mentioned: Wouldn't it be $$n \lt m$$ the condition for non-trivial solution? It seems to me that $$m \lt n$$ is precisely the case we have either only trivial solution or no solution at all.

Hint:

look at.

$$2x=0$$ with $$m=1 , n=1$$

and

$$2x+y=0$$ with $$m=1, n=2$$

what is the equation with non trivial ( i.e. not null) solution?

• Thank you, I can see now. – freehumorist Jan 15 '19 at 20:58

The book's claim is wrong:

$$\begin{cases}x-y=0,\\3x+y=0,\\x+y=0\end{cases}$$ only has a trivial solution.

• Sorry for that. That s on me. I falsely copied the hypothesis. Edited. – freehumorist Jan 15 '19 at 20:58