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

Question

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.

Answer 1

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?

Answer 2

The book's claim is wrong:

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