True or false: Every real homogeneous linear system of equation which has more than one solution has infinite solutions This is a task from a test exam you can find here (in German):
http://docdro.id/QRtdXkM

Is the following statement true or false?
Every real homogeneous linear system of equation that has more than
  one solution, has infinite solutions.

I think the statement is true because a linear system of equations can only have either one solution, no solution or infinite solutions. This statement clearly says "more than one solution $\rightarrow$ infinite solutions" which is true.
Is it really correct like that or there is some special case which can make this statement false?

Indeed, this is even true for non-homogeneous linear systems. Consider the system $Ax=b$, and assume $x_0$ and $x_1$ are solutions. Then for any $x_\lambda = (1-\lambda)x_0+\lambda x_1$ you get
$$Ax_\lambda = A((1-\lambda)x_0 + \lambda x_1) = (1-\lambda)A x_0 + \lambda A x_1 = (1-\lambda) b + \lambda b = b$$
Therefore $x_\lambda$ is also a solution, thus you get infinitely many (indeed even uncountably many) solutions.
The homogeneous system is just the special case for $b=0$. Since $x=0$ is always a solution of a homogeneous linear system, for those you can even write the condition as:

If any real homogeneous linear system of equations has a non-zero solution, it has infinitely many.

