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?
 A: Hint: if the homogeneous system has two distinct solutions, then one of them, call it $v$, is nonzero. What can you say about $\alpha v$ for a scalar $\alpha$?
A: Yes indeed this is true. The way that this is best seen is by noting that if a homogeneous system of equations has more than one solution, then the matrix corresponding to the system of equations, $A$, has a non-trivial kernel.
This means that $\exists \vec{v},\vec{u} \in \ker(A): A\vec{v} = 0$ and $A\vec{u} = 0$. Now it is worth noting that we can take any linear combination of these vectors $\lambda \vec{v} + \mu \vec{u}$ for $\lambda, \mu \in \mathbb{R}$ and we have $A(\lambda \vec{v} + \mu \vec{u}) = \lambda A\vec{v} + \mu A\vec{u} = 0$, so $\lambda \vec{v} + \mu \vec{u}$ also solves this system of equations.
Therefore we have an infinite number of solutions if we have more than one unique solution.
A: Every linear comination over $\mathbb{R}$ of two solutions of a homogeneous linear system will be solution to the homogeneous linear system. One of the solutions in necessarly not the zero vector and then it has infinte non trivial combinations which are solutions.
If you write it as a matrix, $A \in _{\mathbb{R}_{n \times m}}$, represents the homogeneous linear equation system, andfor the vectors $a,b \in \mathbb{R}^{m}$ such that $Aa=0,Ab=0$.
From that you get that for every $\alpha, \beta \in \mathbb{R}$, exists:
$A(\alpha a+\beta b)=\alpha Aa+\beta Ab=0$.
A: 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.

